chord name please?
triology
what would you call it?
Stefaan Himpe
> C - Eb - G - B
>
> what would you call it?
One way to name this chord would be CmMaj7
Joey Goldstein
> C - Eb - G - B
>
> what would you call it?
Cm(maj7)
G+/C, Eb+/C, B+/C
--
Joey Goldstein
<http://www.joeygoldstein.com>
<http://homepage.mac.com/josephgoldstein/AudioClips/audio.htm>
joegold AT primus DOT ca
triology
that's what i thought, but i also thought it was an awkward way of
writing it... i've never seen a chord named that before, but i've used
that particular configuration (minor triad with a major 7th) in more
than one song or musical piece before.
thanks !
Bohgosity BumaskiL
news:a4148017-f852-4d64...@s15g2000yqs.googlegroups.com...
>C - Eb - G - B
>
> what would you call it?
40:48:60:75
http://ecn.ab.ca/~brewhaha/Sound/Cm(Maj7).mp3
It is a five second linear glide up and down a tritave, maintaining the
chord, starting at a C that I can just barely sing at (the fourty is
actually scaled up to 66Hertz), so my basis frequency is 1.65Hz.
GeF
"Bohgosity BumaskiL" <brew...@freenet.edmonton.ab.ca> wrote in message
news:7h6imeF...@mid.individual.net...
Bohgosity BumaskiL
Asssume 40 reprezents a C (that is as low as I can get it without using
decimals). The ratio between Eb and C is 6:5. 40*6/5=48. G is a fifth above
C, so that is sixty. The B is a major seventh above C, so that is
40*15/8=75. 40:48:60:75 in http://en.wikipedia.org/wiki/just_intonation ,
five prime limit, scale as you please. In plain old sine waves:
http://ecn.ab.ca/~brewhaha/Sound/Cm(Maj7).mp3 5s@24kbps. PD trivia.
One person called for a name. How would you number it?
"GeF" <remo...@gerardferini.com> wrote in message
news:1252924...@sicinfo3.epfl.ch...
Hans Aberg
> Asssume 40 reprezents a C (that is as low as I can get it without using
> decimals). The ratio between Eb and C is 6:5. 40*6/5=48. G is a fifth above
> C, so that is sixty. The B is a major seventh above C, so that is
> 40*15/8=75. 40:48:60:75 in http://en.wikipedia.org/wiki/just_intonation ,
> five prime limit, scale as you please. In plain old sine waves:
> http://ecn.ab.ca/~brewhaha/Sound/Cm(Maj7).mp3 5s@24kbps. PD trivia.
> One person called for a name. How would you number it?
Scala <http://www.huygens-fokker.org/scala/> has a comprehensive
implementation of scales and chord.
Hans
Bohgosity BumaskiL
news:h8np6f$62b$1...@news.eternal-september.org...
I could scale a major third by major seconds for a scale. The simple way to
do it would result in fractional Hertz values (starting with 100:60). The
hard way to do it would result in eight pairs of natural numbers.
I am sure it will be worth a look when I get back to my longest piece in
four parts. It has some chords in it that I call vibrato chords (I like the
way they resolve, though), and I am pretty sure I can re-cast all four parts
to eliminate those. It is just a matter of which strategy to employ first.
Hmmm. You know what? I am planning on spending a lot of tomorrow near high
speed access. So, I will pack my USB key to do the downloads, and I see that
scala feeds into ChucK, which someone else recommended.
Chords written with natural numbers (whole numbers if you consider zero to
be a rest, and rests are important) blend, like a barbershop quartet, and
since I am currently writing them with sine waves, even more so. So, what I
would like to do with a package is put a pipe on the soprano part (flutes
are not much different from sine waves), brass on alto part (trumpet for
bright harmonics), a reed on the tenor part, and a string on the bottom
(strings carry harmonics into the stratosphere if you equalize them in). The
point of such a cacoph-ensemble would be to enable singers to pick out their
part from parts that blend.
I figure it will sound like one instrument that can change its timbre.
_______
http://ecn.ab.ca/~brewhaha/Sound/Diamonds.mp3
That starts off with prime numbers above seven, because it is an early work
that I recently extended to a surprise ending that is an octave lower than
you might expect. I think it might get longer, yet. If you hear vocals in
it, that is a happy accident of post-processing.
Bohgosity BumaskiL
eight
' major thirds (5/4). When I thought about it, 5/3 (major sixths) would be
harder,
' and it was quite a bit so. This code to jenerate the natural numbers
results in
' numbers that are inaudible unless my basis frequency is much less than
one.
DIM k AS DOUBLE
DIM t AS DOUBLE
DIM d AS DOUBLE
PRINT
t = 8 ^ 8 * 3
d = 8 ^ 8 * 5
FOR k = 1 TO 8
PRINT d / t,
t = t * 9 / 8
d = d * 9 / 8
PRINT d, t
NEXT k
' Actually, owing to the stratospheric numbers involved,
' I cannot avoid fractional hertz values in my revised task
' of jenerating an oddball scale (of major sixths).
Hans Aberg
> "Hans Aberg" <haberg_...@math.su.se> wrote in message
> news:h8np6f$62b$1...@news.eternal-september.org...
>> Bohgosity BumaskiL wrote:
>>> Asssume 40 reprezents a C (that is as low as I can get it without using
>>> decimals). The ratio between Eb and C is 6:5. 40*6/5=48. G is a fifth
>>> above C, so that is sixty. The B is a major seventh above C, so that is
>>> 40*15/8=75. 40:48:60:75 in http://en.wikipedia.org/wiki/just_intonation ,
>>> five prime limit, scale as you please. In plain old sine waves:
>>> http://ecn.ab.ca/~brewhaha/Sound/Cm(Maj7).mp3 5s@24kbps. PD trivia.
>>> One person called for a name. How would you number it?
>> Scala <http://www.huygens-fokker.org/scala/> has a comprehensive
>> implementation of scales and chord.
>
> I could scale a major third by major seconds for a scale. The simple way to
> do it would result in fractional Hertz values (starting with 100:60). The
> hard way to do it would result in eight pairs of natural numbers.
I'm not sure exactly what you want to do. Traditionally, one generates
the pitch system by iterating a choice of perfect fifths and the octave,
but alternatively one can choose any two linearly independent intervals.
Just intonation does not fall into this pattern: it uses two differently
sized major seconds. So it is hard to do transposable music around it.
The original tuning according to this method is the Pythagorean tuning
which sets the perfect fifth to the interval ration 3/2. During the
Renaissance the quarter-comma meantone was popular, which sets the major
third to the interval ratio 5/4. In terms of ETs (equal temperaments),
Pythagorean tuning is successively better approximated by E12, E41 and
E53, and quarter-comma meantone by E31.
> I am sure it will be worth a look when I get back to my longest piece in
> four parts. It has some chords in it that I call vibrato chords (I like the
> way they resolve, though), and I am pretty sure I can re-cast all four parts
> to eliminate those. It is just a matter of which strategy to employ first.
> Hmmm. You know what? I am planning on spending a lot of tomorrow near high
> speed access. So, I will pack my USB key to do the downloads, and I see that
> scala feeds into ChucK, which someone else recommended.
I made a file where you can play these tunings on the typing keyboard in
ChucK. - See the archives of the user list.
Scala is probably not compatible with ChucK as of today - maybe in the
future.
> Chords written with natural numbers (whole numbers if you consider zero to
> be a rest, and rests are important) blend, like a barbershop quartet, and
> since I am currently writing them with sine waves, even more so. So, what I
> would like to do with a package is put a pipe on the soprano part (flutes
> are not much different from sine waves), brass on alto part (trumpet for
> bright harmonics), a reed on the tenor part, and a string on the bottom
> (strings carry harmonics into the stratosphere if you equalize them in). The
> point of such a cacoph-ensemble would be to enable singers to pick out their
> part from parts that blend.
>
> I figure it will sound like one instrument that can change its timbre.
Exact intervals minimizes the beats among the partials, but offsets are
known as a tremulant effect, and some musical instruments are tunes that
way with various amounts (some organ stops, accordions, pianos and
gamelan orchestras). It is also hard to ensure exact ratios in a piece
throughout - a piece I listened to had some rather sharp beats in it.
And if trying to pivot exact Just chords, common chord sequences may
result in "comma pumps" - the pitch will in reality slip rather than
coming back to the original. Variable pitch instrument more likely play
in some kind of adaptive tuning - one adjusts the pitch on the fly to
get a desired musical effect.
Hans
Hans Aberg
> ' I said that I could come up with natural numbers to reprezent a scale of
> eight
> ' major thirds (5/4).
This is called quarter-comma meantone, and is close to E12.
> When I thought about it, 5/3 (major sixths) would be
> harder,
> ' and it was quite a bit so.
If one set the minor third to 6/5, then that generates another
meant-tone that is close to E19. The octave inversion if the minor third
is the major sixth, and the octave inversion of 6/5 is 2/(6/5) = 5/3.
So the tuning you wanted to derive is the same as this one.
The Wikipedia describes some of this:
http://en.wikipedia.org/wiki/Meantone_temperament
Hans
Bohgosity BumaskiL
"Hans Aberg" <haberg_...@math.su.se> wrote in message
news:h8qota$vte$1...@news.eternal-september.org...
> Bohgosity BumaskiL wrote:
>> "Hans Aberg" <haberg_...@math.su.se> wrote in message
>> news:h8np6f$62b$1...@news.eternal-september.org...
>>> Bohgosity BumaskiL wrote:
>>>> Asssume 40 reprezents a C (that is as low as I can get it without using
>>>> decimals). The ratio between Eb and C is 6:5. 40*6/5=48. G is a fifth
>>>> above C, so that is sixty. The B is a major seventh above C, so that is
>>>> 40*15/8=75. 40:48:60:75 in http://en.wikipedia.org/wiki/just_intonation
>>>> , five prime limit, scale as you please. In plain old sine waves:
>>>> http://ecn.ab.ca/~brewhaha/Sound/Cm(Maj7).mp3 5s@24kbps. PD trivia.
>>>> One person called for a name. How would you number it?
>>> Scala <http://www.huygens-fokker.org/scala/> has a comprehensive
>>> implementation of scales and chord.
>>
>> I could scale a major third by major seconds for a scale. The simple way
>> to do it would result in fractional Hertz values (starting with 100:60).
>> The hard way to do it would result in eight pairs of natural numbers.
>
> I'm not sure exactly what you want to do. Traditionally, one generates the
> pitch system by iterating a choice of perfect fifths and the octave, but
> alternatively one can choose any two linearly independent intervals. Just
> intonation does not fall into this pattern: it uses two differently sized
> major seconds. So it is hard to do transposable music around it.
' More art is in the timing and stereophony of this:
' http://ecn.ab.ca/~brewhaha/Sound/MSixths.mp3 18kbps
' than is in the following code, which shows you a pattern
' in numbers that end this message.
' This is a way to make just intonation
' approximate the western C-scale of naturals.
' To continue the scale into the next octave,
' I would increment k by even numbers from sixteen.
' Some people question whether eleven or thirteen
' should figure into just intonation. I do not
' normally care, preferring to go with what sounds
' right.
DIM k AS INTEGER
DIM LeftBasis AS DOUBLE
DIM RightBasis AS DOUBLE
LeftBasis = 110 / 7
' This makes seven reprezent a C to western ears, and I hav
' westernized ears, because sometimes my tunes work better
' when I adjust my first note to a note on the western scale.
' Notice that LeftBasis equals five thirds of this:
RightBasis = 66 / 7
' The names for those variables do not work here.
' They are only for highlighting the comparison to what
' is in the MP3. 110:66 = 5:3.
' Note that MicroSoft's circa 1992 RealTimeClock-based
' (crude) synthesizer does not do fractional hertz
' values or stereo. Mine does, although I normally
' avoid fractions in favour of naturals.
'
PRINT
FOR k = 7 TO 14
PRINT LeftBasis * k; RightBasis * k
IF k MOD 2 = 0 THEN
SOUND LeftBasis * k, 2
SOUND RightBasis * k, 4
ELSE
SOUND RightBasis * k, 2
SOUND LeftBasis * k, 4
END IF
NEXT k
' I tried literally scaling 5:3 (major sixths) by major seconds (9:8).
' While that did give the impression of Doh-Ray-Mee, there was more
' dissonance in it than there should hav been. What is above sounds
' much cleaner, and it is simpler to calculate.
' The tune should work for any basis frequency, and if I started with
' natural numbers, then the whole series printed would be natural
' numbers (plus M$ would render it exactly).
DATA 60,30,54,27,54,48,54,60
DATA 35,40,45,50,55,60,65,70
' 5*x, x=7...14. I am not using that data here.
It is in that MP3. I was tempted to record the alternating sequence above
into the MP3, and I was afraid that might detract from the fact that the two
progressions are vaguely similar in stereo, and the second one delivers
better harmony, if only because it delivers more variety in the harmony.
> The original tuning according to this method is the Pythagorean tuning
> which sets the perfect fifth to the interval ration 3/2. During the
> Renaissance the quarter-comma meantone was popular, which sets the major
> third to the interval ratio 5/4. In terms of ETs (equal temperaments),
> Pythagorean tuning is successively better approximated by E12, E41 and
> E53, and quarter-comma meantone by E31.
The nice thing about working with my synthesizer in natural numbers, is that
it glides, and when I need to adjust the glide constant, that often tells me
that I am in a transitional part of my tune or going off on a tangent. I
would not get clues like that from ET, because it always clicks.
>> I am sure it will be worth a look when I get back to my longest piece in
>> four parts. It has some chords in it that I call vibrato chords (I like
>> the way they resolve, though), and I am pretty sure I can re-cast all
>> four parts to eliminate those. It is just a matter of which strategy to
>> employ first. Hmmm. You know what? I am planning on spending a lot of
>> tomorrow near high speed access. So, I will pack my USB key to do the
>> downloads, and I see that scala feeds into ChucK, which someone else
>> recommended.
>
> I made a file where you can play these tunings on the typing keyboard in
> ChucK. - See the archives of the user list.
>
> Scala is probably not compatible with ChucK as of today - maybe in the
> future.
The author mentioned a file format.
>> Chords written with natural numbers (whole numbers if you consider zero
>> to be a rest, and rests are important) blend, like a barbershop quartet,
>> and since I am currently writing them with sine waves, even more so. So,
>> what I would like to do with a package is put a pipe on the soprano part
>> (flutes are not much different from sine waves), brass on alto part
>> (trumpet for bright harmonics), a reed on the tenor part, and a string on
>> the bottom (strings carry harmonics into the stratosphere if you equalize
>> them in). The point of such a cacoph-ensemble would be to enable singers
>> to pick out their part from parts that blend.
>>
>> I figure it will sound like one instrument that can change its timbre.
>
> Exact intervals minimizes the beats among the partials, but offsets are
> known as a tremulant effect, and some musical instruments are tunes that
> way with various amounts (some organ stops, accordions, pianos and gamelan
> orchestras). It is also hard to ensure exact ratios in a piece
> throughout - a piece I listened to had some rather sharp beats in it. And
> if trying to pivot exact Just chords, common chord sequences may result in
> "comma pumps" - the pitch will in reality slip rather than coming back to
> the original. Variable pitch instrument more likely play in some kind of
> adaptive tuning - one adjusts the pitch on the fly to get a desired
> musical effect.
>
> Hans
' Committed to Public Domain:
DATA 35,27,2, 24,40,6, 45,27,2, 30,50,4
DATA 55,33,2, 36,60,6, 65,39,3, 42,70,6,0,0,4
' These two sequences sound similar.
' One is more interesting, because it is not a scale.
DATA 60,35,2, 30,40,6, 54,45,2, 27,50,4
DATA 54,55,2, 48,60,6, 54,65,3, 60,70,6, 0,0,8
' The other key to feed this data into my synthesizer and get the same
output as in
' the MP3 is a basis frequency of four. Notice that 440 (the reference
frequency
' for equal temperament) is divisible by four.
Hans Aberg
>> Scala is probably not compatible with ChucK as of today - maybe in the
>> future.
>
He has a sequence file format .seq, and said a few months ago he wanted
to make a make a seq to ck converter. But I do not know the state of it.
Hans
Bohgosity BumaskiL
What I came up with reduces to 7...14 with the same basis frequency. The
right channel of the second sequence in my file sounds like the C-natural
scale to me (the left channel is more creative, and it sounds like a
passable duet). That means that my notes are getting closer together as they
ascend. Thirteen is about the most dissonant note, which is as it should be,
if I will resolve on the easiest harmony to understand next to unison.
The first sequence is simpler; two scales in parallel. They start a major
sixth apart, and they end that way. The complication is that those two parts
change sides, so it might be hard to notice (21, 24, 27...) (35, 40,45...),
which are always a major sixth apart.
_______
http://ecn.ab.ca/~brewhaha/Sound/MSixths.mp3
DATA 35,21,2,24,40,6,45,27,2,30,50,4
DATA 55,33,2,36,60,4,65,39,4,42,70,4,0,0,4
DATA 60,35,2,30,40,6,54,45,2,27,50,4
DATA 54,55,2,48,60,4,54,65,4,60,70,4,0,0,8
Hans Aberg
> I wrote:
>>> ' I said that I could come up with natural numbers to reprezent a scale
>>> of eight
>>> ' major thirds (5/4).
>> This is called quarter-comma meantone, and is close to E12.
Sorry, typo. It is E31 that is close to the quarter-comma meantone
(which in turn sett the major third to the rational interval 5/4).
>>> When I thought about it, 5/3 (major sixths) would be harder,
>>> ' and it was quite a bit so.
>> If one set the minor third to 6/5, then that generates another meant-tone
>> that is close to E19. The octave inversion if the minor third is the major
>> sixth, and the octave inversion of 6/5 is 2/(6/5) = 5/3. So the tuning you
>> wanted to derive is the same as this one.
>>
>> The Wikipedia describes some of this:
>> http://en.wikipedia.org/wiki/Meantone_temperament
> What I came up with reduces to 7...14 with the same basis frequency. The
> right channel of the second sequence in my file sounds like the C-natural
> scale to me (the left channel is more creative, and it sounds like a
> passable duet). That means that my notes are getting closer together as they
> ascend. Thirteen is about the most dissonant note, which is as it should be,
> if I will resolve on the easiest harmony to understand next to unison.
>
> The first sequence is simpler; two scales in parallel. They start a major
> sixth apart, and they end that way. The complication is that those two parts
> change sides, so it might be hard to notice (21, 24, 27...) (35, 40,45...),
> which are always a major sixth apart.
I cannot follow exactly what you do, but my impression was that you
forgot reducing with octaves. One needs two interval, and traditionally
that is the pure fifths and octaves. Just writing these applications on
a line gives
Fbb Cbb Gbb Dbb Abb Ebb Bbb
Fb Cb Gb Db Ab Eb Bb
F C G D A E B
F# C# G# D# A# E# B#
Fx Cx Gx Dx Ax Ex Bx
But one can switch to using the major and minor seconds as vectors in a
two dimensional layout, which gives:
A# B# Cx Dx Ex
A B C# D# E# Fx Gx Ax Bx
Bb C D E F# G# A# B#
Cb Db Eb F G A B C'# D'#
Dbb Ebb Fb Gb Ab Bb C' D' E'
Now, the original tuning is the Pythagorean which sets the pure fifths
exactly to the interval 3/2. It is easiest to understand using E53,
practically the same, whose tonesteps are called "commas": M = 9, m = 4.
Sharps and flats always alter with M - m, which here is 9 - 4 = 5 commas.
I find it useful to use the width of m in cents to pin down these
tunings - it moves fatser than M. So in E53 and the Pythagorean tuning
it is about 90 cents - m is narrow.
In the tuning, the interval ration 5/4 is a syntonic comma off the major
third, which is a bit far for harmony involving this interval. So
during the Renaissance, one used the quarter-comma meantone which sets
it exactly to the interval ration 5/4. The perfect fifth gets a quarter
syntonic comma off.
In this tuning, the interval 7/4 is approximated well, but on the
augmented 6th - not the minor 7th, as many seems to assume. This is
problematic, since it is without the common diatonic scales. So it is
mainly used in as a predominant resolving to a dominant chord (look up
German sixth).
Anyway, this scale give good approximation of some 7-limit intervals.
The next step would be to add the the partials 11/8 and 13/8. Relative
E12, 11/8 is fairly close quarter note - in E72, intervals 5/4, 7/8 and
11/8 lowers relative E12 with 1, 2, and 3 tonesteps. For 13/8, one has
to go up to E144.
A quick computation in E93 = E(3*31), gives good approximations of the
intervals 11/8 and 13/8.
Hans
Bohgosity BumaskiL
> Bohgosity BumaskiL wrote:
>> I wrote:
>>>> ' I said that I could come up with natural numbers to reprezent a scale
>>>> of eight
>>>> ' major thirds (5/4).
>>> This is called quarter-comma meantone, and is close to E12.
>
> Sorry, typo. It is E31 that is close to the quarter-comma meantone (which
> in turn sett the major third to the rational interval 5/4).
Rather than using more than one name for the same ratio, I think it would be
practical to name them after letters like D and T if you are using primarily
two ratios. Five to four is a minor third. In related terminology, there is
no "Eleventh Harmonic" other than 11:1. I do not know what to call 11:8, and
"harmonic" presents a problem in the term that leads to confusion after
reading about the harmonic series in Physics. In addition, being less than
an octave, 11:8 does not fall anywhere near 8:3, where a perfect eleventh
does.
>>>> When I thought about it, 5/3 (major sixths) would be harder,
>>>> ' and it was quite a bit so.
>>> If one set the minor third to 6/5, then that generates another
>>> meant-tone that is close to E19. The octave inversion if the minor third
>>> is the major sixth, and the octave inversion of 6/5 is 2/(6/5) = 5/3. So
>>> the tuning you wanted to derive is the same as this one.
>>>
>>> The Wikipedia describes some of this:
>>> http://en.wikipedia.org/wiki/Meantone_temperament
>
>> What I came up with reduces to 7...14 with the same basis frequency. The
>> right channel of the second sequence in my file sounds like the C-natural
>> scale to me (the left channel is more creative, and it sounds like a
>> passable duet). That means that my notes are getting closer together as
>> they ascend. Thirteen is about the most dissonant note, which is as it
>> should be, if I will resolve on the easiest harmony to understand next to
>> unison.
>>
>> The first sequence is simpler; two scales in parallel. They start a major
>> sixth apart, and they end that way. The complication is that those two
>> parts change sides, so it might be hard to notice (21, 24, 27...) (35,
>> 40,45...), which are always a major sixth apart.
>
> I cannot follow exactly what you do, but my impression was that you forgot
> reducing with octaves. One needs two interval, and traditionally that is
> the pure fifths and octaves. Just writing these applications on a line
> gives
' For an approximation of the C-scale:
basis = 132/7
' Hertz, that is.
for k = 7 to 14
sound k*basis,3
next k
' It is absurdly simple in one way.
' In another way, my tuning has seven intervals with a prime limit of
thirteen,
' and that is just for a scale of naturals. Seven and fourteen are
' an octave apart. Everything is exactly on the harmonic series (2:1, 3:1,
4:1...),
' although to get an approximation of C, that means a fractional
fundamental
' frequency. For an impression of the chromatic western scale from C, start
with
' basis=132/11 and make k range from eleven to twenty-two. A lot of what I
do
' involves greatest common denominators; reducing fractions, so that I know
' when I am starting on the right natural number in a terminal series. I can
know
' that if some of them reduce to accepted harmonies. I do not hav many other
' means of identifying a terminal series, other than "sounds good". In this
case,
' "resembles Doh-Ray-Mee".
_______
http://ecn.ab.ca/~brewhaha/MSixths.mp3
That is what it sounds like with two schemes for harmony.
> Fbb Cbb Gbb Dbb Abb Ebb Bbb
> Fb Cb Gb Db Ab Eb Bb
> F C G D A E B
> F# C# G# D# A# E# B#
> Fx Cx Gx Dx Ax Ex Bx
>
> But one can switch to using the major and minor seconds as vectors in a
> two dimensional layout, which gives:
> A# B# Cx Dx Ex
> A B C# D# E# Fx Gx Ax Bx
> Bb C D E F# G# A# B#
> Cb Db Eb F G A B C'# D'#
> Dbb Ebb Fb Gb Ab Bb C' D' E'
>
> Now, the original tuning is the Pythagorean which sets the pure fifths
> exactly to the interval 3/2. It is easiest to understand using E53,
> practically the same, whose tonesteps are called "commas": M = 9, m = 4.
> Sharps and flats always alter with M - m, which here is 9 - 4 = 5 commas.
>
> I find it useful to use the width of m in cents to pin down these
> tunings - it moves fatser than M. So in E53 and the Pythagorean tuning it
> is about 90 cents - m is narrow.
I would love to be able to use floating point in Nero's WaveEdit, rather
then needing to look up (or calculate) the cents of a ratio.
Bohgosity BumaskiL
with a scale, Dee and Teh. Zii and Sih are better, since neither is used in
music to my knowledge, and they are closely related in phonetics.
Bohgosity BumaskiL
Temperament
' I <brew...@freenet.edmonton.ab.ca> wrote in message
' news:7hg2clF...@mid.individual.net
' this part detailing a series that is slightly more complicated than two
scales that
' move together, always a major sixth apart:
> DATA 35,27,2, 24,40,6, 45,27,2, 30,50,4
' That first twenty-seven should be a twenty-one. 7*3 = 21, 7*5=35 -- a
' constant 5:3, in parts that change sides. I did not specify a lot of
things in that
' post, owing to the fact that other people will use newer languages. Space
' permits.
' Free verse is the hardest stuff to write tunes for,
' because I probably cannot stick with one beat.
' http://ecn.ab.ca/~brewhaha/Sound/MSixths.mp3 Tune.
' CC-BY: The same license is in the ID3 tags, except for these
' details:
' You can use my tune for these lyrics in person, no fees, no trouble.
' Just drop two occurances of the word "out", expand a contraction,
' and contact me if: you do it a few times, you get permission from
' the Sherman Brothers to record, or you write your own verse to the
' same beat. You do not need permission from me or the Sherman Brothers
' to record. It won't hurt anything if you apply for it before you
distribute it,
' though. Figuring out how to contact me should enable you to do a good job
of
' attribution, especially here, where links are preferred:
' Part, basis, channel volume -- relative to one, no matter how many.
DATA 1,3.7714285718571,.97, 2,3.7714285718571,.97
' Like Basis = 132/35 in crude version of this synth.
OPTION BASE 1
DIM TwoPi AS DOUBLE
DIM pi AS DOUBLE
DIM temp AS DOUBLE
DIM Velocity(4) AS DOUBLE
DIM Acceleration(4) AS DOUBLE
DIM Angle(4) AS DOUBLE
DIM Phase(4) AS DOUBLE
DIM TerminalAngle(4) AS DOUBLE
DIM DropAngle(4) AS DOUBLE
DIM ChanAmp(4) AS SINGLE
DIM Harmonics(4) AS SINGLE
DIM BeatsPerSecond AS SINGLE
DIM note(4) AS DOUBLE
DIM LastNote(4) AS DOUBLE
DIM PhaseDir(4) AS INTEGER
DIM Part(4) AS INTEGER
DIM length AS INTEGER
DIM Glide AS INTEGER
DIM Samples AS LONG
DIM t AS LONG
DIM SampleRate AS LONG
DIM k AS INTEGER
DIM g AS INTEGER
DIM amp AS INTEGER
DIM Basis AS INTEGER
DIM NumHarmonics AS INTEGER
DIM test AS STRING
' For some reason, my version of QBASIC does not do those typings properly
' unless I put them one to a line. I did not check for that problem with
FreeBasic.
' Notice that I wrote no fractions. That is because if 7.3 sounds better
' in my context than either seven or eight, then I divide my basis
' frequency by three and multiply all of my natural numbered notes
' by three. That has no effect on my output, and my input is nothing
' but naturals...uh...plus zero for a rest. (I just noticed that
' "whole note" is ambiguous, as in "whole note rest").
SampleRate = 8000
OPEN "MSixths.bat" FOR OUTPUT AS #1
PRINT #1, "e:\progra~1\freebasic\fbc -lang qb d:\basic\MSixths.bas"
PRINT #1, "MSixths.exe"
PRINT #1, "cd \sox"
PRINT #1, "sox -c 2 -r"; SampleRate; " -sw MSixths.raw MSixths.wav stat"
CLOSE #1
' If you do not know what the above MSixths.bat file would try to do
' if you ran it, then you should ask someone who knows about scripts
' on the internet if you really want to know what script synthesis is about.
' It requires "Sound Exchange", and FreeBasic to be installed.
test = "0"
' I frequently flip that.
NumHarmonics = 2
BeatsPerSecond = 8
pi = 3.141592653589793#
TwoPi = pi * 2
' Committed to Public Domain:
DATA 35,21,2,24,40,6,45,27,2,30,50,4
DATA 55,33,2,36,60,4,65,39,4,42,70,4,0,0,4
DATA 60,35,2,30,40,6,54,45,2,27,50,4
DATA 54,55,2,48,60,4,54,65,4,60,70,4,0,0,8
' Corruption is inevitable. Wear sunscreen.
IF test = "0" THEN
OPEN "\sox\MThirds.raw" FOR OUTPUT AS #4
END IF
g = -1
FOR k = 1 TO NumHarmonics
READ Part(k), Harmonics(k), ChanAmp(k)
note(k) = 0
LastNote(k) = 0
g = -g
IF g = 1 THEN
ChanAmp(k) = ChanAmp(k) * 32767 * 2 / NumHarmonics
ELSE
ChanAmp(k) = ChanAmp(k) * 32767 * 2 / NumHarmonics
END IF
NEXT k
Glide = 30
' This does not handle asynchrony or syncopation: It assumes that
' if one channel is on, then so are the others. Main section after
' initialization. (That might actually be easy to fix).
length = 3
FOR g = 7 TO 24
READ note(1), note(2), length
' SOUND note(1) * 6, length * 2
' That is for the quick and dirty synth internal to BASIC.
' PRINT note(1), note(2), length
IF test = "0" THEN
IF note(1) = 0 AND LastNote(1) = 0 THEN
Samples = SampleRate * length / BeatsPerSecond
GOSUB 300
GOTO 75
END IF
IF note(1) = 0 AND LastNote(1) <> 0 THEN
GOSUB 250
FOR k = 1 TO NumHarmonics
LastNote(k) = note(k)
NEXT k
Samples = SampleRate * length / BeatsPerSecond
GOSUB 300
GOTO 75
END IF
IF note(1) <> 0 AND LastNote(1) = 0 THEN
FOR k = 1 TO NumHarmonics
LastNote(k) = note(k)
NEXT k
Samples = SampleRate * length / BeatsPerSecond
GOSUB 275
GOTO 75
END IF
IF note(1) <> 0 AND LastNote(1) <> 0 THEN
IF Glide > 0 THEN
Samples = SampleRate * length /
BeatsPerSecond / Glide
GOSUB 300
END IF
Samples = SampleRate * length / BeatsPerSecond
FOR k = 1 TO NumHarmonics
LastNote(k) = note(k)
NEXT k
GOSUB 300
GOTO 75
END IF
END IF
75 NEXT g
END
' *Neat* Silencer -- finishes a wave like 275 starts one.
250
FOR k = 1 TO NumHarmonics
temp = Angle(k) / TwoPi
Angle(k) = (temp - FIX(temp)) * TwoPi
IF Angle(k) > pi * 3 / 2 THEN
TerminalAngle(k) = pi * 3.5
PhaseDir(k) = 2
DropAngle(k) = pi * 2.5
ELSEIF Angle(k) > pi / 2 THEN
TerminalAngle(k) = pi * 2.5
PhaseDir(k) = 1
DropAngle(k) = pi * 3 / 2
ELSE
TerminalAngle(k) = pi * 3 / 2
PhaseDir(k) = -1
DropAngle(k) = pi / 2
END IF
NEXT k
260
FOR k = 1 TO NumHarmonics
Phase(k) = SIN(Angle(k))
SELECT CASE PhaseDir(k)
CASE 2
IF Angle(k) >= DropAngle(k) THEN
Phase(k) = Phase(k) / 2 + .5
END IF
CASE 1
IF Angle(k) >= DropAngle(k) THEN
Phase(k) = Phase(k) / 2 - .5
END IF
CASE -1
IF Angle(k) >= DropAngle(k) THEN
Phase(k) = Phase(k) / 2 + .5
END IF
END SELECT
IF Angle(k) < TerminalAngle(k) THEN
Angle(k) = Angle(k) + Velocity(k)
ELSE
Angle(k) = TerminalAngle(k)
END IF
NEXT k
GOSUB 400
FOR k = 1 TO NumHarmonics
IF Angle(k) < TerminalAngle(k) THEN GOTO 260
NEXT k
FOR k = 1 TO NumHarmonics
Angle(k) = Angle(k) - Angle(k)
NEXT k
RETURN
275
' This starts a wave from zero, using half the amplitude, a start
' from where sin(angle) = -1, and a bias of half. This cuts
' a leading click that I can hear on some equipment with some tunes.
FOR k = 1 TO NumHarmonics
Velocity(k) = TwoPi * LastNote(Part(k)) * Harmonics(Part(k)) /
SampleRate
Acceleration(k) = (TwoPi * note(Part(k)) * Harmonics(Part(k)) /
SampleRate - Velocity(k)) / Samples
PhaseDir(k) = 1
Angle(k) = 3 / 2 * pi
NEXT k
280
FOR k = 1 TO NumHarmonics
IF PhaseDir(k) = 1 THEN
Phase(k) = SIN(Angle(k)) / 2 + .5
ELSE
Phase(k) = SIN(Angle(k))
END IF
Angle(k) = Angle(k) + Velocity(k)
Velocity(k) = Velocity(k) + Acceleration(k)
NEXT k
Samples = Samples - 1
GOSUB 400
FOR k = 1 TO NumHarmonics
IF PhaseDir(k) = 1 THEN
Phase(k) = (Phase(k) - .5) * 2
END IF
IF Phase(k) > SIN(Angle(k)) THEN
PhaseDir(k) = -1
END IF
NEXT k
FOR k = 1 TO NumHarmonics
IF PhaseDir(k) = 1 GOTO 280
NEXT k
300
' Calculate constants of change for write loop.
FOR k = 1 TO NumHarmonics
Velocity(k) = TwoPi * LastNote(Part(k)) * Harmonics(Part(k)) /
SampleRate
Acceleration(k) = (TwoPi * note(Part(k)) * Harmonics(Part(k)) /
SampleRate - Velocity(k)) / Samples
NEXT k
' Main write loop. Static Phases problem solved at 250.
FOR t = 1 TO Samples
FOR k = 1 TO NumHarmonics
Phase(k) = SIN(Angle(k))
NEXT k
GOSUB 400
FOR k = 1 TO NumHarmonics
Angle(k) = Angle(k) + Velocity(k)
Velocity(k) = Velocity(k) + Acceleration(k)
NEXT k
NEXT t
RETURN
400
' Write a sample.
temp = temp - temp
FOR k = 1 TO NumHarmonics - 1 STEP 2
temp = temp + Phase(k) * ChanAmp(k)
NEXT k
amp = CINT(temp)
PRINT #4, CHR$(amp AND 255);
PRINT #4, CHR$((amp AND 65280) / 256);
temp = temp - temp
FOR k = 2 TO NumHarmonics STEP 2
temp = temp + Phase(k) * ChanAmp(k)
NEXT k
amp = CINT(temp)
PRINT #4, CHR$(amp AND 255);
PRINT #4, CHR$((amp AND 65280) / 256);
RETURN
' _______
' http://ecn.ab.ca/~brewhaha/ BrewJay's Babble Bin
Hans Aberg
>>> I wrote:
>>>>> ' I said that I could come up with natural numbers to reprezent a scale
>>>>> of eight
>>>>> ' major thirds (5/4).
>>>> This is called quarter-comma meantone, and is close to E12.
>> Sorry, typo. It is E31 that is close to the quarter-comma meantone (which
>> in turn sett the major third to the rational interval 5/4).
>
> Rather than using more than one name for the same ratio, I think it would be
> practical to name them after letters like D and T if you are using primarily
> two ratios. Five to four is a minor third.
A major third. The Just intonation minor third is 6/5.
> In related terminology, there is
> no "Eleventh Harmonic" other than 11:1. I do not know what to call 11:8, and
> "harmonic" presents a problem in the term that leads to confusion after
> reading about the harmonic series in Physics. In addition, being less than
> an octave, 11:8 does not fall anywhere near 8:3, where a perfect eleventh
> does.
The Pythagorean tuning only uses 2 and 3, meantones closes in on the
5/4, and as a byproduct the 7/8. The other interval 11/8 and 13/8 are
foreign to Western music. By an interval like 27/25 which can be use to
describe the intermediate pitch in Persian music - it is two syntonic
commas above the Pythagorean m, is close to 14/12, and 13/11, so it is
possible to get them that way.
> I would love to be able to use floating point in Nero's WaveEdit, rather
> then needing to look up (or calculate) the cents of a ratio.
In ChucK, that is no problem. I made an interface (a class), were you
can switch better using an interval ratio, log2 (or log_n), or cents. It
quickly became inconvenient to enter the intervals other than as is.
Hans
Hans Aberg
I'm not sure what intervals you are have in mind here. Traditionally,
one generates diatonic scales by iterating perfect fifths and octaves.
These scales are the built up by major and minor seconds, often called
whole and half tonesteps, those the latter need not be one half in szie
for the former. This only works for Western music: meantones. Just
intonation uses two major seconds of different sizes. Turkish, Persian
and Arab music uses intermediate intervals, which can be described by
neutral seconds - intervals between the minor and major seconds.
Hans
Bohgosity BumaskiL
> Bohgosity BumaskiL wrote:
>>>> I wrote:
>>>>>> ' I said that I could come up with natural numbers to reprezent a
>>>>>> scale of eight
>>>>>> ' major thirds (5/4).
>>>>> This is called quarter-comma meantone, and is close to E12.
>>> Sorry, typo. It is E31 that is close to the quarter-comma meantone
>>> (which in turn sett the major third to the rational interval 5/4).
>>
>> Rather than using more than one name for the same ratio, I think it would
>> be
>> practical to name them after letters like D and T if you are using
>> primarily two ratios. Five to four is a minor third.
>
> A major third. The Just intonation minor third is 6/5.
You got me there. I was coming back to this thread so soon, because I
detected the error in my memory. My keyboard is always QWERTY, so the names
for the numbers do not stick in my head very well, so I am still looking
most of them up in tables at wikipedia, after use. I do not always care if I
do not find them, either.
>> In related terminology, there is no "Eleventh Harmonic" other than 11:1.
>> I do not know what to call 11:8, and "harmonic" presents a problem in the
>> term that leads to confusion after reading about the harmonic series in
>> Physics. In addition, being less than an octave, 11:8 does not fall
>> anywhere near 8:3, where a perfect eleventh does.
>
> The Pythagorean tuning only uses 2 and 3, meantones closes in on the 5/4,
> and as a byproduct the 7/8. The other interval 11/8 and 13/8 are foreign
> to Western music.
I would not be too sure about that. The barbershop tradition passes chords
along mostly by ear, partly because doing a good job of writing harmony in
the most usable terms is hard. When accidentals were invented, some singers
derided them as being for beginners. In a PDF I was reading, the numbers
were low, and the chords were major or dominant. I worked out Cm(Maj7) to be
40:48:60:75, so when a chorus sings about minor chords, I am not surprised
that pianos only offer clues, and synthesizers are typically all but out to
lunch, unless the harmonic series gets into speaker paper and naturally
resonates with a tune, or unless it resonates in human perception, and
enough guys agree on how. On a real piano, I am sure the sound board and the
strings hold obscure harmonies. Do not ask me to prove that they ever become
prominent, though, because I am not that good at experimental Physics.
http://ecn.ab.ca/~brewhaha/Sound/Cm(Maj7).mp3
That is a linear glide up and down a tritave, holding the chord, and the
gliding (acceleration of angle change) is precise down to samples, so like
ChucK, it is strongly timed. What ChucK can probably do beyond my source is
simulations of different instruments, and I think that would help singers
pick out their part, plus divert me from the habit of making up instruments
with parallel harmonics. In recording, I could put a singer and the
instrument they are following on opposite sides, and I think I will do that
in a piece incomplete.
> By an interval like 27/25 which can be use to describe the intermediate
> pitch in Persian music - it is two syntonic commas above the Pythagorean
> m, is close to 14/12, and 13/11, so it is possible to get them that way.
There is a 27:50 in my complex harmony for the C-scale, which takes the
twenty-five in your figure an octave higher, to a grave major seventh. I
remember writing a grave fourth recently, too. I think it's in "A Tune for
Tigger's Song", in series, not in concert, because that is a monochromatic
tune.
http://en.wikipedia.org/wiki/List_of_intervals_in_5-limit_just_intonation
Hans Aberg
>> The Pythagorean tuning only uses 2 and 3, meantones closes in on the 5/4,
>> and as a byproduct the 7/8. The other interval 11/8 and 13/8 are foreign
>> to Western music.
>
> I would not be too sure about that. The barbershop tradition passes chords
> along mostly by ear, partly because doing a good job of writing harmony in
> the most usable terms is hard. When accidentals were invented, some singers
> derided them as being for beginners. In a PDF I was reading, the numbers
> were low, and the chords were major or dominant. I worked out Cm(Maj7) to be
> 40:48:60:75, so when a chorus sings about minor chords,
This is a Just minor seventh chord, which is in the 5-limit. The idea is
that exact ratios minimize the beats among the partials. On variable
pitch instruments, that can be done on the fly.
> I am not surprised
> that pianos only offer clues, and synthesizers are typically all but out to
> lunch, unless the harmonic series gets into speaker paper and naturally
> resonates with a tune, or unless it resonates in human perception, and
> enough guys agree on how. On a real piano, I am sure the sound board and the
> strings hold obscure harmonies. Do not ask me to prove that they ever become
> prominent, though, because I am not that good at experimental Physics.
Pianos, unless tuned in some differently, are just tuned in E12. Plucked
strings are inharmonic, so one uses a stretch tuning compensating for
that. Strings on the same note are typically tuned slightly apart in
pitch, providing a slow tremulant.
E12 is just the first ET approximation of Pythagorean tuning, the
following are E41, E53, which is so close to Pythagorean that the
difference can't be heard. One way to find this series is to compute the
continued fraction convergents of log2(3/2).
>> By an interval like 27/25 which can be use to describe the intermediate
>> pitch in Persian music - it is two syntonic commas above the Pythagorean
>> m, is close to 14/12, and 13/11, so it is possible to get them that way.
>
> There is a 27:50 in my complex harmony for the C-scale, which takes the
> twenty-five in your figure an octave higher, to a grave major seventh. I
> remember writing a grave fourth recently, too. I think it's in "A Tune for
> Tigger's Song", in series, not in concert, because that is a monochromatic
> tune.
> http://en.wikipedia.org/wiki/List_of_intervals_in_5-limit_just_intonation
27/25 = 133.237 cents behaves as an intermediate pitch in Pythagorean
tunings, but when moving towards meantone tunings, it closes into the
minor seconds, and sounds more like it. In E19, it is only 6.922 cents
off, which is about a half of what 5/4 is off in E12.
Hans
Bohgosity BumaskiL
> Bohgosity BumaskiL wrote:
>> I suggested the pair of letters to reprezent two intervals used to come
>> up with a scale, Dee and Teh. Zii and Sih are better, since neither is
>> used in music to my knowledge, and they are closely related in phonetics.
>
> I'm not sure what intervals you are have in mind here.
The two or three ratios you would use to construct a diatonic scale.
The scale I use is probably not diatonic, whatever that means.
> Traditionally, one generates diatonic scales by iterating perfect fifths
> and octaves. These scales are the built up by major and minor seconds,
> often called whole and half tonesteps, those the latter need not be one
> half in szie for the former. This only works for Western music: meantones.
> Just intonation uses two major seconds of different sizes. Turkish,
> Persian and Arab music uses intermediate intervals, which can be described
> by neutral seconds - intervals between the minor and major seconds.
In just intonation, meaning ratios, it is possible to write and bridge
between all tuning systems. The trick is realizing the the tune you want to
port or incorporate probably spans less than three octaves and uses less
than nine notes. Once you hav worked out what those notes are in natural
numbers...The temptation for me in such a process is to elaborate, so that
when a note is repeated in the orijinal, I use two different notes and
different timing. I use two different notes and different timing when I sing
"Ode To Joy", for example.
I am not sure if it is a diatonic scale that I use. It goes 8:7, 9:8, 10:9,
11:10 (that is the one that sounds farthest off to my ears)...14:13. It is
currently in the right channel of the second phrase in
http://ecn.ab.ca/~brewhaha/Sound/MSixths.mp3 (18kbps). That will probably
change, though, if I can make four-part harmony for something *like* the
C-natural scale (It will hav a better beat, if I finish it). I said also
that I could do something like the chromatic C-scale, and my numbers were
one-off. I should hav said 12...24 were my notes, and my basis frequency
would be 11 (132/12) Hertz. IOW, the numbers twelve to twenty four,
multiplied by a basis frequency, would be an approximation of the chromatic
C-scale, eksept that ratios become smaller as they go up, and eksept that
naturals on such a scale do not correspond. IOW, 14:12 <> 8:7.
I hav no intention of porting this to an acoustic western instrument (except
maybe a guitar, if my notes number six) --simulated, yes, acoustic, no--
because I often find myself writing numbers higher than twenty-four when I
write tunes.
Hans Aberg
>>> I suggested the pair of letters to reprezent two intervals used to come
>>> up with a scale, Dee and Teh. Zii and Sih are better, since neither is
>>> used in music to my knowledge, and they are closely related in phonetics.
>> I'm not sure what intervals you are have in mind here.
>
> The two or three ratios you would use to construct a diatonic scale.
> The scale I use is probably not diatonic, whatever that means.
Ancient Greek music built scales from three types of tetrachords:
diatonic, chromatic and enharmonic. Similar ideas are in use today in
Arab music, see ajnas <http://www.maqamworld.com/>, and also Turkish and
Persian music. One gets the intermediate pithces by different ways of
dividing perfect fourths and minor and major thirds.
>> Traditionally, one generates diatonic scales by iterating perfect fifths
>> and octaves. These scales are the built up by major and minor seconds,
>> often called whole and half tonesteps, those the latter need not be one
>> half in szie for the former. This only works for Western music: meantones.
>> Just intonation uses two major seconds of different sizes. Turkish,
>> Persian and Arab music uses intermediate intervals, which can be described
>> by neutral seconds - intervals between the minor and major seconds.
>
> In just intonation, meaning ratios, it is possible to write and bridge
> between all tuning systems. The trick is realizing the the tune you want to
> port or incorporate probably spans less than three octaves and uses less
> than nine notes. Once you hav worked out what those notes are in natural
> numbers...The temptation for me in such a process is to elaborate, so that
> when a note is repeated in the orijinal, I use two different notes and
> different timing. I use two different notes and different timing when I sing
> "Ode To Joy", for example.
Traditionally, Just intonation is the 5-limit only. You can see two
different ways of generating it here:
http://en.wikipedia.org/wiki/Just_intonation
The first one, the traditional, departs from the Just major chord
1:5/4:3/2 = 4:5:6, and constructs a scale so that the tonic, dominant
and subdominant (in key C this is C, G and F) play this chord. Then one
gets major seconds of two different sizes, and one of the minor chords
go out of tune.
The other method on that page does this for the Just minor chord
!:6/5:3/2. Then one of the major chords go out of tune.
So it is not possible to have chords that are exact rational intervals
in all positions. So either, these must be adapted, avoided, or accepted
as not being fully convergent.
> I am not sure if it is a diatonic scale that I use. It goes 8:7, 9:8, 10:9,
> 11:10 (that is the one that sounds farthest off to my ears)...14:13. It is
> currently in the right channel of the second phrase in
> http://ecn.ab.ca/~brewhaha/Sound/MSixths.mp3 (18kbps).
Just intonation is traditionally only the 5-limit. Quarter-comma
meantone has notes close to the 7-limit. Relative to this, the interval
11/8 is close to an exact quarter-note, and quite foreign music.
The interval is quite close to the interval 27/25, and the latter can be
gotten from Pythagorean tuing by raising the minor second a double
syntonic comma. This can be used as the intermediate pitch in Persian
music, and I think Arab music, too, though one should note that there is
no generally accepted tuning system.
> That will probably
> change, though, if I can make four-part harmony for something *like* the
> C-natural scale (It will hav a better beat, if I finish it). I said also
> that I could do something like the chromatic C-scale, and my numbers were
> one-off. I should hav said 12...24 were my notes, and my basis frequency
> would be 11 (132/12) Hertz. IOW, the numbers twelve to twenty four,
> multiplied by a basis frequency, would be an approximation of the chromatic
> C-scale, eksept that ratios become smaller as they go up, and eksept that
> naturals on such a scale do not correspond. IOW, 14:12 <> 8:7.
If approximations are OK, then E72 approximates the 11-limit: E36
contains the 7/4 and 27/25 intermediate intervals, and E24 the interval
11/8.
Hans
Bohgosity BumaskiL
4:5:6 is one that I calculated to be CEG (going up). Six and four are a
perfect fifth , or G:C, while E is a major third above C, making it
reprezent five, so I am not sure how you get an F in there, above the G. A
series of perfect fourths is 27:36:48:64, and the longer the series you
want, the higher your initial power of three (27=3^3) needs to be. Four part
harmony is hard enough for me.
> The other method on that page does this for the Just minor chord
> !:6/5:3/2. Then one of the major chords go out of tune.
>
> So it is not possible to have chords that are exact rational intervals in
> all positions.
Just intonation, and since I used 23 in a four-part composition, rational
intonation, does not always fit neatly where your hands can reach on a
twelve key board. The way I look at it, just intonation is a list of target
ratios. If I get a good mix of perfect, major, and minor ratios, then I am
starting my composition with the right number. If I get nothing but obscure
ratios, then I might be offering too much of a challenge to my listeners.
The trick is that once a composition is under way, there will be only about
nine or even fewer distinct notes, so they can be worked out to a series of
natural numbers--always--any set of fractions can be re-worked into natural
numbers, except that to be musical, those numbers will *probably* be low, so
rather than try to work by sheer math, it is probably easier to work by ear
and a good guess at where to start, if you are re-writing Mozart in just
intonation, for example.
The understanding I need then is whether a lot of prime numbers should
be in a composition, and usually, not.
http://ecn.ab.ca/~brewhaha/Sound/Diamonds.mp3 is an exception. I am delaying
my source production on that, because I orijinally wrote the numbers into a
fractint formula, too, and I want the source to jenerate a fractint formula
with the extension I wrote into that tune, recently, then cross-post into
sci.fractal. I ignored zeros for graphics output--had to put them back in to
understand the tune. Maybe I should write them as ones (which are usually
subsonic).
> So either, these must be adapted, avoided, or accepted as not being fully
> convergent.
>
>> I am not sure if it is a diatonic scale that I use. It goes 8:7, 9:8,
>> 10:9, 11:10 (that is the one that sounds farthest off to my
>> ears)...14:13. It is currently in the right channel of the second phrase
>> in http://ecn.ab.ca/~brewhaha/Sound/MSixths.mp3 (18kbps).
>
> Just intonation is traditionally only the 5-limit. Quarter-comma meantone
> has notes close to the 7-limit. Relative to this, the interval 11/8 is
> close to an exact quarter-note, and quite foreign music.
I am reading that acute is the converse of grave, both terms reflecting
something that is farther from normal than diminished or augmented. I am
glad you kept that in the thread. Twelve to eight is a perfect fifth, so
eleven to eight is an acute undecimal fifth or something, and not an
eleventh at all. It is perhaps a complement to a perfect eleventh (8:3).
I hope you understand that my demonstration output, when it is finished
that is, will include thirteen in what sounds like Doh-Ray-Me (I am not sure
whether eleven is sharp or flat of the western scale), so I am inclined to
include a prime limit of thirteen, if not more, in the definition of just
intonation.
In most cases, I do not care about "notes per octave", "prime limit", or
"odd limit", as long as it sounds right. I tried to rewrite "Lohruh" (and
"Lohruh, Regruntled") with lower numbers, and it did not sound right. So,
either I did not pick the right lower number for starting, or it was a
terminal series to begin with, and I won't make it better. I should check if
I applied a cut on the volume of upper parallel harmonics to that one,
especially since I am not currently writing ringtones.
> The interval is quite close to the interval 27/25, and the latter can be
> gotten from Pythagorean tuing by raising the minor second a double
> syntonic comma. This can be used as the intermediate pitch in Persian
> music, and I think Arab music, too, though one should note that there is
> no generally accepted tuning system.
>
>> That will probably change, though, if I can make four-part harmony for
>> something *like* the C-natural scale (It will hav a better beat, if I
>> finish it). I said also that I could do something like the chromatic
>> C-scale, and my numbers were one-off. I should hav said 12...24 were my
>> notes, and my basis frequency would be 11 (132/12) Hertz. IOW, the
>> numbers twelve to twenty four, multiplied by a basis frequency, would be
>> an approximation of the chromatic C-scale, eksept that ratios become
>> smaller as they go up, and eksept that naturals on such a scale do not
>> correspond. IOW, 14:12 <> 8:7.
>
> If approximations are OK, then E72 approximates the 11-limit: E36 contains
> the 7/4 and 27/25 intermediate intervals, and E24 the interval 11/8.
>
> Hans
_______
http://ecn.ab.ca/~brewhaha/
Bohgosity BumaskiL
> Just intonation is traditionally only the 5-limit. Quarter-comma meantone
> has notes close to the 7-limit. Relative to this, the interval 11/8 is
> close to an exact quarter-note, and quite foreign music.
Alberta is east of Japan. "11/8 undecimal semi-augmented fourth" is in a
Jerman document of all musical intervals. -- "lesser undecimal tritone" on
wikipedia. There were a few terms for it, one being "Major Fourth". Someone
deprecated using major and minor for the perfect ratios. It is part of a
tune that I published in June at
http://groups.google.com/group/rec.music.theory/msg/7359c4bfd56bacc0 . The
ratio occurs on the fifth note of the synthesizer rendition and on the first
occurance of "a" (again, the fifth note) of my vocals, as 40:55. I used a
septimal tritone in that, too, as 49:35. Neither got very much emphasis.
Both occurred before bits of unison at the end of a phrase.
Hans Aberg
>> Traditionally, Just intonation is the 5-limit only. You can see two
>> different ways of generating it here:
>> http://en.wikipedia.org/wiki/Just_intonation
>> The first one, the traditional, departs from the Just major chord
>> 1:5/4:3/2 = 4:5:6, and constructs a scale so that the tonic, dominant and
>> subdominant (in key C this is C, G and F) play this chord. Then one gets
>> major seconds of two different sizes, and one of the minor chords go out
>> of tune.
>
> 4:5:6 is one that I calculated to be CEG (going up). Six and four are a
> perfect fifth , or G:C, while E is a major third above C, making it
> reprezent five, so I am not sure how you get an F in there, above the G. A
> series of perfect fourths is 27:36:48:64, and the longer the series you
> want, the higher your initial power of three (27=3^3) needs to be. Four part
> harmony is hard enough for me.
Use the octave inversions, too. First start at C, then G is 3/2 above C.
F is the inversion 2/3 shifted into the octave, that is, 4/3.
Then take the Just major chord 1:5/4:3/2 = 4:5:6 on C, F and G. It
generates a major scale with major seconds of two different sizes;
between C-D and D-E.
Then the minor chords at D, E and A. One of the goes out of tune.
So it is not possible to do exact Just harmony on that idea.
One idea is to temper out the out the two major seconds so that they
become equal. Then all chords will strictly speaking go out of tune, but
only with a little.
Or you can avoid the chord that goes out of tune, or you can try to
adjust it on the fly, leading to an adaptive tuning. This last one is
probably what musicians on variable pitch instruments do.
>> The other method on that page does this for the Just minor chord
>> !:6/5:3/2. Then one of the major chords go out of tune.
>>
>> So it is not possible to have chords that are exact rational intervals in
>> all positions.
>
> Just intonation, and since I used 23 in a four-part composition, rational
> intonation, does not always fit neatly where your hands can reach on a
> twelve key board. The way I look at it, just intonation is a list of target
> ratios. If I get a good mix of perfect, major, and minor ratios, then I am
> starting my composition with the right number. If I get nothing but obscure
> ratios, then I might be offering too much of a challenge to my listeners.
So problem above arouse by the requriement of doing exact harmony on all
notes.
> The trick is that once a composition is under way, there will be only about
> nine or even fewer distinct notes, so they can be worked out to a series of
> natural numbers--always--any set of fractions can be re-worked into natural
> numbers, except that to be musical, those numbers will *probably* be low, so
> rather than try to work by sheer math, it is probably easier to work by ear
> and a good guess at where to start, if you are re-writing Mozart in just
> intonation, for example.
And it happened with only 7 notes. If you exclude a minor chord, then it
can be done.
> The understanding I need then is whether a lot of prime numbers should
> be in a composition, and usually, not.
You can try close intervals like 14/13, 27/25, and 13/12, to see which
ones you find sound best. High primes may not be important in harmony,
because the partials are weak. A trumpet player can play up the the 16th
partial. So it is probably no point going higher.
> http://ecn.ab.ca/~brewhaha/Sound/Diamonds.mp3 is an exception. I am delaying
> my source production on that, because I orijinally wrote the numbers into a
> fractint formula, too, and I want the source to jenerate a fractint formula
> with the extension I wrote into that tune, recently, then cross-post into
> sci.fractal. I ignored zeros for graphics output--had to put them back in to
> understand the tune. Maybe I should write them as ones (which are usually
> subsonic).
Ultimately it is just the ears that decide what is right.
>> Just intonation is traditionally only the 5-limit. Quarter-comma meantone
>> has notes close to the 7-limit. Relative to this, the interval 11/8 is
>> close to an exact quarter-note, and quite foreign music.
>
> I am reading that acute is the converse of grave, both terms reflecting
> something that is farther from normal than diminished or augmented. I am
> glad you kept that in the thread. Twelve to eight is a perfect fifth, so
> eleven to eight is an acute undecimal fifth or something, and not an
> eleventh at all. It is perhaps a complement to a perfect eleventh (8:3).
The quarter-comma just comes from the fact that the M3 = 4 P5 - 2 P8.
One wants to adjust down the M3 of Pythagorean tuning to exactly 5/4, a
difference whuich is called the syntonic comma. The then P5 which is 3/2
must be lowered by a 1/4 of that comma.
> I hope you understand that my demonstration output, when it is finished
> that is, will include thirteen in what sounds like Doh-Ray-Me (I am not sure
> whether eleven is sharp or flat of the western scale), so I am inclined to
> include a prime limit of thirteen, if not more, in the definition of just
> intonation.
>
> In most cases, I do not care about "notes per octave", "prime limit", or
> "odd limit", as long as it sounds right. I tried to rewrite "Lohruh" (and
> "Lohruh, Regruntled") with lower numbers, and it did not sound right. So,
> either I did not pick the right lower number for starting, or it was a
> terminal series to begin with, and I won't make it better. I should check if
> I applied a cut on the volume of upper parallel harmonics to that one,
> especially since I am not currently writing ringtones.
Only you can decide what sounds right to you. Theories about 5-limit,
etc, is used to create a certain type of music. - Romm for
experimentation :-).
The notion of a scale degree is though very important: one can have a
scale with variable scale degrees, like in melodic minor. Oriental
microtonal scales have essentially the same scale degree system, but
combined with intermediate pitches.
I don't know how wide applicability such a system has.
Hans
Hans Aberg
Since Medieval times one has experimented with all sorts of intervals.
Perhaps this site has something sbout it:
http://www.medieval.org/
But the traditional Just intonation only uses the 5-limit. If one
tempers it out to a quarter comma meantone tuning, that is, set M3 to
5/4 and also the octave as usual to 2, then the interval 7/4 is
approximated as a augmented 6th. It is only used as a predominant -look
up "German sixth" chord.
In E72, the interval 11/8 is very exactly a E12 quartertone. Arab music,
one agrees that the intermediate pitch should be lowered, though
traditionally that is based on the Pythagorean tuning (use E53, M = 9, m
= 4, and an intermediate pitch suggestion is n = 6).
So 11/8 does fit here either. But it is certainly possible to make
harmonically convergent chords with it. But there is a problem if one
want to fit it into a melodic pattern. One idea might to use it as an
alternate scale degree to P4 = 4/3.
Hans
Bohgosity BumaskiL
> Bohgosity BumaskiL wrote:
>>> Traditionally, Just intonation is the 5-limit only. You can see two
>>> different ways of generating it here:
>>> http://en.wikipedia.org/wiki/Just_intonation
>>> The first one, the traditional, departs from the Just major chord
>>> 1:5/4:3/2 = 4:5:6, and constructs a scale so that the tonic, dominant
>>> and subdominant (in key C this is C, G and F) play this chord. Then one
>>> gets major seconds of two different sizes, and one of the minor chords
>>> go out of tune.
>>
>> 4:5:6 is one that I calculated to be CEG (going up). Six and four are a
>> perfect fifth , or G:C, while E is a major third above C, making it
>> reprezent five, so I am not sure how you get an F in there, above the G.
>> A series of perfect fourths is 27:36:48:64, and the longer the series you
>> want, the higher your initial power of three (27=3^3) needs to be. Four
>> part harmony is hard enough for me.
(snip)
>> Just intonation, and since I used 23 in a four-part composition, rational
>> intonation, does not always fit neatly where your hands can reach on a
>> twelve key board. The way I look at it, just intonation is a list of
>> target ratios. If I get a good mix of perfect, major, and minor ratios,
>> then I am starting my composition with the right number. If I get nothing
>> but obscure ratios, then I might be offering too much of a challenge to
>> my listeners.
>
> So problem above arouse by the requriement of doing exact harmony on all
> notes.
I hav not run into the problem in a way that is insurmountable. It really is
the point of a script synthesizer. All I need to do is start with a better
number. In Loruh, that was pointless. As far as I can tell, the high prime
numbers were supposed to be in Lohruh (last time I checked, that was my
top-rated piece, without a complicated post-process), and so was a perfect
fifth that occurred as thirty-six to twenty-four supposed to be in it.
http://groups.google.com/group/rec.music.makers.synth/msg/3ecef34e4516ee1a
>> The trick is that once a composition is under way, there will be only
>> about nine or even fewer distinct notes, so they can be worked out to a
>> series of natural numbers--always--any set of fractions can be re-worked
>> into natural numbers, except that to be musical, those numbers will
>> *probably* be low, so rather than try to work by sheer math, it is
>> probably easier to work by ear and a good guess at where to start, if you
>> are re-writing Mozart in just intonation, for example.
>
> And it happened with only 7 notes. If you exclude a minor chord, then it
> can be done.
Any series of fractions can be given a common denominator. At that point,
you can ignore the denominator and arbitrarily change it to one. Note that
you may not be writing the numerator as a bigger figure than the
denominator, which is usual practice, because you might be moving pitch
down. If the numbers are very high, then your basis frequency will be
between zero and one. If the numbers are below a hundred, as mine typically
are, then you will want to scale pitches up to make them audible. If you run
into what seems like an insurmountable problem, then there are many
compromises you can make, and a run of primes is not out of the question.
>> The understanding I need then is whether a lot of prime numbers
>> should be in a composition, and usually, not.
>
> You can try close intervals like 14/13, 27/25, and 13/12, to see which
> ones you find sound best. High primes may not be important in harmony,
> because the partials are weak. A trumpet player can play up the the 16th
> partial. So it is probably no point going higher.
>
>> http://ecn.ab.ca/~brewhaha/Sound/Diamonds.mp3 is an exception. I am
>> delaying my source production on that, because I orijinally wrote the
>> numbers into a fractint formula, too, and I want the source to jenerate a
>> fractint formula with the extension I wrote into that tune, recently,
>> then cross-post into sci.fractal. I ignored zeros for graphics
>> output--had to put them back in to understand the tune. Maybe I should
>> write them as ones (which are usually subsonic).
>
> Ultimately it is just the ears that decide what is right.
Pauses are essential. In that particular case, I did not want zeros in my
graphic, so I will either explicitly ignore them or write them as ones.
Answering whether the background for
http://ecn.ab.ca/~brewhaha/WP_CRYSTAL.HTM is related would take some
experimentation and understanding of my source code -- still not finished
integrating a source fork, because I had housework and website maintenance.
>>> Just intonation is traditionally only the 5-limit. Quarter-comma
>>> meantone has notes close to the 7-limit. Relative to this, the interval
>>> 11/8 is close to an exact quarter-note, and quite foreign music.
>>
>> I am reading that acute is the converse of grave, both terms
>> reflecting something that is farther from normal than diminished or
>> augmented. I am glad you kept that in the thread. Twelve to eight is a
>> perfect fifth, so eleven to eight is an acute undecimal fifth or
>> something, and not an eleventh at all. It is perhaps a complement to a
>> perfect eleventh (8:3).
>
> The quarter-comma just comes from the fact that the M3 = 4 P5 - 2 P8. One
> wants to adjust down the M3 of Pythagorean tuning to exactly 5/4, a
> difference whuich is called the syntonic comma. The then P5 which is 3/2
> must be lowered by a 1/4 of that comma.
When I went back to my composition, I found that there were two names for
the same ratio on wikipedia. One of them does not present a problem with
either Physics or musical terminology. Whether you actually hear a "lesser
undecimal tritone" as three tones while only two play, is perhaps a personal
or contextual matter. There was an article on "Major Fourth". I did not do
anything about that other than label a link to it as a "semi-augmented
fourth", and I probably won't do anything more.
>> I hope you understand that my demonstration output, when it is
>> finished that is, will include thirteen in what sounds like Doh-Ray-Me (I
>> am not sure whether eleven is sharp or flat of the western scale), so I
>> am inclined to include a prime limit of thirteen, if not more, in the
>> definition of just intonation.
>>
>> In most cases, I do not care about "notes per octave", "prime limit",
>> or "odd limit", as long as it sounds right. I tried to rewrite "Lohruh"
>> (and "Lohruh, Regruntled") with lower numbers, and it did not sound
>> right. So, either I did not pick the right lower number for starting, or
>> it was a terminal series to begin with, and I won't make it better. I
>> should check if I applied a cut on the volume of upper parallel harmonics
>> to that one, especially since I am not currently writing ringtones.
>
> Only you can decide what sounds right to you. Theories about 5-limit, etc,
> is used to create a certain type of music. - Romm for experimentation :-).
You need some of the easier harmonies to understand in a piece. 2:1, for
example, is where I started in HeLoh. I did not pull a 25:16 (a classic
augmented fifth -- 24:16 is a perfect fifth), until the tune was well under
way.
> The notion of a scale degree is though very important: one can have a
> scale with variable scale degrees, like in melodic minor. Oriental
> microtonal scales have essentially the same scale degree system, but
> combined with intermediate pitches.
>
> I don't know how wide applicability such a system has.
>
> Hans
>
"pitch correction" in Nero's WaveEdit actually wrecks "A Tune for Tigger's
Song", no matter which of three variants on western tuning I use. If I
select the right key in Pythagorean carefully, then maybe that won't happen.
I won't know. My work on that tune is done.
I look at it this way, occasionally remembering the look of disdain that a
choir director aimed at synthesizers: When we made the synth, we gave up
resonant sound boards and a series that naturally fits as standing waves on
a string. Even the piano-forte's intended feature, to be able to play softly
or loudly, only comes with more expensive models of synth. The potential in
a script synth, is to fully specify what a group of woodshedders *might*
come up with, or even go beyond them and detail chords that they would not
hit without an example, or only explain what is written.
I can't say much about Oriental tuning, except that sometimes I like it.
There is a just intonation specified for a twenty-four note scale. I do not
believe that it would be usable in every tune that some people can think of
or sing, and singing is really where composition should start.
Hans Aberg
>>>> Traditionally, Just intonation is the 5-limit only. You can see two
>>>> different ways of generating it here:
>>>> http://en.wikipedia.org/wiki/Just_intonation
>>>> The first one, the traditional, departs from the Just major chord
>>>> 1:5/4:3/2 = 4:5:6, and constructs a scale so that the tonic, dominant
>>>> and subdominant (in key C this is C, G and F) play this chord. Then one
>>>> gets major seconds of two different sizes, and one of the minor chords
>>>> go out of tune.
>> So problem above arouse by the requirement of doing exact harmony on all
>
> I hav not run into the problem in a way that is insurmountable. It really is
> the point of a script synthesizer. All I need to do is start with a better
> number. In Loruh, that was pointless. As far as I can tell, the high prime
> numbers were supposed to be in Lohruh (last time I checked, that was my
> top-rated piece, without a complicated post-process), and so was a perfect
> fifth that occurred as thirty-six to twenty-four supposed to be in it.
> http://groups.google.com/group/rec.music.makers.synth/msg/3ecef34e4516ee1a
>> And it happened with only 7 notes. If you exclude a minor chord, then it
>> can be done.
>
> Any series of fractions can be given a common denominator. At that point,
> you can ignore the denominator and arbitrarily change it to one. Note that
> you may not be writing the numerator as a bigger figure than the
> denominator, which is usual practice, because you might be moving pitch
> down. If the numbers are very high, then your basis frequency will be
> between zero and one. If the numbers are below a hundred, as mine typically
> are, then you will want to scale pitches up to make them audible. If you run
> into what seems like an insurmountable problem, then there are many
> compromises you can make, and a run of primes is not out of the question.
If you leave the fixed values and adjusting them as you go along, you
are doing an adaptive tuning.
In Western (CPP - common practise period) harmony theory, one wants to
pivot pitches so some of the pitches of the old chord remains in the new
one. If one is doing that in Just intonation, the pitch will drift in
certain common combinations. So some adjustment is necessary. One method
is tempering - spreading the drift on several intervals.
> When I went back to my composition, I found that there were two names for
> the same ratio on wikipedia. One of them does not present a problem with
> either Physics or musical terminology. Whether you actually hear a "lesser
> undecimal tritone" as three tones while only two play, is perhaps a personal
> or contextual matter. There was an article on "Major Fourth". I did not do
> anything about that other than label a link to it as a "semi-augmented
> fourth", and I probably won't do anything more.
You might look for names of chords in Scala, some stuff is posted
online. Som names may be older, but others have been added rather ad hoc
in modern times, I think. Best to fit them in some context.
> You need some of the easier harmonies to understand in a piece. 2:1, for
> example, is where I started in HeLoh. I did not pull a 25:16 (a classic
> augmented fifth -- 24:16 is a perfect fifth), until the tune was well under
> way.
Such intervals are tied to Western CPP harmony. If your musicis not seem
to be strongly tied to that, then you might try putting intervals like
13/8, 11/8, and 7/4. It may depend on the instrument used - the clarinet
is in harmonic above the 7th partial in the high an low ranges.
> "pitch correction" in Nero's WaveEdit actually wrecks "A Tune for Tigger's
> Song", no matter which of three variants on western tuning I use. If I
> select the right key in Pythagorean carefully, then maybe that won't happen.
> I won't know. My work on that tune is done.
They give slightly different styles. Roughly Pythagorean or E53 (E41)
are Medieval, Oriental. Meantone or E31 is Renaissance and better for
CPP, and some folk like Scottish. E12 is general modern pop and jazz.
> I look at it this way, occasionally remembering the look of disdain that a
> choir director aimed at synthesizers: When we made the synth, we gave up
> resonant sound boards and a series that naturally fits as standing waves on
> a string. Even the piano-forte's intended feature, to be able to play softly
> or loudly, only comes with more expensive models of synth. The potential in
> a script synth, is to fully specify what a group of woodshedders *might*
> come up with, or even go beyond them and detail chords that they would not
> hit without an example, or only explain what is written.
>
> I can't say much about Oriental tuning, except that sometimes I like it.
> There is a just intonation specified for a twenty-four note scale. I do not
> believe that it would be usable in every tune that some people can think of
> or sing, and singing is really where composition should start.
A possible tuning for Arab and Persian music is Pythagorean, where the
intermediate pitch is 27/25. In E53, major and minor seconds m = 4, M =
9, and the intermediate pitch can be described by a neutral n = 6. This
means that the microtonal accidents are: raising from m to n, that is 2
commas, and lowering from M to n, which is 3 commas.
This comes from different ways of dividing the minor and major third
intervals, starting with n, and the remaining part is a large second N =
7 and a plustone P = 12 commas.
You might check some maqams on the <http://maqamworld.com> pages. The
give other information on how one is playing. There is no general
accepted one, so the above is just a suggestion. The microtonal accents
are from E24, but the intermediate pitch should be lower than an exact
quarternote. The suggestion above comes from Farhat's thesis.
If you want to try E12 compatible, then you might try E36, where m = 3,
M = 6, and n = 4. Microtonal accent raising 1, lowering 2 tone steps.
Turkish music divides (in E53 theoretical description) the perfect
fourth as n P n, giving n = 5, P = 12, with some variations, due to the
fact that it wasn't done in a theoretically consistent way.
Hans
Bohgosity BumaskiL
(...)
> Then take the Just major chord 1:5/4:3/2 = 4:5:6 on C, F and G. It
> generates a major scale with major seconds of two different sizes; between
> C-D and D-E.
1:1 4:3 3:2 is the same as 4:4 16:12 6:4. To put all three over twelve:
12:12 16:12 18:12. At that point, you can ignore the denominator: 12:16:18,
then divide all three numbers by 2 to get 6:8:9 in reduced form. Note that
nine to six is a perfect fifth, just like G:C, and nine to eight is a major
second, while eight to six is a perfect fourth. In Equal Temperament, you
can get the same triad in different keys, with that three-member ratio as a
constant. In practice, you would hav to multiply all three numbers by at
least five Hertz to make them audible; eleven Hz for singable.
Hans Aberg
>> Then take the Just major chord 1:5/4:3/2 = 4:5:6 on C, F and G. It
>> generates a major scale with major seconds of two different sizes; between
>> C-D and D-E.
> (...)
> 1:1 4:3 3:2 is the same as 4:4 16:12 6:4. To put all three over twelve:
> 12:12 16:12 18:12. At that point, you can ignore the denominator: 12:16:18,
> then divide all three numbers by 2 to get 6:8:9 in reduced form.
In Scala, this notation. I take it is the same thing as what is called
projective coordinates in math. The notation x_0:...:x_n is the same
thing as the tuple (x_0, ..., x_n) only that one instroduces an
equivalence relation
(x_0, ..., x_n) ~ (y_0, ..., y_n)
if and only if there is a non-zero scalar r such that
(x_0, ..., x_n) = r(y_0, ..., y_n)
So 1:5/4:3/2 is short for (1, 5/4, 3/2) but equivalent if we multiply by
4: 4(1, 5/4, 3/2) = 4*(1, 5/4, 3/2) = (4*1, 4*5/4, 4*3/2) = (4, 5, 6).
This the formal calulation, but one can then just write
1:5/4:3/2 = 4*(1:5/4:3/2) = 4:5:6
Then 1:4/3:3/2 = 6*(1:4/3:3/2) = 6:8:9.
> Note that
> nine to six is a perfect fifth, just like G:C, and nine to eight is a major
> second, while eight to six is a perfect fourth.
If you look at the derivation here
http://en.wikipedia.org/wiki/Just_intonation#Diatonic_scale
you can see that the one on A is 1:Ts:Tt, wheres the one on D is
1:ts:Tt. So one of them is not the Just minor chord - check that it is
the one on D.
> In Equal Temperament, you
> can get the same triad in different keys, with that three-member ratio as a
> constant. In practice, you would hav to multiply all three numbers by at
> least five Hertz to make them audible; eleven Hz for singable.
ETs as well as any extended meantone or Pythagorean tuning (notes A B C
D E F G plus all notes gotten from them applying any number of sharps
and flats) tempered out this difference, so that error becomes equal in
any playing position.
One the cannot only have two intervals exact as chosen. Typically the
octave is set to 2. If one want to approximate major third 5/4 better,
then the pure fifth 3/2 becomes worse, but only at a speed a 1/4 of the
change in the major third, since M3 = 4P4 - 2P8. So when M3 is exactly
5/4, is has moved a by definition a syntonic comma, and P5 a quarter of
that, hence it is called historically the quarter-comma meantone.
Historically, a meantone is a choice of 12 notes from the the extended
meantone. This leaves a gap, called the wolf. It really sounds that bad.
But there is really no need to reduce to 12 notes in computer generated
music.
Hans
Bohgosity BumaskiL
6:8:9 is C:F:G.
To verify, six to four is a perfect fifth, and so is nine to six -- G:C in
both cases.
E is the mean of G and C. It is also a major third above C, and a minor
third below G.
F is not the mean of G and C. It is a major second below G, and a perfect
fourth above C. The math is the same as when you add or subtract fractions,
except that instead of adding or subtracting fractions, once all ratios are
above same denominator, ignore denominator, then concatenate into one ratio
with three or more parts.
_______
http://ecn.ab.ca/~brewhaha/ BrewJay's Babble Bin
"Hans Aberg" <haberg_...@math.su.se> wrote in message
news:hcda6f$1qk$1...@news.eternal-september.org...
> Bohgosity BumaskiL wrote:
>>> Then take the Just major chord 1:5/4:3/2 = 4:5:6 on C, F and G. It
>>> generates a major scale with major seconds of two different sizes;
>>> between C-D and D-E.
>> (...)
>> 1:1 4:3 3:2 is the same as 4:4 16:12 6:4. To put all three over twelve:
>> 12:12 16:12 18:12. At that point, you can ignore the denominator:
>> 12:16:18, then divide all three numbers by 2 to get 6:8:9 in reduced
>> form.
>
> In Scala, this notation. I take it is the same thing as what is called
> projective coordinates in math. The notation x_0:...:x_n is the same thing
> as the tuple (x_0, ..., x_n) only that one instroduces an equivalence
> relation
> (x_0, ..., x_n) ~ (y_0, ..., y_n)
> if and only if there is a non-zero scalar r such that
> (x_0, ..., x_n) = r(y_0, ..., y_n)
>
> So 1:5/4:3/2 is short for (1, 5/4, 3/2) but equivalent if we multiply by
> 4: 4(1, 5/4, 3/2) = 4*(1, 5/4, 3/2) = (4*1, 4*5/4, 4*3/2) = (4, 5, 6).
>
> This the formal calulation, but one can then just write
> 1:5/4:3/2 = 4*(1:5/4:3/2) = 4:5:6
Which is C:E:G, not C:F:G.
> Then 1:4/3:3/2 = 6*(1:4/3:3/2) = 6:8:9.
Different ratios -- C:F:G.
Agreed. I typically use notes as high as sixty, and that means about thirty
notes per octave. My tunes do not use them all, and typically span one
octave. I used a wolf fifth -- 40:27 in
http://ecn.ab.ca/~brewhaha/Sound/HeLoh.mp3
It does not hav much stress, and it is there. It is not that bad in context.
I do not follow many rules in duet harmony. I just write two tunes that fit
the same beat, then transpose them to end in unison. Then, I check to see if
they should be an octave apart or a fifth apart, adjusting the basis
frequency of each part. Unison seems to be a good way to start and end a
duet.
LJS
<brewh...@freenet.edmonton.ab.ca> wrote:
> 4:5:6 is C:E:G.
> 6:8:9 is C:F:G.
>
> To verify, six to four is a perfect fifth, and so is nine to six -- G:C in
> both cases.
> E is the mean of G and C. It is also a major third above C, and a minor
> third below G.
> F is not the mean of G and C. It is a major second below G, and a perfect
> fourth above C. The math is the same as when you add or subtract fractions,
> except that instead of adding or subtracting fractions, once all ratios are
> above same denominator, ignore denominator, then concatenate into one ratio
> with three or more parts.
>
> "Hans Aberg" <haberg_20080...@math.su.se> wrote in message
> It does not hav much stress, and it is there. It is not that bad in context.
> I do not follow many rules in duet harmony. I just write two tunes that fit
> the same beat, then transpose them to end in unison. Then, I check to see if
> they should be an octave apart or a fifth apart, adjusting the basis
> frequency of each part. Unison seems to be a good way to start and end a
> duet.
Do I understand that you are using these ratios as a means to use mean
tone scale and other non 12-tet scales (or maybe them also) to be
generated on the spot so that if you modulate, the new key will be
established by these ratios?
If I understand correctly, and I am not sure that I do, this would
allow the same techniques of modulation for "mean tone" music found in
12-tet music of the CPP to work in the same manner.
IF my understanding is in line with what you are saying, then I would
ask if you have a means of "getting rid of the wolf" (your name isn't
Peter is it? lol) as it seems that as you modulate through keys that
the wolf will cause your return to tonic to be slightly off from the
original pitch.
I have no problem with that and I had brought up this thought in an
old electronic music class I had at Oberlan around 1990. At the time,
there did not seem to have the capacity to do this with the tools that
they had at their electronic studio. I suspect that it could be done
with "Max" but this program was fairly new to them at the time and I
did not get to investigate that concept any further.
I understand that Guesualdo did this with his vocal compositions back
when it was about as popular as the other "interesting" hobbies that
he had in the privacy of his own castle. In his case, he seems to have
just allowed the pitch move around to follow his modulations.
In either case, it sounds like an interesting project or research.
LJS
Hans Aberg
> 4:5:6 is C:E:G.
> 6:8:9 is C:F:G.
In Just intonation, the variation given on the Wikipedia page that makes
sure all major chords are built up exact rational interval ratios. (Some
use "Just" to denote anything that is made up by rational intervals.
In other tuning systems, these are a bit off from the exact rational ratios.
> To verify, six to four is a perfect fifth, and so is nine to six -- G:C in
> both cases.
A Just perfect fifth is 3/2, and Just perfect fourth is the octave
inversion 4/3 = 2/(3/2).
> E is the mean of G and C.
No. To understand Pythagorean tuning, I find it easy to use E53, where m
= 4, M = 9, and #, b alter with M - m = 5. A E53 tonestep is very close
to a syntonic comma, the latter can be defined as the difference between
the (lower) Just major third 5/4 and the (higher) Pythagorean major third.
In E53, major third is M + M = 18, whereas the Just major third 5/4 is
quite close to 17. The Just perfect 5th is very close to 3M + m = 31.
So the half of that 31 is 15.5.
> It is also a major third above C, and a minor
> third below G.
> F is not the mean of G and C.
It is the octave inversion, as I wrote above.
> It is a major second below G, and a perfect
> fourth above C.
Yes, but also note that Just intonation makes use of two major seconds
of different sizes. This give one of them, 9/8. The other Just major
second is the difference between the Just major third 5/4 and this major
second, (5/4)/(9/8) = 10/9.
> The math is the same as when you add or subtract fractions,
> except that instead of adding or subtracting fractions, once all ratios are
> above same denominator, ignore denominator, then concatenate into one ratio
> with three or more parts.
It is just convenient to write intervals as additively, as music depends
so much on the ability to transpose. An interval is a transposable ratio
of pitches. A pitch is quite closely related to frequencies, though it
may be altered somewhat through stretch tuning, dynamics. And the aer is
not able to transpose above about 3.5 kHz.
>> In Scala, this notation. I take it is the same thing as what is called
>> projective coordinates in math. The notation x_0:...:x_n is the same thing
>> as the tuple (x_0, ..., x_n) only that one instroduces an equivalence
>> relation
>> (x_0, ..., x_n) ~ (y_0, ..., y_n)
>> if and only if there is a non-zero scalar r such that
>> (x_0, ..., x_n) = r(y_0, ..., y_n)
>>
>> So 1:5/4:3/2 is short for (1, 5/4, 3/2) but equivalent if we multiply by
>> 4: 4(1, 5/4, 3/2) = 4*(1, 5/4, 3/2) = (4*1, 4*5/4, 4*3/2) = (4, 5, 6).
>>
>> This the formal calulation, but one can then just write
>> 1:5/4:3/2 = 4*(1:5/4:3/2) = 4:5:6
>
> Which is C:E:G, not C:F:G.
>
>> Then 1:4/3:3/2 = 6*(1:4/3:3/2) = 6:8:9.
>
> Different ratios -- C:F:G.
Yes. I just wanted to indicate the underlying formal math, as it may
help doing the computations.
>> ETs as well as any extended meantone or Pythagorean tuning (notes A B C D
>> E F G plus all notes gotten from them applying any number of sharps and
>> flats) tempered out this difference, so that error becomes equal in any
>> playing position.
>>
>> One the cannot only have two intervals exact as chosen. Typically the
>> octave is set to 2. If one want to approximate major third 5/4 better,
>> then the pure fifth 3/2 becomes worse, but only at a speed a 1/4 of the
>> change in the major third, since M3 = 4P4 - 2P8. So when M3 is exactly
>> 5/4, is has moved a by definition a syntonic comma, and P5 a quarter of
>> that, hence it is called historically the quarter-comma meantone.
>>
>> Historically, a meantone is a choice of 12 notes from the the extended
>> meantone. This leaves a gap, called the wolf. It really sounds that bad.
>> But there is really no need to reduce to 12 notes in computer generated
>> music.
> Agreed. I typically use notes as high as sixty, and that means about thirty
> notes per octave.
Do you mean ETs?
> My tunes do not use them all, and typically span one
> octave. I used a wolf fifth -- 40:27 in
> http://ecn.ab.ca/~brewhaha/Sound/HeLoh.mp3
> It does not hav much stress, and it is there. It is not that bad in context.
> I do not follow many rules in duet harmony. I just write two tunes that fit
> the same beat, then transpose them to end in unison. Then, I check to see if
> they should be an octave apart or a fifth apart, adjusting the basis
> frequency of each part. Unison seems to be a good way to start and end a
> duet.
The "wolf" shows up historically when cutting down a extended meantone
system to a 12-tone meantone system. That is, from the notes A B C D E F
G, plus what one can get by applying arbitrary number of sharps # and
flats b, one selects 12 notes.
One can simply avoid the problem by not choosing 12 pitches. It is
possible to make it playable on a 2-dimensional keyboard by this layout:
A# B# Cx Dx Ex
A B C# D# E# Fx Gx Ax Bx
Bb C D E F# G# A# B#
Cb Db Eb F G A B C'# D'#
Dbb Ebb Fb Gb Ab Bb C' D' E'
This keyboard layout just expresses that the notes are generated by two
intervals, the minor second m, and the major second M, plus that # b
alter with the interval M - m.
But it does not work with Just intonation having two major seconds of
different sizes. So in Just intonation, one looses an easy way to
transpose between keys.
Hans
Bohgosity BumaskiL
news:a71b7b3d-5100-49b3...@n35g2000yqm.googlegroups.com
replied to brew...@ecn.ab.ca
(snip)
> Do I understand that you are using these ratios as a means to use mean
> tone scale and other non 12-tet scales (or maybe them also) to be
> generated on the spot so that if you modulate, the new key will be
> established by these ratios?
There was a nice animation of a guitar string and the harmonic series that I
could find from wikipedia. I use free-form just intonation, which is exactly
the harmonic series, naming desirable ratios. The first line in my code is
my basis frequency. The way I compose, being on a twelve tone scale is
something of an accident. I start with a guess at the first note. Eighteen,
thirty and twenty-seven often work. I look at the first two or four ratios
that sound right in relation to the tune I hav been rehearsing in my head
for twenty minutes or singing for years, and ask does it reduce. 36:48
reduces, so I reduce the fraction, and go from there, raising my basis
frequency to fit whatever key is in my head. If I write a fraction, then I
scale all of my notes upward, and drop my basis frequency, again so that the
first note matches what is in my head. I prefer high basis frequencies.
All of my notes are effectively natural numbers, inherently on the harmonic
series, with zero for a rest. Most of my output is serial tunes with
parallel harmonics (all parts move together). When I go for complex duets or
four-part harmony, I drop the parallel harmonics and use a version of my
synthesizer that does part separations, automatically breaking four-parts
into four sectionals, six duets, and a quartet. I figure if the duets all
work, then so will the quartet.
> If I understand correctly, and I am not sure that I do, this would
> allow the same techniques of modulation for "mean tone" music found in
> 12-tet music of the CPP to work in the same manner.
Allow for, yes, if I adapt my synthesizer to change my basis frequency
during the tune.
I will not do that; too complicated; allows for tunes not entirely on the
harmonic series. There are plenty of quirky things I can do with obscure
ratios without resorting to modulation of my basis frequency.
> IF my understanding is in line with what you are saying, then I would
> ask if you have a means of "getting rid of the wolf" (your name isn't
> Peter is it? lol) as it seems that as you modulate through keys that
> the wolf will cause your return to tonic to be slightly off from the
> original pitch.
I do not write wolf notes by intention. They are incidental, and if my ears
do not hear the howl, then I do not care. Wolf notes hav their place.
> I have no problem with that and I had brought up this thought in an
> old electronic music class I had at Oberlan around 1990. At the time,
> there did not seem to have the capacity to do this with the tools that
> they had at their electronic studio. I suspect that it could be done
> with "Max" but this program was fairly new to them at the time and I
> did not get to investigate that concept any further.
I wrote my synthesizer before I realized what it was. I called my tuning
"linear". That delayed my discovery that other people do things like this
(script synth). Just Intonation is basically my school of thought and
practice. Writing music with nothing but naturals also meets the math, until
you need a rest, which is zero, a whole number.
> I understand that Guesualdo did this with his vocal compositions back
> when it was about as popular as the other "interesting" hobbies that
> he had in the privacy of his own castle. In his case, he seems to have
> just allowed the pitch move around to follow his modulations.
> In either case, it sounds like an interesting project or research.
It's partly research, especially when it comes to going through my source
and finding names for the ratios that I am using. It gets deeper and much
more difficult with duets and quartets. In another way, it is a lot of trial
and error. Sometimes I can see patterns in a terminal series. Sometimes it
is nothing but play by ear.
It does not matter how much you write. What matters is how much you ChucK.
(uuuu. I see a logo if Princeton ever advertizez it).
Bohgosity BumaskiL
(...)
> It is just convenient to write intervals as additively, as music depends
> so much on the ability to transpose.
In a script synth, transposition is simple:
Change the fundamental frequency. I call it a basis frequency in my code.
Equal temperament is fine for acoustic instruments.
In computer music, though, there are too many limits in it, for me.
Hans Aberg
>> It is just convenient to write intervals as additively, as music depends
>> so much on the ability to transpose.
> (...)
>
> In a script synth, transposition is simple:
> Change the fundamental frequency. I call it a basis frequency in my code.
> Equal temperament is fine for acoustic instruments.
> In computer music, though, there are too many limits in it, for me.
For instruments od different inharmonicity with a compensation stretch
tuning, there is only one frequency they agree on, the tuning frequency.
So one should then transpose using intervals, and then calculate the
frequency with the stretch tuning applied. I think though that synths
ignore the issue - that is the case of Scala.
Hans
Orangeboxman
> tuning, there is only one frequency they agree on, the tuning frequency.
> So one should then transpose using intervals, and then calculate the
> frequency with the stretch tuning applied. I think though that synths
> ignore the issue - that is the case of Scala.
>
> Hans
Tuning frequency being fundamental frequency or center frequency?
Pitch height (a human precept) maps to neither fundamental frequency
nor center frequency in a reliable way.
Hans Aberg
>> For instruments od different inharmonicity with a compensation stretch
>> tuning, there is only one frequency they agree on, the tuning frequency.
>> So one should then transpose using intervals, and then calculate the
>> frequency with the stretch tuning applied. I think though that synths
>> ignore the issue - that is the case of Scala.
> Tuning frequency being fundamental frequency or center frequency?
>
> Pitch height (a human precept) maps to neither fundamental frequency
> nor center frequency in a reliable way.
The tuning frequency that of the source used for the tuning, such as a
tuning fork or an note on the oboe. When tuning, one listens for the
beats, so it then becomes fairly exact. A guitar is transposing, so its
A is actually one octave lower than that of the tuning fork. When
listening for beats, one is actually using the second partial. This
means that the fundamental will actually be slightly lower than one half
the frequency of the tuning fork.
Hans
Bohgosity BumaskiL
> Bohgosity BumaskiL wrote:
>> 4:5:6 is C:E:G.
>> 6:8:9 is C:F:G.
>
> In Just intonation, the variation given on the Wikipedia page that makes
> sure all major chords are built up exact rational interval ratios. (Some
> use "Just" to denote anything that is made up by rational intervals.
>
> In other tuning systems, these are a bit off from the exact rational
> ratios.
>
>> To verify, six to four is a perfect fifth, and so is nine to six -- G:C
>> in both cases.
>
> A Just perfect fifth is 3/2, and Just perfect fourth is the octave
> inversion 4/3 = 2/(3/2).
It does not matter how I reprezent a perfect fifth: I do not need to be
using reduced fractions. For example, if my fundamental frequency is 12Hz,
and I write a perfect fifth as 30:20, then output will be identical if I
write
it as 15:10, and raise basis frequency to 24Hz; Identical. When I *am*
writing reduced fractions for notes in a duet, then I know I am on the right
track with my tune.
>> E is the mean of G and C.
>
> No. To understand Pythagorean tuning, I find it easy to use E53, where m =
> 4, M = 9, and #, b alter with M - m = 5. A E53 tonestep is very close to a
> syntonic comma, the latter can be defined as the difference between the
> (lower) Just major third 5/4 and the (higher) Pythagorean major third.
>
> In E53, major third is M + M = 18, whereas the Just major third 5/4 is
> quite close to 17. The Just perfect 5th is very close to 3M + m = 31.
>
> So the half of that 31 is 15.5.
>
>> It is also a major third above C, and a minor third below G.
>> F is not the mean of G and C.
>
> It is the octave inversion, as I wrote above.
Let us assume that you want a perfect CEFG chord. In that case, I get
12:15:16:18, if G:F is major. Notice that one number is odd. If I temper
fifteen to fourteen, then it reduces to 6:7:8:9, which is not readily
available on piano. I suspect that it is more assonant than CEFG, though,
and less assonant than 2:3:4:5 -- CGCE, which I am told is a triad. I
openned my video: http://ecn.ab.ca/~brewhaha/font/Saffron_Karaoke_Duet.wmv
with a transposition of CGCE in round hertz figures.
>> It is a major second below G, and a perfect fourth above C.
>
> Yes, but also note that Just intonation makes use of two major seconds of
> different sizes. This give one of them, 9/8. The other Just major second
> is the difference between the Just major third 5/4 and this major second,
> (5/4)/(9/8) = 10/9.
Okay, you are losing me, here, because that makes G:F minor.
I am using a tuning in which 50:25 is a perfect octave, and 8:4 is a perfect
octave, while 15:10 is just as much a perfect fifth as 3:2. I used to call
it linear. I could call it "pure naturals", and the main body of people
using different source code for the same thing calls their school of thought
"just intonation", which is not only a way of tuning pianos with a prime
limit of five. It is more of an attempt to understand why barbershop singers
tend to sink their key and remain harmonious.
My method is to write two tunes that stand alone, then transpose them to be
harmonious -- by rounded division in trials. My ears often demand higher
numbers,
sometimes awkward prime numbers, while it is easier to find harmony in lower
numbers, so I do trial divisions, then listen and tweak. If you get into
this, then takes steps to keep your numbers low.
Hans Aberg
>>> 4:5:6 is C:E:G.
>>> 6:8:9 is C:F:G.
>> In Just intonation, the variation given on the Wikipedia page that makes
>> sure all major chords are built up exact rational interval ratios. (Some
>> use "Just" to denote anything that is made up by rational intervals.
>>
>> In other tuning systems, these are a bit off from the exact rational
>> ratios.
>>
>>> To verify, six to four is a perfect fifth, and so is nine to six -- G:C
>>> in both cases.
>> A Just perfect fifth is 3/2, and Just perfect fourth is the octave
>> inversion 4/3 = 2/(3/2).
>
> It does not matter how I reprezent a perfect fifth: I do not need to be
> using reduced fractions. For example, if my fundamental frequency is 12Hz,
> and I write a perfect fifth as 30:20, then output will be identical if I
> write
> it as 15:10, and raise basis frequency to 24Hz; Identical.
Yes, with the projective coordinate ":" notation, also 20:30 = 10:15 =
2:3. This refers to two notes. By contrast, an interval like 3/2 is a
ration of pitches, or rather, how they are tuned, since one may apply
stretch tuning.
> When I *am*
> writing reduced fractions for notes in a duet, then I know I am on the right
> track with my tune.
I find using fractions more intuitive, for example Just major triad
1:5/4:3/2 instead of 4:5:6 as is listed in Scala. After some time,
perhaps one gets used to the latter.
>>> E is the mean of G and C.
>> No. To understand Pythagorean tuning, I find it easy to use E53, where m =
>> 4, M = 9, and #, b alter with M - m = 5. A E53 tonestep is very close to a
>> syntonic comma, the latter can be defined as the difference between the
>> (lower) Just major third 5/4 and the (higher) Pythagorean major third.
>>
>> In E53, major third is M + M = 18, whereas the Just major third 5/4 is
>> quite close to 17. The Just perfect 5th is very close to 3M + m = 31.
>>
>> So the half of that 31 is 15.5.
Yes, and 1.5 tonesteps in E31 is 58.064 cents off, which is too far off
for good harmony - even a new note:
Experimenting in Scala, listening for the beats of the partials on a
pipe organ patch, I could hear differences of simultaneous pitches
slightly below to 0.1 cents. The perfect 5th 3/2 is approximated in E53
with 0.07 cents, and that is not really detectable, at least not in a
music piece.
For really good harmony, at most few cents might be acceptable, but the
5/4 is 13.686 cents off in E12, which works. But in E53, 5/4 is more
than 20 cents off from the Pythagorean major third(though approximated
with a few cents in E53), which is too much for good harmony so switch
to another tuning than the Pythagorean, then as they did during the
Renaissance, giving quarter-comma meantone and E31.
Increase the step further, and it starts to sound as a new,
intermediate, pitch. This is then like in Arab music. Try this, perhaps
somewhere from 20-30 cents.
Now, the 58.064 cents mentioned above is more like a new intermediate
pitch: perhaps close to the interval 11/8.
In E72, the interval 3/2 is approximated within the E12 subset. The
interval 5/4 lowers one E72 tonestep, 7/4 two tonesteps, and 11/8 is
three tonesteps - an E12 quartertone. This may be useful for finding out
about where these intervals are located, though a computer might just as
well use exact intervals.
>>> It is also a major third above C, and a minor third below G.
>>> F is not the mean of G and C.
>> It is the octave inversion, as I wrote above.
>
> Let us assume that you want a perfect CEFG chord. In that case, I get
> 12:15:16:18, if G:F is major. Notice that one number is odd. If I temper
> fifteen to fourteen, ...
Those seemingly small number effects may have a significant effect on
harmony. The interval 15/14 is 119.442 cents.
> ...then it reduces to 6:7:8:9, which is not readily
> available on piano. I suspect that it is more assonant than CEFG, though,
> and less assonant than 2:3:4:5 -- CGCE, which I am told is a triad. I
> openned my video: http://ecn.ab.ca/~brewhaha/font/Saffron_Karaoke_Duet.wmv
> with a transposition of CGCE in round hertz figures.
You may try out some chords with exact interval like 11/8 of a pipe
organ patch (as they are supposed to be harmonic and have rich overtone
support). The partials line up, giving harmonic convergence.
Western harmonic consonance/dissonance concepts seems referring to the
convergence in a specific tuning. CPP (common practice period) harmony
seems to have sprung out of ideas of the quarter-comma meantone tuning
which sets the major third exactly to the interval 5/4 (and close to
E31), popular during the Renaissance.
This tuning also approximates the interval 7/4 well, but on the
augmented sixth. This interval is the used in predominant chords, but is
hard to otherwise fit into the major/minor diatonic scales system.
So chords like 6:7:8:9 are just as good, but are harder to make music
of. The interval 7/6 is on key 7 in E31, two tonesteps over the major
second, so an augmented second.
So you do not change it at all, if you do not want to.
>>> It is a major second below G, and a perfect fourth above C.
>> Yes, but also note that Just intonation makes use of two major seconds of
>> different sizes. This give one of them, 9/8. The other Just major second
>> is the difference between the Just major third 5/4 and this major second,
>> (5/4)/(9/8) = 10/9.
>
> Okay, you are losing me, here, because that makes G:F minor.
In Just intonation, the interval from F to G is 9/8, the same as from C
to D. But from D to E, it is 10/9, otherwise C to E would not be 5/4.
>>> ... I typically use notes as high as sixty, and that means about
>>> thirty notes per octave.
>> Do you mean ETs?
>
> I am using a tuning in which 50:25 is a perfect octave, and 8:4 is a perfect
> octave, while 15:10 is just as much a perfect fifth as 3:2.
As described above, 50:25 = 8:4 = 2:1, two pitches in an interval ratio 2.
> I used to call
> it linear. I could call it "pure naturals", and the main body of people
> using different source code for the same thing calls their school of thought
> "just intonation", which is not only a way of tuning pianos with a prime
> limit of five. It is more of an attempt to understand why barbershop singers
> tend to sink their key and remain harmonious.
I think it is often called "Just (intonation)" and 5-limit, and so on. I
prefer to use "rational interval (ratio)", as Just intonation is
historically (I think) a specific tuning.
> My method is to write two tunes that stand alone, then transpose them to be
> harmonious -- by rounded division in trials. My ears often demand higher
> numbers,
> sometimes awkward prime numbers, while it is easier to find harmony in lower
> numbers, so I do trial divisions, then listen and tweak. If you get into
> this, then takes steps to keep your numbers low.
>
> http://ecn.ab.ca/~brewhaha/ BrewJay's Babble Bin
Large prime numbers may not so important, simply because harmony depends
on the partials lining up. And high partials may be weak or inharmonic.
Hans
Bohgosity BumaskiL
> Bohgosity BumaskiL wrote:
>> 4:5:6 is C:E:G.
>> 6:8:9 is C:F:G.
>
> In Just intonation, the variation given on the Wikipedia page that makes
> sure all major chords are built up exact rational interval ratios. (Some
> use "Just" to denote anything that is made up by rational intervals.
I am becoming inclined towards that. It is desirable to hit major, and minor
ratios, especially if your tune will venture into obscure territory, like
tritones, tridecimal
ratios and undecimal ratios. It is hard to avoid ratios with seven in them,
though.
> In other tuning systems, these are a bit off from the exact rational
> ratios.
>
>> To verify, six to four is a perfect fifth, and so is nine to six -- G:C
>> in both cases.
>
> A Just perfect fifth is 3/2, and Just perfect fourth is the octave
> inversion 4/3 = 2/(3/2).
I do not need to use reduced fractions. For example, if my basis frequency
is 12Hz, and I write a perfect fifth as 30:20, then output will be identical
if I write 12:8, and
raise basis frequency to 30Hz; Identical. When I write *named* reduced
fractions for notes in a duet, then I know I am on a good track with my
relation.
Conversely, if I write obscure ratios, then I can't be sure that I am on the
best track.
>> E is the mean of G and C.
>
> No. To understand Pythagorean tuning, I find it easy to use E53, where m =
> 4, M = 9, and #, b alter with M - m = 5. A E53 tonestep is very close to a
> syntonic comma, the latter can be defined as the difference between the
> (lower) Just major third 5/4 and the (higher) Pythagorean major third.
1 to 4/3, 1 to 3/2 -- 3:4, 3:9/2 -- 6:8:9 -- CFG.
1 to 5/4, 1 to 3/2 -- 4:5:6 -- CEG, and E is the mean of G and C.
Because this is not statistics, that is not a coincidence, either.
> In E53, major third is M + M = 18, whereas the Just major third 5/4 is
> quite close to 17. The Just perfect 5th is very close to 3M + m = 31.
>
> So the half of that 31 is 15.5.
>
>> It is also a major third above C, and a minor third below G.
>> F is not the mean of G and C.
>
> It is the octave inversion, as I wrote above.
Let us assume that you want a perfect CEFG chord. In that case, I get
12:15:16:18, if G:F is major. Notice that one number is odd. If I temper
fifteen to fourteen, then it reduces to 6:7:8:9, which is not readily
available on piano. I suspect that it is more assonant than CEFG, though,
and less assonant than 2:3:4:5 -- CECG, which I am told is a triad. I opened
my video: http://ecn.ab.ca/~brewhaha/font/Saffron_Karaoke_Duet.wmv with a
transposition of CECG.
>> It is a major second below G, and a perfect fourth above C.
>
> Yes, but also note that Just intonation makes use of two major seconds of
> different sizes. This give one of them, 9/8. The other Just major second
> is the difference between the Just major third 5/4 and this major second,
> (5/4)/(9/8) = 10/9.
Okay, you are losing me, here, because that makes G:F minor.
> > The math is the same as when you add or subtract fractions,
>> except that instead of adding or subtracting fractions, once all ratios
>> are above same denominator, ignore denominator, then concatenate into one
>> ratio with three or more parts.
>
> It is just convenient to write intervals as additively, as music depends
> so much on the ability to transpose. An interval is a transposable ratio
> of pitches. A pitch is quite closely related to frequencies, though it may
> be altered somewhat through stretch tuning, dynamics. And the aer is not
> able to transpose above about 3.5 kHz.
aer? If I use basis frequencies in the hundreds of Hertz, and notes in the
tens (like usual), then I will be in that range.
I am using reference figures for perfect harmony, a tuning in which 50:25 is
an octave, and 8:4 is an octave, while 15:10 is just as much a perfect fifth
as 3:2. I used to call it linear. I could call it "pure naturals", and the
main body of people using different source code for the same thing calls
their school of thought "just intonation", which is not just a way of tuning
pianos with a prime limit of five.
>> My tunes do not use them all, and typically span one octave. I used a
>> wolf fifth -- 40:27 in http://ecn.ab.ca/~brewhaha/Sound/HeLoh.mp3
>> It does not hav much stress, and it is there. It is not that bad in
>> context.
>> I do not follow many rules in duet harmony. I just write two tunes that
>> fit the same beat, then transpose them to end in unison. Then, I check to
>> see if they should be an octave apart or a fifth apart, adjusting the
>> basis frequency of each part. Unison seems to be a good way to start and
>> end a duet.
>
> The "wolf" shows up historically when cutting down a extended meantone
> system to a 12-tone meantone system. That is, from the notes A B C D E F
> G, plus what one can get by applying arbitrary number of sharps # and
> flats b, one selects 12 notes.
For me, more than one wolf _is_ to avoid. Typically, and not strictly, it is
ratios with prime numbers. 40:3^3 is one of them. That does not mean
avoiding them in every case. That particular one has many factors, so in "A
Tune for Tigger's Song", where I used it, if I were inclined to write
harmony for it, and I am not, it would not be hard.
> One can simply avoid the problem by not choosing 12 pitches. It is
> possible to make it playable on a 2-dimensional keyboard by this layout:
> A# B# Cx Dx Ex
> A B C# D# E# Fx Gx Ax Bx
> Bb C D E F# G# A# B#
> Cb Db Eb F G A B C'# D'#
> Dbb Ebb Fb Gb Ab Bb C' D' E'
>
> This keyboard layout just expresses that the notes are generated by two
> intervals, the minor second m, and the major second M, plus that # b alter
> with the interval M - m.
>
> But it does not work with Just intonation having two major seconds of
> different sizes. So in Just intonation, one looses an easy way to
> transpose between keys.
My method is to write two tunes that stand alone, then transpose them to be
harmonious -- by rounded division. My ears often demand higher numbers,
sometimes awkward prime numbers, while it is easier to find harmony in lower
numbers, so I do trial divisions, then listen and tweak. If you get into
this, then takes steps to keep your numbers low.
I never even tried to get twelve notes per octave before I decided to
re-write Doh-Ray-Mee in two or four parts, and I learned that the easy way
to transpose a tune is to change one number, my
http://en.wikipedia.org/wiki/fundamental_frequency .
http://ecn.ab.ca/~brewhaha/ Barbieshop Singers: Sweet Adelines
Hans Aberg
[You seem to somehow made a double post. So I add some details here.]
>>> 4:5:6 is C:E:G.
>>> 6:8:9 is C:F:G.
>> In Just intonation, the variation given on the Wikipedia page that makes
>> sure all major chords are built up exact rational interval ratios. (Some
>> use "Just" to denote anything that is made up by rational intervals.
>
> I am becoming inclined towards that. It is desirable to hit major, and minor
> ratios, especially if your tune will venture into obscure territory, like
> tritones, tridecimal
> ratios and undecimal ratios. It is hard to avoid ratios with seven in them,
> though.
There is a conflict between melodic treatment and harmonic treatment. If
the pitches of the melody adjust against those of the chords, or if it
say is in the foreground and does not need to adjust, one is freer in
choosing chords. The conflicts arise when one assumes that the pitches
from the chords should be a subset of the those that the melody uses.
So perhaps one can experiment with a freer treatment of chords and melody.
>> In other tuning systems, these are a bit off from the exact rational
>> ratios.
>>
>>> To verify, six to four is a perfect fifth, and so is nine to six -- G:C
>>> in both cases.
>> A Just perfect fifth is 3/2, and Just perfect fourth is the octave
>> inversion 4/3 = 2/(3/2).
>
> I do not need to use reduced fractions. For example, if my basis frequency
> is 12Hz, and I write a perfect fifth as 30:20, then output will be identical
> if I write 12:8, and
> raise basis frequency to 30Hz; Identical.
They are identical mathematically, too. 30:20 = 12:8 = 3:2. But these
are pairs of pitches, not ratios. Scala writes them from smaller to
higher, which gives them a unique representation. Like 2:3, which then
is equal to 1:3/2.
> When I write *named* reduced
> fractions for notes in a duet, then I know I am on a good track with my
> relation. Conversely, if I write obscure ratios, then I can't be sure that I am
on the
> best track.
I think reduced fractions is easier to use. So I write the Just major as
1:5/4:3/2 instead of 4:5:6, but they are mathematically equal - it is
just a preference.
>>> E is the mean of G and C.
>> No. To understand Pythagorean tuning, I find it easy to use E53, where m =
>> 4, M = 9, and #, b alter with M - m = 5. A E53 tonestep is very close to a
>> syntonic comma, the latter can be defined as the difference between the
>> (lower) Just major third 5/4 and the (higher) Pythagorean major third.
>
> 1 to 4/3, 1 to 3/2 -- 3:4, 3:9/2 -- 6:8:9 -- CFG.
> 1 to 5/4, 1 to 3/2 -- 4:5:6 -- CEG, and E is the mean of G and C.
> Because this is not statistics, that is not a coincidence, either.
Ah, I misunderstood. The reason probably only that you try to do
harmony, making the partials lining up for stability. The partials form
an arithmetic series. Try to make a sound generator with overtones from
something radically different from an arithmetic series. It may ba hard
to determine which pitch it is.
Historically, in Medieval times one had the Pythagorean tuning and the
diatonic modes. Two major seconds is quite close to 5/4, so if 1:3/2 is
filled in with that, one gets the Just major 4:5:6. Arab music didn't
try harmony, but added an intermediate interval instead.
But you might try filling in 1:4/3 the same way: this gives 6:7:8. OPen
for experimentation.
Hans
Bohgosity BumaskiL
> Bohgosity BumaskiL wrote:
> [You seem to somehow made a double post. So I add some details here.]
>>>> 4:5:6 is C:E:G.
>>>> 6:8:9 is C:F:G.
>>> In Just intonation, the variation given on the Wikipedia page that makes
>>> sure all major chords are built up exact rational interval ratios. (Some
>>> use "Just" to denote anything that is made up by rational intervals.
>>
>> I am becoming inclined towards that. It is desirable to hit major, and
>> minor ratios, especially if your tune will venture into obscure
>> territory, like tritones, tridecimal
>> ratios and undecimal ratios. It is hard to avoid ratios with seven in
>> them, though.
>
> There is a conflict between melodic treatment and harmonic treatment. If
> the pitches of the melody adjust against those of the chords, or if it say
> is in the foreground and does not need to adjust, one is freer in choosing
> chords. The conflicts arise when one assumes that the pitches from the
> chords should be a subset of the those that the melody uses.
I hav found that melodic ratios tend to be in harmony. In other words, if my
left channel is jumping by a fifth, then there is a higher chance that in a
nearby note, I should be using a fifth in concert. Maybe I should go back to
my most recent work and guess how that might work. I could write the series
ratio for the left channel, the series ratio for the right channel, then the
ratio between channels.
> So perhaps one can experiment with a freer treatment of chords and melody.
>
>>> In other tuning systems, these are a bit off from the exact rational
>>> ratios.
>>>
>>>> To verify, six to four is a perfect fifth, and so is nine to six -- G:C
>>>> in both cases.
>>> A Just perfect fifth is 3/2, and Just perfect fourth is the octave
>>> inversion 4/3 = 2/(3/2).
>>
>> I do not need to use reduced fractions. For example, if my basis
>> frequency
>> is 12Hz, and I write a perfect fifth as 30:20, then output will be
>> identical
>> if I write 12:8, and
>> raise basis frequency to 30Hz; Identical.
>
> They are identical mathematically, too. 30:20 = 12:8 = 3:2. But these are
> pairs of pitches, not ratios. Scala writes them from smaller to higher,
> which gives them a unique representation. Like 2:3, which then is equal to
> 1:3/2.
At the clause ", not ratios", you lost me. 30:20 is an octave above 15:10,
exactly, so these numbers are both pairs of pitches AND they form ratios
with each other. I hav found melody to be freer with the ratios. To get
harmony, I find that I sometimes compromise on melody. Theoretically, there
are so many options, that I do not need to compromise, and I do not feel
that I hav sufficient time, intuition, or method with numbers to explore all
of those options.
The idea for me is to reduce punctuation and numbers. Trying to make slashes
and colons mean something different goes nowhere. The only difference for me
is that colons say that the division is optional. What I want to end up with
is essentially a terminal series, divided by the same punctuation. IOW,
1:3/2 is 2:3. Traditionally, higher numbers are first in the series, and
since I type stereo in, 2:3 is lojisitically different from 3:2, so the only
purpose behind writing the higher number first is so that I can easily look
it up on huygens-fokker.
>> When I write *named* reduced
>> fractions for notes in a duet, then I know I am on a good track with my
>> relation. Conversely, if I write obscure ratios, then I can't be sure
>> that I am
> on the
>> best track.
>
> I think reduced fractions is easier to use. So I write the Just major as
> 1:5/4:3/2 instead of 4:5:6, but they are mathematically equal - it is just
> a preference.
One of them is a lot easier to read, because it uses only one kind of
punctuation,
plus it is three numbers instead of six. 4:5:6 goes directly into my code.
Fractions do not.
>>>> E is the mean of G and C.
>>> No. To understand Pythagorean tuning, I find it easy to use E53, where m
>>> =
>>> 4, M = 9, and #, b alter with M - m = 5. A E53 tonestep is very close to
>>> a
>>> syntonic comma, the latter can be defined as the difference between the
>>> (lower) Just major third 5/4 and the (higher) Pythagorean major third.
>>
>> 1 to 4/3, 1 to 3/2 -- 3:4, 3:9/2 -- 6:8:9 -- CFG.
>> 1 to 5/4, 1 to 3/2 -- 4:5:6 -- CEG, and E is the mean of G and C.
>> Because this is not statistics, that is not a coincidence, either.
>
> Ah, I misunderstood. The reason probably only that you try to do harmony,
> making the partials lining up for stability. The partials form an
> arithmetic series. Try to make a sound generator with overtones from
> something radically different from an arithmetic series. It may ba hard to
> determine which pitch it is.
>
> Historically, in Medieval times one had the Pythagorean tuning and the
> diatonic modes. Two major seconds is quite close to 5/4, so if 1:3/2 is
> filled in with that, one gets the Just major 4:5:6. Arab music didn't try
> harmony, but added an intermediate interval instead.
>
> But you might try filling in 1:4/3 the same way: this gives 6:7:8. OPen
> for experimentation.
1:4/3 * 3= 3:4 You can get the six and eight out of that (scale it again by
two), and only two numbers are in it, so the seven is missing.
Hans Aberg
>> There is a conflict between melodic treatment and harmonic treatment. If
>> the pitches of the melody adjust against those of the chords, or if it say
>> is in the foreground and does not need to adjust, one is freer in choosing
>> chords. The conflicts arise when one assumes that the pitches from the
>> chords should be a subset of the those that the melody uses.
>
> I hav found that melodic ratios tend to be in harmony. In other words, if my
> left channel is jumping by a fifth, then there is a higher chance that in a
> nearby note, I should be using a fifth in concert. Maybe I should go back to
> my most recent work and guess how that might work. I could write the series
> ratio for the left channel, the series ratio for the right channel, then the
> ratio between channels.
This leads to one type of music. In CPP (Common Practice Music) one
wants to exploit the 5-limit, with some 7-limit, terming such ratios a
"stable" or "consonant", with the idea that the opposite, "dissonances"
or are "unstable", and "wants to move towards consonances".
In other types of music, one may a small amount of departure from a
rational ration favorable, as it produces beats that makes the sound
more interesting. If the treatment is more like pitched rhythms, thne
one might avoid such rations altogether. Gamelan and African music may
have examples of this.
> They are identical mathematically, too. 30:20 = 12:8 = 3:2. But these are
>> pairs of pitches, not ratios. Scala writes them from smaller to higher,
>> which gives them a unique representation. Like 2:3, which then is equal to
>> 1:3/2.
>
> At the clause ", not ratios", you lost me. 30:20 is an octave above 15:10,
> exactly, so these numbers are both pairs of pitches AND they form ratios
> with each other.
That is not the notation used in say Scala. There, 30:20 is short for
the frequency pair (30f, 20f), and 15:10 for the frequency pair (15g,
10g), where f, g are some reference frequencies. So setting 2f = g, they
will produce the same frequency pairs - the same two-note harmony.
> I hav found melody to be freer with the ratios. To get
> harmony, I find that I sometimes compromise on melody. Theoretically, there
> are so many options, that I do not need to compromise, and I do not feel
> that I hav sufficient time, intuition, or method with numbers to explore all
> of those options.
That is also a general treatment in music: the more melodic, the less
importance of interval ratios.
> The idea for me is to reduce punctuation and numbers. Trying to make slashes
> and colons mean something different goes nowhere. The only difference for me
> is that colons say that the division is optional. What I want to end up with
> is essentially a terminal series, divided by the same punctuation. IOW,
> 1:3/2 is 2:3. Traditionally, higher numbers are first in the series, and
> since I type stereo in, 2:3 is lojisitically different from 3:2, so the only
> purpose behind writing the higher number first is so that I can easily look
> it up on huygens-fokker.
If you mean
http://www.huygens-fokker.org/docs/intervals.html
he is using a "/" here. These are rational numbers, which is why the
higher number comes first.
Scala also has a list of chords - open the keyboard, and there should be
a button for choosing chords. This leads to a list with chords using the
":" notation.
This thoery does not distinguish between right-left channels. If you
want to have true harmony, then one should be able to hear the beats
between the partials, and if the notes are truly separated between the
channels, the beats cannot be heard - try this out. Possibly you might
develop a freer relation between the rations if the channels area
separated. But you cannot ensure that such a channels separation can be
prevail in a listener situation.
>> I think reduced fractions is easier to use. So I write the Just major as
>> 1:5/4:3/2 instead of 4:5:6, but they are mathematically equal - it is just
>> a preference.
>
> One of them is a lot easier to read, because it uses only one kind of
> punctuation,
> plus it is three numbers instead of six. 4:5:6 goes directly into my code.
> Fractions do not.
Use whatever you find easiest. It is possible to mix them, though.
>>>>> E is the mean of G and C.
>>>> No. To understand Pythagorean tuning, I find it easy to use E53, where m
>>>> =
>>>> 4, M = 9, and #, b alter with M - m = 5. A E53 tonestep is very close to
>>>> a
>>>> syntonic comma, the latter can be defined as the difference between the
>>>> (lower) Just major third 5/4 and the (higher) Pythagorean major third.
>>> 1 to 4/3, 1 to 3/2 -- 3:4, 3:9/2 -- 6:8:9 -- CFG.
>>> 1 to 5/4, 1 to 3/2 -- 4:5:6 -- CEG, and E is the mean of G and C.
>>> Because this is not statistics, that is not a coincidence, either.
>> Ah, I misunderstood. The reason probably only that you try to do harmony,
>> making the partials lining up for stability. The partials form an
>> arithmetic series. Try to make a sound generator with overtones from
>> something radically different from an arithmetic series. It may ba hard to
>> determine which pitch it is.
>>
>> Historically, in Medieval times one had the Pythagorean tuning and the
>> diatonic modes. Two major seconds is quite close to 5/4, so if 1:3/2 is
>> filled in with that, one gets the Just major 4:5:6. Arab music didn't try
>> harmony, but added an intermediate interval instead.
>>
>> But you might try filling in 1:4/3 the same way: this gives 6:7:8. OPen
>> for experimentation.
>
> 1:4/3 * 3= 3:4 You can get the six and eight out of that (scale it again by
> two), and only two numbers are in it, so the seven is missing.
I seemed you filled in a tone which is the average. So 3:4 = 6:8, and
the arithmetic mean of 6 and 8 is 7, giving 6:7:8. I have no idea how
such a method relates to music, other than a formal technique of
producing new notes.
But there are summation and difference tones, produced in the cochlea -
sometimes called "acoustic", though they do not exists acoustically. The
difference tone is stronger, the summation tone rather weak. But for
6:8, the difference tone is 2 = 8-2, and the summation tone is 14 = 6+8.
Then 14 is an octave above the arithmetic mean 7.
Hans
LJS
<brewh...@freenet.edmonton.ab.ca> wrote:
> "LJS" <ljsche...@gmail.com> wrote in message
>
> news:a71b7b3d-5100-49b3...@n35g2000yqm.googlegroups.com
> replied to brewh...@ecn.ab.ca
Well, I understand some of this and it seems interesting. Do you know
of a program called MAX?
http://en.wikipedia.org/wiki/Max_(software)
I think this is the program I am talking about but it is all that I
could find. I used it at a workshop in Oberlin just a little bit as it
was a short workshop and there was lots to do.
The program I am talking about is like an engineering program. In it
you actually build the "circuts" with modules that are supplied and
that can also be created with tools that they have.
In effect, you could build a synthesizer that would allow you to set
your parameters as you describe and then by simply reading the next
note in your composition it could used that set of parameters for
tuning the harmony to that note as a root or fundamental or to any
other event or information that you want to assign to it. The program
is so versatile that it is really difficult to explain. I would think
that if you e-mailed the electronic workshop at Oberlin college. (I
don' t remember the name they use for the electronic music dept, but
it should be on the Oberlin site) they could give you more
information.
All I can say for sure is that if you had access to this program and
you learn how to use it, I suspect that your life will be much easier.
Once you would "program" or build your machine to do what you
described in your posts one time, you can then duplicate on any level
as well as to change any or all of the parameters and combination of
parameters with a few simple keystrokes.
You owe it to yourself to see what this program is all about. You
sound as if it was, unlike the Windows 7 adds in the US, truly
written especially for YOU.
I will try to understand more of your ideas and chime in when I can.
LJS