Basic question - rational of octave
Etienne Marais
Hello, some very simply questions (I think)
A vibrating item makes sound, the higher the rate of vibraion
(frequency), the higher the pitch.
1) Why, following this 'physical' view of music, do notes that sound
alike reoccur periodacly, and what is the physical attribute that make
them sound the same, even if the pitch is not. (ie A and octave higher
A)
2) How did it come to be that an octave got divided into 12 parts in
the 1/2 step 'system'. Are there advantages, or merely so in
historical perspective ?
3) And why the odd 7 letters, plus 5 relative notes for this 12*1/2
step division. Do the 'relatives' (ie black notes on piano) have any
specific behaviour/attribute (especially seen from a phsysics point of
view)
4) Can this division be into any number of notes, theoretically ?
5) Would an instrument, ie a 'new' piano with 20 devisions per octave
produce better sound, ie is history straining musical possibility ?
6) WHY is it that the notes in scalars go together ?
7) What are some other attributes of sound (and possible the human
ear/hearing process) that I may find usefull to REALLY understand
music theory before I go further ?
Thanks !
Etienne Marais
Matt
In article <353c6370...@nntp.up.ac.za>,
Etienne Marais <s941...@student.up.ac.za> wrote:
>Hello, some very simply questions (I think)
>
>A vibrating item makes sound, the higher the rate of vibraion
>(frequency), the higher the pitch.
>
>1) Why, following this 'physical' view of music, do notes that sound
>alike reoccur periodacly, and what is the physical attribute that make
>them sound the same, even if the pitch is not. (ie A and octave higher
>A)
Partly the structure of our cochleas, and partly an adaptation in the
first few months of our lives to trying to share utterances between
adults and children. Octave transposition provides some semblance of
analogy between the intonation pattern of adult and child.
>2) How did it come to be that an octave got divided into 12 parts in
>the 1/2 step 'system'. Are there advantages, or merely so in
>historical perspective ?
It started out as a way to make straight frets on lutes. 12 was the
smallest number that proved tolerable for the style of music that was
already popular at the time.
Advantages are dependent on the musical style, and include the possiblity
of playing equally well (or poorly) in all keys, of modulating through
a cycle of 12 keys, and of enharmonically reinterpreting notes and chords.
>3) And why the odd 7 letters, plus 5 relative notes for this 12*1/2
>step division. Do the 'relatives' (ie black notes on piano) have any
>specific behaviour/attribute (especially seen from a phsysics point of
>view)
This is two questions. The 7 notes of a scale refer to conventions of
music-making within scales for most of the last 1200 years. Physically
speaking, white notes and black notes have no special significance, and
are grouped that way on a keyboard by convention and as a way of
making some keys convenient to play.
>4) Can this division be into any number of notes, theoretically ?
Yes, with more or less pleasing or grating sounds. Easley Blackwood
created compositions in a large number of different equal temperments.
>5) Would an instrument, ie a 'new' piano with 20 devisions per octave
>produce better sound, ie is history straining musical possibility ?
This depends on what you like. Most folks would greatly prefer a
piano with 19 notes per octave. Such an instrument would have a large
number of keys and strings to be tuned, and might be prohibitively
expensive and difficult to play. But you can do as many people do now
and emulate it in software.
>6) WHY is it that the notes in scalars go together ?
Notes in scales go together because we are accustomed to hearing them
together.
>7) What are some other attributes of sound (and possible the human
>ear/hearing process) that I may find usefull to REALLY understand
>music theory before I go further ?
This is a very broad question. There's some acoustics fans here.
Matt
--
Matt Fields, A.Mus.D. http://www-personal.umich.edu/~fields/complist.html
Featured addresses: hostm...@INREACH.COM sel...@NETVIGATOR.COM
TwelveToneToyBox!!!! http://www-personal.umich.edu/~fields/TTTB
Philippe Chose
Matt wrote:
>
> In article <353c6370...@nntp.up.ac.za>,
> Etienne Marais <s941...@student.up.ac.za> wrote:
> >Hello, some very simply questions (I think)
> >
> >A vibrating item makes sound, the higher the rate of vibraion
> >(frequency), the higher the pitch.
> >
> >1) Why, following this 'physical' view of music, do notes that sound
> >alike reoccur periodacly, and what is the physical attribute that make
> >them sound the same, even if the pitch is not. (ie A and octave higher
> >A)
>
> Partly the structure of our cochleas, and partly an adaptation in the
> first few months of our lives to trying to share utterances between
> adults and children. Octave transposition provides some semblance of
> analogy between the intonation pattern of adult and child.
>
If we consider complex sounds produced by instruments,
A and octave higher sounds almost the same, mainly because
they share many hamonics in their spectrum.
M. Schulter
Etienne Marais <s941...@student.up.ac.za> wrote:
: 1) Why, following this 'physical' view of music, do notes that sound
: alike reoccur periodacly, and what is the physical attribute that make
: them sound the same, even if the pitch is not. (ie A and octave higher
: A)
Hello, there. The quick answer would be that the ratio of 2:1 is the
simplest unequal ratio, and produces the "equisonal" octave where the two
notes seem almost to blend as one.
: 2) How did it come to be that an octave got divided into 12 parts in
: the 1/2 step 'system'. Are there advantages, or merely so in
: historical perspective ?
Here there are actually two questions: why divide an octave into 12 parts,
and why are these parts taken as equal in one of the most popular current
temperaments, 12-tone equal temperament or 12tet for short?
First, why 12 tones per octave? The complete chromatic scale with its 12
tones seems to have been established by the early 14th century, when
keyboard pieces from the Robertsbridge Codex call for all 12 notes,
including the accidentals (Bb, Eb, F#, C#, G#). How do we get 12?
Basically, this result follows from the model of tuning current in that
period, and very congenial to late Gothic music, although we should
remember that performers of non-fixed-pitched instruments rarely follow
precisely any neat mathematical tuning <grin>. (By fixed-pitch I mean
instruments such as keyboard and harp.)
In this Pythagorean tuning, which goes back perhaps 2500 years in China
and Greece and possibly much earlier, we tune a scale in perfect fifths,
which have a ratio of 3:2. This chain of fifths can produce a seven-note
diatonic scale or, when extended, a 12-note chromatic scale:
Eb Bb F C G D A E B F# C# G#
-------------------
diatonic scale
In a medieval context -- or a similar modern context, for that matter --
this works very nicely, because we get just fifths (3:2) and fourths
(4:3), also just major seconds (9:8) and minor sevenths (16:9), and rather
wide major thirds and sixths (81:64, 27:16) and narrow minor thirds and
sixths (32:27, 128:81) that add some dynamic tension to cadences resolving
to sonorities with stable octaves, fifths, and fourths.
However, there's one catch: 12 just fifths don't quite come out evenly to
an octave. We have an extra 24 cents (100 cents being an equally-tempered
half-tone, and 1200 cents a just octave) left over, referred to as a
Pythagorean comma.
The 14th-century solution is simple: put the comma between Eb and G#, two
notes unlikely to be used together in this period! We get one "Wolf" fifth
(and fourth) -- but since the notes don't get used together, it's a rather
academic problem, and the tuning is otherwise ideal for this music.
(This might be an example of how a tuning can serve purposes not
necessarily anticipated by its developers; early expositors of the
Pythagorean system weren't necessarily planning to accommodate the
part-music of the late Gothic, but the system happens to fit this practice
very nicely, which may be one reason why theorists accepted it as a matter
of course.)
However, by the 15th century, there was a problem: as composers started to
treat thirds and sixths as the predominant sonorities in a piece, the wide
Pythagorean versions of these intervals became not so pleasing. Also,
sooner or later, some theorists would consider the "Wolf" fifth a flaw
worth "fixing." This led to various new or modified tunings.
One branch of this evolution leads to meantone: make the fifths a bit
narrower to permit ideally euphonious thirds, now that thirds are "solid"
harmonic points of rest rather than unstable intervals in motion.
Another branch leads to equal temperament: solve the problem of the "Wolf"
fifth by making each fifth just a _bit_ narrower -- more precisely, 1/12
of a comma, or just less than 2 cents. Thus a pure fifth is about 702
cents; in equal temperament, each fifth is precisely 700 cents.
The result is a kind of very artful compromise which some writers in the
early modern era, interestingly, confused with a pure Pythagorean
temperament. The fifths and fourths are almost just, the major thirds
somewhat wide (400 cents, compare with 408 cents Pythagorean and 386 cents
just), and the minor thirds somewhat small (300 cents, compare with 294
cents Pythagorean and 316 cents just).
It's a neat solution, although by no means the only one: much Gothic music
would be ideal in Pythagorean, and Renaissance music in meantone or (with
non-fixed-pitch instruments or singers) may some kind of fluid
approximation of just intonation.
Most respectfully,
Margo Schulter
msch...@value.net
ghost
>In article <353c6370...@nntp.up.ac.za>,
>Etienne Marais <s941...@student.up.ac.za> wrote:
>>3) And why the odd 7 letters, plus 5 relative notes for this 12*1/2
>>step division. Do the 'relatives' (ie black notes on piano) have any
>>specific behaviour/attribute (especially seen from a phsysics point of
>>view)
>This is two questions. The 7 notes of a scale refer to conventions of
>music-making within scales for most of the last 1200 years.
Western European classical music doesn't go back 1200 years.
The earlier forms of theoretical, notated music it derives from may
go back that far, but, as various discussions here & elsewhere show,
no-one is in agreement on how to interpret those earlier forms
of notated music. And no-one was around with a tape-recorder to record
what was really being done. (And if they were, the classical musicians
upon hearing it would all chime in "isn't that sound just an artifact
of the taping process"; that's what they say when they hear traditional
music that doesn't sound like they think it should.)
Other cultures use other scales.
>>4) Can this division be into any number of notes, theoretically ?
>Yes, with more or less pleasing or grating sounds. Easley Blackwood
>created compositions in a large number of different equal temperments.
I thought his whole point was using different *just* temperaments,
so that for each seperate piece of music he composed
he was dividing the octave into larger & larger
#s of notes with smaller & smaller interval-sizes between notes.
He did not, however, use *all* the notes generated by dividing by any
large factor in a single piece of music (neither would a traditional
music that uses smaller interval sizes use, in one piece of music,
*all* the possible intervals generated by whatever scheme they use
to generate intervals).
I heard the music he composed with this scheme on an NPR special where
they interviewed him, & none of it sounded like any kind of
equal-temperament-music to me. It sounded like he had specifically organized
each piece on a different particular scale to sound like a different
particular kind of traditional music.
>>6) WHY is it that the notes in scalars go together ?
>Notes in scales go together because we are accustomed to hearing them
>together.
Nope. This is, I think, the fundamental (pun intended) inaccurate
supposition of Western European classical music; that there is no
underlying factor defining scales used in this or any other culture
other than "social recognition".
Matt
In article <6hjdgd$c...@necco.harvard.edu>, ghost <j...@deas.harvard.edu> wrote:
>In article <6hi03c$mur$1...@news.eecs.umich.edu> fie...@zip.eecs.umich.edu (Matt) writes:
>
>>In article <353c6370...@nntp.up.ac.za>,
>>Etienne Marais <s941...@student.up.ac.za> wrote:
>
>>>3) And why the odd 7 letters, plus 5 relative notes for this 12*1/2
>>>step division. Do the 'relatives' (ie black notes on piano) have any
>>>specific behaviour/attribute (especially seen from a phsysics point of
>>>view)
>
>>This is two questions. The 7 notes of a scale refer to conventions of
>>music-making within scales for most of the last 1200 years.
>
>Western European classical music doesn't go back 1200 years.
Western Europoean classical music is founded on western singing
styles which go back further.
>The earlier forms of theoretical, notated music it derives from may
>go back that far, but, as various discussions here & elsewhere show,
>no-one is in agreement on how to interpret those earlier forms
>of notated music.
You're just making this up. Sources and proportions tell us
quantitatively the meaning of notated music back to about 600,
and continuous practice of religious musics conforms with the sources.
> And no-one was around with a tape-recorder to record
>what was really being done. (And if they were, the classical musicians
>upon hearing it would all chime in "isn't that sound just an artifact
>of the taping process"; that's what they say when they hear traditional
>music that doesn't sound like they think it should.)
????
>Other cultures use other scales.
Sure.
>>>4) Can this division be into any number of notes, theoretically ?
>
>>Yes, with more or less pleasing or grating sounds. Easley Blackwood
>>created compositions in a large number of different equal temperments.
>
>I thought his whole point was using different *just* temperaments,
>so that for each seperate piece of music he composed
>he was dividing the octave into larger & larger
>#s of notes with smaller & smaller interval-sizes between notes.
>He did not, however, use *all* the notes generated by dividing by any
>large factor in a single piece of music (neither would a traditional
>music that uses smaller interval sizes use, in one piece of music,
>*all* the possible intervals generated by whatever scheme they use
>to generate intervals).
Large amounts of the music made with 12-tone equal temperment doesn't
use all 12 notes.
>I heard the music he composed with this scheme on an NPR special where
>they interviewed him, & none of it sounded like any kind of
>equal-temperament-music to me. It sounded like he had specifically organized
>each piece on a different particular scale to sound like a different
>particular kind of traditional music.
Hmmm. What would make a piece sound like equal-temperment-music?
Serialism?
>>>6) WHY is it that the notes in scalars go together ?
>
>>Notes in scales go together because we are accustomed to hearing them
>>together.
>Nope. This is, I think, the fundamental (pun intended) inaccurate
>supposition of Western European classical music; that there is no
>underlying factor defining scales used in this or any other culture
>other than "social recognition".
The usual supposition passed around by western musicians, as by
others, is that *their* particular scale system was handed to the
world by a deity, and all other cultures can only put on a pale
immitation of it. The finding that notes go together differently
to different peoples purely as a result of what they're familiar with
involves stepping outside any particular tradition and looking at
music as a human activity.
Manfred40
Etienne Marais wrote in message <353c6370...@nntp.up.ac.za>...
>Hello, some very simply questions (I think)
>
>>3) And why the odd 7 letters, plus 5 relative notes for this 12*1/2
>step division. Do the 'relatives' (ie black notes on piano) have any
>specific behaviour/attribute (especially seen from a phsysics point of
>view)
Though I've not seen it written about, I find some significance in the black
keys being modes of the common pentatonic.
If you are interested enough to make me dig through some piles, there is a
dry, old (1970's) textbook on physics and music which will answer many of
your questions. Let me know if you want me to dig for the title
K C Moore
In article <6hmj46$9n4$1...@supernews.com> Manf...@juno.com "Manfred40" writes:
> [ ... ]
> If you are interested enough to make me dig through some piles, there is a
> dry, old (1970's) textbook on physics and music which will answer many of
> your questions. Let me know if you want me to dig for the title
If you want the best information in psycho-acoustics, you may need to
get something a bit more recent. Starting with Plomp and Levelt in
1965 (Journal of the Acoustical Soc. of America, No 38) there has been
a succession of papers clarifying and refining the Helmholtz theory of
consonance. These might not have influenced a 1970 textbook.
Other interesting authors:
Pierce, JASA 40, 1968.
Kameoka and Kuriyagawa, JASA 45, 1968.
Slaymaker, JASA 47, 1970
Geary, JASA 67.
Mathews & Pierce, JASA 68, 1980.
Sethares, JASA 94 (3) Pt 1, 1993.
The standard textbook to give you the basics is:
A H Benade, Fundamentals of Musical Acoustics, (NY, OUP, 1976).
--
Ken Moore
k...@hpsl.demon.co.uk
ghost
>In article <6hjdgd$c...@necco.harvard.edu>, ghost <j...@deas.harvard.edu> wrote:
>>In article <6hi03c$mur$1...@news.eecs.umich.edu> fie...@zip.eecs.umich.edu (Matt) writes:
>>>This is two questions. The 7 notes of a scale refer to conventions of
>>>music-making within scales for most of the last 1200 years.
>>Western European classical music doesn't go back 1200 years.
>Western Europoean classical music is founded on western singing
>styles which go back further.
Yeah, but you can't sing in those styles, & probably neither can I.
"Founded on" is not the same as "unchanging lineal descent".
>>The earlier forms of theoretical, notated music it derives from may
>>go back that far, but, as various discussions here & elsewhere show,
>>no-one is in agreement on how to interpret those earlier forms
>>of notated music.
>You're just making this up. Sources and proportions tell us
>quantitatively the meaning of notated music back to about 600,
>and continuous practice of religious musics conforms with the sources.
If you follow strictly the proportions given for materials, & reconstruct
exactly (no mean feat) instruments constructed according to those
proportions given, you might be able to come up with a *stringed* instrument
you could play that would give you sounds that were somewhat like what the
notators were aiming for. I wouldn't be too sure of your success, though.
All bets are off with any instrument that isn't meant to be played very
mechanically, so you're pretty limited to keyboard instruments here;
instruments you *can't* tune on the fly, consciously or subconsciously.
You can follow the history & how the tuning schemes changed on keyboard
instruments back *up* from the year 600, but you have no real knowledge
of how anything on other instruments sounded going *back* from that date.
Unless you leave the classical realm, that is. And once you leave the
classical realm, all the carefully notated tuning schemes go
right out the window.
Traditional musicians pass down tuning schemes, but oral traditions
are not the traditions you're citing. And even I am not going to claim
that oral traditions exist *completely* unchanged for 2,500 years.
>> And no-one was around with a tape-recorder to record
>>what was really being done. (And if they were, the classical musicians
>>upon hearing it would all chime in "isn't that sound just an artifact
>>of the taping process"; that's what they say when they hear traditional
>>music that doesn't sound like they think it should.)
>????
I heard this comment made, very loudly, at a meeting I attended last spring.
The recording the patronizing crew was referring to was not
an artifact-laden thing made on one of Edison's prototype recorders;
it was fairly recent, & very representative of musicians still alive
& making music in exactly the fashion captured on tape.
>>Other cultures use other scales.
>Sure.
OK. Any "theory of how music works" has to account for *all* scales
that have a human culture that will voluntarily & without a great deal of
outside drill be able to use them. That rules *out* a lot of variations on
the main scale used in Western European classical music (but certainly
doesn't rule out *all* scale-variations used in WEC), & rules *in* lots
of traditional scales (some mutually incompatible).
You have to ask "can the people in this cultural/sociological group
reproduce this music on this scale, without drill or prodding?"
In other words, is the scale memorable & does it follow some logical
pattern based in the interaction of the physics of sound production
& human perception of sound (even if you can't shake out of the system
what the actual logical pattern is)?
Etienne, who's refs I've accidentally jettisoned, asked:
>>>>4) Can this division be into any number of notes, theoretically ?
>>>Yes, with more or less pleasing or grating sounds. Easley Blackwood
>>>created compositions in a large number of different equal temperments.
>>I thought his whole point was using different *just* temperaments,
>>so that for each seperate piece of music he composed
>>he was dividing the octave into larger & larger
>>#s of notes with smaller & smaller interval-sizes between notes.
>>He did not, however, use *all* the notes generated by dividing by any
>>large factor in a single piece of music (neither would a traditional
>>music that uses smaller interval sizes use, in one piece of music,
>>*all* the possible intervals generated by whatever scheme they use
>>to generate intervals).
>Large amounts of the music made with 12-tone equal temperment doesn't
>use all 12 notes.
So what? We *were* discussing whether Blackwood was demonstrating
different *equal* temperaments, as you claim, or demonstrating different
*just* temperaments, as I'm sure I heard. You're dodging the issue,
& picking up on an illustrative comment of mine as though it were the
whole point. We can't both be right.
>>I heard the music he composed with this scheme on an NPR special where
>>they interviewed him, & none of it sounded like any kind of
>>equal-temperament-music to me. It sounded like he had specifically organized
>>each piece on a different particular scale to sound like a different
>>particular kind of traditional music.
>Hmmm. What would make a piece sound like equal-temperment-music?
Most music actually *played* in equal-temperament, without the musicians
trying to mollify the consequences on the fly, sounds like music written in
equal-temperament to me (& sounds pretty crummy, from a little to a lot
off-pitch from the scale used in a lot of forms of Western European traditional
music). When people try to harmonize long-held open-intervals using values
for those intervals derived from the equal-temperament scale they sound
especially wretched.
Etienne asked:
>>>>6) WHY is it that the notes in scalars go together ?
>>>Notes in scales go together because we are accustomed to hearing them
>>>together.
>>Nope. This is, I think, the fundamental (pun intended) inaccurate
>>supposition of Western European classical music; that there is no
>>underlying factor defining scales used in this or any other culture
>>other than "social recognition".
>The usual supposition passed around by western musicians, as by
>others, is that *their* particular scale system was handed to the
>world by a deity, and all other cultures can only put on a pale
>immitation of it. The finding that notes go together differently
>to different peoples purely as a result of what they're familiar with
>involves stepping outside any particular tradition and looking at
>music as a human activity.
Nope. Its a "human activity" all right, but all your sarcasm aside,
the humans who engage in this activity are constructed according the
identical physiological patterns, as far as anyone has been able to
determine. And the physics those physiological systems are set up
to perceive doesn't change from culture to culture, either.
Matt
Okay, JMF, explain the physical acoustical foundation behind slendro
and pelog.
--
Matt Fields, DMA http://listen.to/mattaj TwelveToneToyBox http://start.at/tttb
Featured addresses: hostm...@INREACH.COM sel...@NETVIGATOR.COM
James Martin
>Other cultures use other scales.
>>Notes in scales go together because we are accustomed to hearing them
>>together.
>Nope. This is, I think, the fundamental (pun intended) inaccurate
>supposition of Western European classical music; that there is no
>underlying factor defining scales used in this or any other culture
>other than "social recognition".
So why do they (notes in scales) go together, if not on account of social
conditioning (i.e. social construction)? You've already said that other
cultures use other scales, just as we could note that cultures use
different languages. Doesn't it follow that different cultures have
different ideas about how chords and scales fit together
(consonance/dissonance, etc.)? Do you take our ideas about harmony and
melody to be cultural or natural?
James Martin
>The usual supposition passed around by western musicians, as by
>others, is that *their* particular scale system was handed to the
>world by a deity, and all other cultures can only put on a pale
>immitation of it. The finding that notes go together differently
>to different peoples purely as a result of what they're familiar with
>involves stepping outside any particular tradition and looking at
>music as a human activity.
I'm glad you said this (and said it so well). Now I know I'm not crazy.
james
Matt
Well, you know, every so often somebody starts throwing around the
"Music all comes from the harmonic series" notion, and I ask why
classical music uses the major scale so much more than Lydian b7.
The latter resembles the harmonic series a whole lot more than the
former.
A440A
James writes:
>Do you take our ideas about harmony and
>melody to be cultural or natural?
They are culturally based as all get-out!
Think "slendro", or "gamelan".
I, for one, am perfectly happy in this Western state of saturated
culturization, I love a good fifth. It is Friday, which begins the weekend,
a perfect interval, no?
Regards,
Ed Foote
Precision Piano Works
Nashville, Tenn. USA
http://www.airtime.co.uk/forte/history/edfoote.html
John Sheehy
fie...@zip.eecs.umich.edu (Matt) writes:
>Well, you know, every so often somebody starts throwing around the
>"Music all comes from the harmonic series" notion, and I ask why
>classical music uses the major scale so much more than Lydian b7.
>The latter resembles the harmonic series a whole lot more than the
>former.
Makes you wonder why carpenters cut wood with saws and such, and don't
just split it along the grain.
Taking a TX81Z, and microtuning the white keys to the first 7 unique
members of the overtone series, was a very sobering experience to this
once idealistic young man.
--
<>>< ><<> ><<> <>>< ><<> <>>< <>>< ><<>
John P Sheehy <jsh...@ix.netcom.com>
><<> <>>< <>>< ><<> <>>< ><<> ><<> <>><
Bill Bailer
I am familiar with most of which is cited above. The Benade is one of
the best books on acoustics of the *sources* of music, but has little
to say about *why* acoustics is relevant to music specifically. By far
the most valuable book that I have seen, which is best reference on the
scientific foundations of music is:
Introduction to the Physics and Psychophysics of Music, by Juan G.
Roederer, published by Springer-Verlag.
I have the second edition, 1979, but there may be newer editions.
This book was a revelation to me -- many of the issues argued in this
group are answered in this book. If you are interested in ANY aspect of
music, theory, composition, acoustics, aesthetics, etc., you must get
this book. It is the only concise source of information on the
scientific foundations of music (since Helmholz) that I have ever seen.
It is a thin paper back, well written by an extraordinary
teacher-scientist-musician.
Bill Bailer
wba...@cris.com, Rochester NY USA, tel:716-473-9556
Acoustics, piano technology, music theory, JSBach
Manfred40
what happened?
Brian Megilligan
On 21 Apr 1998 11:31:56 GMT, fie...@zip.eecs.umich.edu (Matt) wrote:
>
>>2) How did it come to be that an octave got divided into 12 parts in
>>the 1/2 step 'system'. Are there advantages, or merely so in
>>historical perspective ?
>
>It started out as a way to make straight frets on lutes. 12 was the
>smallest number that proved tolerable for the style of music that was
>already popular at the time.
> Advantages are dependent on the musical style, and include the possiblity
>of playing equally well (or poorly) in all keys, of modulating through
>a cycle of 12 keys, and of enharmonically reinterpreting notes and chords.
>
Is this the best historical answer? Wouldn't the idea of 12 tones in an octave
have been worked out by Greek theorists who, for example, starting with the
perfect fifth, would have found themselves at the same (theoretical) pitch 11
tones later?
Brian
------------------------------------------------------------------------
Take the extra "z" out of "softjazzz" and you'll have my e-mail address.
Brian Megilligan
On Sat, 25 Apr 1998 06:05:11 GMT, jsh...@ix.netcom.com (John Sheehy) wrote:
>fie...@zip.eecs.umich.edu (Matt) writes:
>
>>Well, you know, every so often somebody starts throwing around the
>>"Music all comes from the harmonic series" notion, and I ask why
>>classical music uses the major scale so much more than Lydian b7.
>>The latter resembles the harmonic series a whole lot more than the
>>former.
>
>Makes you wonder why carpenters cut wood with saws and such, and don't
>just split it along the grain.
>
>Taking a TX81Z, and microtuning the white keys to the first 7 unique
>members of the overtone series, was a very sobering experience to this
>once idealistic young man.
I'm very curious about this whole thing...mind if I explore this further with
y'all? How can we safely assume that an outgrowth of scales in tonal music or
any music based on the overtone series would result in a lydian b7 scale? I
understand the argument...that those pitches are present in the overtone series,
but aren't they so substantially weaker that they're almost theoretical anyway?
It has been my perception (right or wrong) that the strongest of the overtones
are the octave, fifth, and octave, anything beyond that (other than the third
perhaps) cannot be really "heard." I know I don't really have a thorough
understanding of this stuff so please help me out.
I guess to sum up my question: Why would it be a surprise that western music is
based on the overtone series since the strongest (and maybe most important)
overtones are the octave and the fifth? Why would we assume that the rest of
the series, the #4 and b7, being so much higher up and weaker should necessarily
pull any weight over the fifths and octaves when it comes to constructing a
scale?
Thanks for your time.
M. Schulter
Brian Megilligan <soft...@concentric.net> wrote:
: Is this the best historical answer? Wouldn't the idea of 12 tones in an
: octave
: have been worked out by Greek theorists who, for example, starting with the
: perfect fifth, would have found themselves at the same (theoretical) pitch
: 11 tones later?
Hello, there.
While I'm not sure at just what point such a tuning system was extended to
all 12 tones either in China or in the tradition moving from Classic
Greece through Rome and the Middle Ages, I tend to give this answer also.
The one qualification is that 12 fifths gives us just a bit more than an
even octave; it's an interval known in medieval theory as a hexatone, or
six whole tones or (9:8)^6. It comes to roughly 1224 cents, and that extra
24 cents is called a Pythagorean comma.
In medieval tuning, the practical solution is to put that anomalous 24
cents between the two notes least likely to appear together: Eb and G#.
Since those two notes are in fact _very_ unlikely to get used together in
a Gothic context, we can relax and enjoy all those just fifths and
fourths.
BTW, I've seen reports here that in some Arabic systems, for example,
notes beyond the first twelve in this chain of fifths do get used,
resulting in what might be termed "microtonal" intervals available as
scale elements.
A440A
Greetings,
inre; where did the octave division come from?
The TTSTTTST arrangement of the frequencies is the result of tuning
by fifths. The octave is used for compacting the resulting "strings" into one
scale with the note values in sequential order. Today, tones and semitones
are regarded as having only one size each, but this was not always the case.
In Western music, the development of a 12 note octave seems to have arisen
between 900 and 1300 A.D. It followed the refinement of organs,where we can
see the black keys are late-coming intruders. That instruments still survive
allows us to trace a history of increasing notes per octave.
The keyboards prior to this time had no accidentals. The ability to
produce smaller increments of musically usable sound followed the regulation of
wind-pressure in earlier organs. Without even pressure, the notes were so
variable that anything smaller than a diapente or so would be unbearably
dissonant. By 1365, the organ at Halberstadt had a 7/5 keyboard arrangement,
and though more accidentals were added in other instruments, (split keys,etc),
problems with human manipulation of them effectively rendered them
impractical.
The combination of a usable number of notes, and the musical
"disjuncture" that occurs between the octaves and fifths produced the various
forms of tuning that keyboards have evolved with. Knowing these historical
intonational eras is critical to understanding the music that came from them.
Today, equal temperament is about all that most people hear. Musically,
we are living in a "one-size-fits-all" world. Having been talked about for
centuries, the actual attainment of ET is a recent phenomenom, and I predict
in a hundred years, will be viewed as a large detour from which music had to
recover.
We could have the purity of meantone without any wolves, if we have 54
notes per octave. We just can't figure out how to play an instrument with that
many levers. The computer's are now closing in on making purity possible in
all the keys, but we have to ask ourselves, "Is this good?". Do we want a
music devoid of dissonance that comes from the commas? The presence of
dissonance has been a constant throughout musical history.
To the poster that suggested that 12 note octaves were the result of lutes
and their fretting, I must disagree. The lutes( along with the organs) were
the first instruments to be capable of ET. This followed the publication of
the Mersenne ratios in the early 1600's. Fret placement could be done by
math, and lo and behold, an ET would result. Mersenne himself said that the
numbers would be of no use to claviers, as their pitch had to be judged by ear.
Matt
In article <3542ad22....@news.concentric.net>,
Brian Megilligan <soft...@concentric.net> wrote:
>On 21 Apr 1998 11:31:56 GMT, fie...@zip.eecs.umich.edu (Matt) wrote:
>
>>
>>>2) How did it come to be that an octave got divided into 12 parts in
>>>the 1/2 step 'system'. Are there advantages, or merely so in
>>>historical perspective ?
>>
>>It started out as a way to make straight frets on lutes. 12 was the
>>smallest number that proved tolerable for the style of music that was
>>already popular at the time.
>> Advantages are dependent on the musical style, and include the possiblity
>>of playing equally well (or poorly) in all keys, of modulating through
>>a cycle of 12 keys, and of enharmonically reinterpreting notes and chords.
>>
>
>
>have been worked out by Greek theorists who, for example, starting with the
>perfect fifth, would have found themselves at the same (theoretical) pitch 11
>tones later?
>
No, they would have found themselves a Pythagorian comma away from the
same pitch. In classical thought, the octave is a 2:1 tuning ratio and
the fifth is a 3:2 tuning ratio. To what integer power greater than
zero can you raise 3 to get an integer power of 2?
Matt
Just a note that keyboards with 15 keys per octave have been an
on-again-off-again concern for about 400 years. These involve some
somewhat more subtle issues than stacks of fifths, but the point again is
that they sound "in tune" in a wide variety of keys.
Matt
In article <3543aec5....@news.concentric.net>,
A couple of notes here.
1. Go into a resonant space and hum a single pitch. Slowly shift your
vowel from deepest uu to brightest iiy, while conciously trying to
listen for high-pitched sounds. You will find harmonics up to about
the 23rd partial clearly audible. If you cannot manage this, find a
large music library and listen to just about any recording of David
Hykes. Also, look up how many harmonics are used in identifying vowel
sounds of tenor, baritone, and bass voices (I think you'll find that
the psychoacoustic literature says up to 16 or so).
2. The interesting finding is that much of western music is
structured around major scales, which contain the artifices of the 4th
and the 6th that are not from the harmonic series of the tonic. More
than once, folks here have proposed that we consider the harmonic
series of the subdominant and not the tonic as generating the
materials of a tonality--a hypothesis which flies in the face of
attempts to explain musical structure, consonance, and dissonance
in terms of the assumption fundamental==tonic.
And a large chunk of the remaining western music is structured
around minor scales, containing the additional artifice of the minor
third.
So the real curiousity is not that the harmonic series plays a role in
the structure of western music, but rather that it plays so small a
role. We've become so accustomed to our artifices that we tend to
imagine they're perfectly natural and universal, just like FORTRAN!
Matt
In article <199804261143...@ladder01.news.aol.com>,
A440A <a4...@aol.com> wrote:
> To the poster that suggested that 12 note octaves were the result of lutes
>and their fretting, I must disagree. The lutes( along with the organs) were
>the first instruments to be capable of ET.
And worse than that, if you didn't fret them that way, you either had
wild tuning problems or had to build very complex frets (or, of
course, leave the frets off altogether). Somewhere on the web there's
pictures of Harry Partch's solutions to this. Each fret on his guitars
passes under only one course of strings.
K C Moore
If you recall that, as late as the 18th C, fingering charts for woodwind
instruments distinguished C# from Db and that even in the early 20th C
French academic training was to write using a complete chromatic of at
least 31 notes (7 each naturals, sharps, flats, 5 each double flats &
double sharps) you may agree with me that the concept of the 12 note
octave became widely accepted during the 19th C as a result of the
enormous success of the piano and pianist composers. Chopin was
probably the earliest major exploiter of enharmonic ambiguities,
though Mozart and Schubert preceded him and Liszt and Wagner
followed his example enthusiastically.
--
Ken Moore
k...@hpsl.demon.co.uk
John Chalmers
Two points: The division of the octave into 12 pitch areas dates
to the Babylonians in the 2nd millennium B.C.E. (M.L. West has
an excellent paper on the topic.). Twelve tone equal temperament
was first calculated in China by Prince Tsai Yu in the late 16th
century and very soon thereafter in Europe by Simon Stevin and
others although approximations were known already. The cycle of
fifths or fourths approximates a system of 12 semitones and these
seem to have become the basis for music in the Near East at a very
early date and to have spread from there to China, Greece, etc.
(Chinese claim priority and a date of about 2700 BCE, but such
early dates are probably mythical as the story contains supernatural
elements, and the inventor's name is actually a title.)
Guitars with moveable frets under each string are made in Germany.
I recently saw and heard one played by John Schneider at MicroFest
at Pierce College in Woodland Hills, California. It was tuned as
Harry Partch's Adapted Guitar.
--John
John Sheehy
soft...@concentric.net (Brian Megilligan) writes:
>Is this the best historical answer? Wouldn't the idea of 12 tones in an octave
>have been worked out by Greek theorists who, for example, starting with the
>perfect fifth, would have found themselves at the same (theoretical) pitch 11
>tones later?
12 tones later, yes (11 tones only gives you all 12, it doesn't return
to the starting note). The idea of taking an entire scale directly from
the lower, relevant part of the overtone series seems to only be popular
in recent times (possibly due to the popularization of the series by
people like Helmholtz). The pythagorean series could theoretically be
taken from the overtone series, if you look at it as partials 1, 3, 9,
27, 81, 243, 729, etc, but anything beyond 25 or so is pretty much lost
on us, in terms of significance.
BTW, the endpoints of such a cycle of perfect 3:2 5ths overlap by 23
cents (percent of a 12TET semitone, where there are 1200 cents to an
octave). ET 5ths are 700 cents, and 3:2 is log(3/2)/log(2^(1/1200)) =
701.955 cents. A cycle of 12 ET 5ths takes you 700*12 = 8400 cents,
which is 7 octaves (8400/1200 = 7). That extra 1.955 cents in the pure
3:2 multiplied 12 times gives you 12*1.955 = 23.46 cents past the octave
mark.
K C Moore
In article <6hvesr$ceo$1...@news.eecs.umich.edu>
fie...@zip.eecs.umich.edu "Matt" writes:
> [ ... ]
> So the real curiousity is not that the harmonic series plays a role in
> the structure of western music, but rather that it plays so small a
> role. We've become so accustomed to our artifices that we tend to
> imagine they're perfectly natural and universal, just like FORTRAN!
At last the light dawns on me! my reservations about ET12 result from
having been brought up on Algol 60! (APL, Algol68 & SISAL added later).
Scots bagpipes have a half flattened 4th, like an 11th harmonic. I'm
sure that happens in other ethnic musics. Examples anyone?
--
Ken Moore
k...@hpsl.demon.co.uk
M. Schulter
A440A <a4...@aol.com> wrote:
: In Western music, the development of a 12 note octave seems to have arisen
: between 900 and 1300 A.D. It followed the refinement of organs,where we can
: see the black keys are late-coming intruders. That instruments still survive
: allows us to trace a history of increasing notes per octave.
Thanks for another informative post on tuning-related matterns, and I
might just add the fine distinction that Bb has a kind of special status,
being counted by Guido d'Arezzo (c. 1030) and his followers as a part of
the basic gamut of music, or what came to be termed _musica recta_,
unlike other accidentals, either written or unwritten and supplied by
performers, which came to be known as _musica ficta_ (maybe best
translated as "invented" notes).
: dissonant. By 1365, the organ at Halberstadt had a 7/5 keyboard arrangement,
: and though more accidentals were added in other instruments, (split keys,etc),
: problems with human manipulation of them effectively rendered them
: impractical.
Two interesting bits of evidence seem to indicate that your date of 1300
quoted above might be a good guess for the 7/5 keyboard. First, Jacobus of
Liege, in his _Speculum musicae_ (c. 1325), notes that keyboards are
common which have all twelve semitones, observing as a matter of course
that these semitones are unequal (i.e. 256:243 or 90 cents for c#-d or
eb-d, and 114 cents or 2187:2048 for c-c# or eb-e).
Also, the first known Western European keyboard compositions from the
Robertsbridge Codex (c. 1325-1340?) call for all 12 notes, suggesting that
keyboards had them available.
This ties in nicely with Halberstadt, an instrument I've heard of but am
not well acquainted with -- in fact, if one favors a slightly later dating
of Robertsbridge which I've seen proposed by at least one recent scholar,
then it might be just about contemporary with this organ.
ghost
In article <35569e71...@nntp.ix.netcom.com> jsh...@ix.netcom.com (John Sheehy) writes:
>The idea of taking an entire scale directly from
>the lower, relevant part of the overtone series seems to only be popular
>in recent times (possibly due to the popularization of the series by
>people like Helmholtz). The pythagorean series could theoretically be
>taken from the overtone series, if you look at it as partials 1, 3, 9,
>27, 81, 243, 729, etc, but anything beyond 25 or so is pretty much lost
>on us, in terms of significance.
I say that taking a scale from the overtone series is exactly what all
forms of traditional music do. They just don't all take the *same* scale.
There's lots of overtones to choose from (there's an infinite amount).
You then get to choose to form your scale out of groups of notes that
compliment each other in some way (such as by sharing members of their
own harmonic series with each other).
Nothhing locks you into the *low* end of the series, even though you're more
likely to actually *hear* the low end of the series when sounding a
particular fundamental note. The higher end of the series still
theoretically exists for that note, & if you while experimenting with
whatever you use to make sounds happen to hit on something that is based
on a note from the higher end of the overtone series, you'll recognize it
as being in the same family as your fundamental even though it isn't one of
the notes that ring out loudly whenever you sound that fundamental.
And not knowing, in a quantitative physical way, about the existance of the
overtone series doesn't prevent you from *hearing* it, even if you
can't describe & explain, in the terminology of your day,
what it is you're hearing.
ghost
In article <6hqk1k$3...@picayune.uark.edu> jema...@comp.uark.edu (James Martin) writes:
I'd said:
>>Other cultures use other scales.
Matt Field had said:
>>>Notes in scales go together because we are accustomed to hearing them
>>>together.
I'd said:
>>Nope. This is, I think, the fundamental (pun intended) inaccurate
>>supposition of Western European classical music; that there is no
>>underlying factor defining scales used in this or any other culture
>>other than "social recognition".
James Martin says:
>So why do they (notes in scales) go together, if not on account of social
>conditioning (i.e. social construction)? You've already said that other
>cultures use other scales, just as we could note that cultures use
>different languages. Doesn't it follow that different cultures have
>different ideas about how chords and scales fit together
>(consonance/dissonance, etc.)? Do you take our ideas about harmony and
>melody to be cultural or natural?
I don't want to drag the "are there underlying constructs of spoken language
that are universal?" discussion into this. There might be,
but I believe the way various forms of traditional music are constructed
is going to be a lot easier to disect than the way various languages are
constructed. Please note I didn't say "easy", I said "easier than".
I think that different forms of traditional music choose their scales
based on "families of notes that go together", but that what makes one
family more appealing than any other family *is* the cultural-conditioning
part of the choice.
Notes can be grouped into families in various ways:
(a) harmonics of a given fundamental that graph out to similar curves
(I think these will be found to share each other's harmonic series to a
great extent, but don't have a reference to cite right now)
(b) picking a favored harmonic interval, like 5ths, & climbing up by that
interval.
(this is not a contradiction of <a>; you can, if you like, discard
any points you generate this way which don't also fit description <a>.)
If you're not going to ever sound 2 notes simultaneously, though,
having notes that harmonize beyond the dying strains of the 1st
is not so important.
(c) picking notes that when sounded simultaneously produce specific
beat patterns, with the stricture that the beats produced be in a
tightly recurring pattern. Usually beats from simultaneously sounded
notes is what you're trying to avoid but if you can get a beat pattern
under control it becomes part of the rhythm of the piece.
[I've read that some people think that "a pleasing beat pattern"
is what some forms of traditional music are going for to a certain
extent, but I'm not so sure that's true.]
[Its late & I'm tired or I'd think this out more thoroughly]
HRH Tritone King
Matt <fie...@zip.eecs.umich.edu> wrote in article
<6hve04$c1j$1...@news.eecs.umich.edu>...
[...]
> To what integer power greater than
> zero can you raise 3 to get an integer power of 2?
That is a question that I've been grappling with for quite some time now,
but have never found an answer.
Do you have the answer?
> --
> Matt Fields, DMA http://listen.to/mattaj TwelveToneToyBox
http://start.at/tttb
> Featured addresses: hostm...@INREACH.COM sel...@NETVIGATOR.COM
--
Appreciating any feedback about this or other mode-related issues,
HRH Tritone King
Matt
In article <6i16h7$d...@bgtnsc03.worldnet.att.net>,
HRH Tritone King <Trito...@worldnet.att.net> wrote:
>Matt <fie...@zip.eecs.umich.edu> wrote in article
><6hve04$c1j$1...@news.eecs.umich.edu>...
>
>[...]
>> To what integer power greater than
>> zero can you raise 3 to get an integer power of 2?
>
>That is a question that I've been grappling with for quite some time now,
>but have never found an answer.
>
>Do you have the answer?
Yes, it can't be done, and that's a rudimentary theorem of number theory
(uniqueness of prime factorization).
--
Matt Fields, DMA http://listen.to/mattaj TwelveToneToyBox http://start.at/tttb
"There is no constitutional requirement that the incremental cost of sending
massive quantities of unsolicited advertisements must be borne by the
recipients." -- Judge Graham, Compuserve vs. Cyber Promotions
ghost
>Yes, it can't be done, and that's a rudimentary theorem of number theory
>(uniqueness of prime factorization).
Yeah, but what do primes have to do with the physics of sound?
Using increasingly large primes for your note-generating
ratios will give you increasingly smaller & smaller intervals, which is
good because some of these notes you'll generate that way fall on
harmonics (will fall on them several octaves above the octave you want to
generate notes in, but you can use the ratio to reflect the note back).
But others don't. I don't really think "primeness" of a number
has a whole bunch to do with the physics of sound or the physiology of
human perception of sound.
*Integers* have a bunch to do with the physics of sound, because a
vibrating string produces harmonics according to integer divisions of
the length of the string (resulting in integer multiples of the frequency
generated by the original string length). There's no "only primes"
rule; *all* integers are involved. The only reason, as far as I can see
it, that people like to use odd numbers (after using "2", the only even
prime), & from among the odd numbers, primes for their note-generating
ratios is that you get fewer already-generated notes that way.
Your octaves are generated by powers-of-2, which, beyond "2^1",
are *not* primes.
Seriously: If anyone has any physiology articles to cite that say
that human hearing responds differently to something having to do with
prime numbers than it does to something having to do with non-primes,
please cite it.
K C Moore
In article <893619...@hpsl.demon.co.uk>
k...@hpsl.demon.co.uk "K C Moore" writes:
> Scots bagpipes have a half flattened 4th, like an 11th harmonic.
^^^^^^^^^
Sorry, should be "sharpened" (relative to perfect 4th).
--
Ken Moore
k...@hpsl.demon.co.uk
Matt
In article <6i2bsh$5...@necco.harvard.edu>, ghost <j...@deas.harvard.edu> wrote:
>In article <6i1qii$fop$1...@news.eecs.umich.edu> fie...@zip.eecs.umich.edu (Matt) writes:
>>Yes, it can't be done, and that's a rudimentary theorem of number theory
>>(uniqueness of prime factorization).
JMF is clearly trolling here, has deleted the yes/no question and is
now going to go on a straw-man rampage. Bulk of straw-man-pushing
deleted below.
>Yeah, but what do primes have to do with the physics of sound?
The question was what integer power of 3 is an integer power of 2 and
the answer is none. Corrolary: no matter how many Pythagorian-tuned
fifths you stack, you'll never exactly match frequency ratios with
compounded octaves, though you may get as close as you like.
[stuff on a different topic snipped]
>*Integers* have a bunch to do with the physics of sound, because a
>vibrating string produces harmonics according to integer divisions of
A *driven* vibrating string, yes. But not a plucked string ringing freely.
The modes of vibration of a freely ringing string are measurably wide of
integer ratios.
[...other stuff again on a different topic...]
>Seriously: If anyone has any physiology articles to cite that say
>that human hearing responds differently to something having to do with
>prime numbers than it does to something having to do with non-primes,
>please cite it.
That is a different topic.
ghost
>In article <6i2bsh$5...@necco.harvard.edu>, ghost <j...@deas.harvard.edu> wrote:
>>In article <6i1qii$fop$1...@news.eecs.umich.edu> fie...@zip.eecs.umich.edu (Matt) writes:
>>>Yes, it can't be done, and that's a rudimentary theorem of number theory
>>>(uniqueness of prime factorization).
>JMF is clearly trolling here,
No, I'm not, & keep your interpetations of my questions free of
of that kind of perjorative.
>has deleted the yes/no question and is
>now going to go on a straw-man rampage. Bulk of straw-man-pushing
>deleted below.
I'd like to hear how any "theory of music" accounts for all music,
not just specially-manicured music, designed to fit into a particular
theory. In other words, the theory, if its really going to explain
"how music works", should fit any music, rather than having a theory that
only fits specific music, music which has been pruned to fit the theory.
>>*Integers* have a bunch to do with the physics of sound, because a
>>vibrating string produces harmonics according to integer divisions of
>A *driven* vibrating string, yes. But not a plucked string ringing freely.
>The modes of vibration of a freely ringing string are measurably wide of
>integer ratios.
Oh really? Not the way I've read it. As your vibration dies down you'll
possibly stray wide of the original series, but most players damp out their
strings long before then.
>>Seriously: If anyone has any physiology articles to cite that say
>>that human hearing responds differently to something having to do with
>>prime numbers than it does to something having to do with non-primes,
>>please cite it.
>That is a different topic.
Sure is. Has nothing to do with "prime number theory".
Has lots to do with "how & what humans hear".
Have anything informative to say on it?
Mike Dusseault
In article <6hvesr$ceo$1...@news.eecs.umich.edu>,
fie...@zip.eecs.umich.edu (Matt) writes:
> In article <3543aec5....@news.concentric.net>,
> Brian Megilligan <soft...@concentric.net> wrote:
>>On Sat, 25 Apr 1998 06:05:11 GMT, jsh...@ix.netcom.com (John Sheehy) wrote:
>>
>>>fie...@zip.eecs.umich.edu (Matt) writes:
>>>
>>>>Well, you know, every so often somebody starts throwing around the
>>>>"Music all comes from the harmonic series" notion, and I ask why
>>>>classical music uses the major scale so much more than Lydian b7.
>>>>The latter resembles the harmonic series a whole lot more than the
>>>>former.
I think the places we derivate from the "perfect" is relevant itself.
The minor third has a certain sound to me in part because the harmonic
series doesn't coincide early on. Still it seems to me impossible not
to figure out the minor third eventually once you learn about thirds
since it's the remaining interval between the third and the fifth.
So my guess is we found it, listened to it, found a use for it's sound
and the rest is history. So although I think the harmonic series is
the prime reason why we percieve music as we do, I don't think that
means our scales must follow it exactly. Let's just say that a scale
that *completely* ignores and goes against the harmonic series will
lack a certain character and contrast.
--- snip ---
> 2. The interesting finding is that much of western music is
> structured around major scales, which contain the artifices of the 4th
> and the 6th that are not from the harmonic series of the tonic. More
> than once, folks here have proposed that we consider the harmonic
> series of the subdominant and not the tonic as generating the
> materials of a tonality--a hypothesis which flies in the face of
> attempts to explain musical structure, consonance, and dissonance
> in terms of the assumption fundamental==tonic.
The fourth makes perfect sense to me when you consider the harmonic
series. The tonic is part of the fourth's harmonic series, rather
than the other way around. That's how I hear it anyways (I think).
Let's see...
Take a C major scale. The tonic, C0, would contain as it's first harmonics
C1, G1, C2, E2, G2 or the octave, fifth, another octave, third and fifth.
The fourth would be F0. That note's first few harmonics would be
F1, C1, F2, A3, C3. Note that C figures as F's 5th. I think that is
the fundamental relationship of a tonic to it's fourth.
We could do the same for the sixth. The sixth of C is A. Let's see
what the harmonic series is for A: A1, E1, A2, C#2, E2, G2, A3... still
no C, only a C sharp. Not only do the harmonics not coincide, at least
early on, but now we have a C#2 in the series for F that would create
a physiologically dissonant interval of a minor second with C. To me,
that covers a lot of the character of the sixth interval.
I don't think we always have to get everything to "line up" perfectly.
Sometimes it's the imperfections that make it interesting. The
interactions between the overtones of two notes affects greatly the
sound and color of the interval.
So in response, I don't see why we can't consider the fourth's harmonic
series to learn why it works as it does, at least when considering the
fourth and how it sounds against the tonic.
> And a large chunk of the remaining western music is structured
> around minor scales, containing the additional artifice of the minor
> third.
>
> So the real curiousity is not that the harmonic series plays a role in
> the structure of western music, but rather that it plays so small a
> role. We've become so accustomed to our artifices that we tend to
> imagine they're perfectly natural and universal, just like FORTRAN!
I think the harmonic series does not dictate the scale to us, but
rather dictates how intervals will sound. If your goal is maxiumum
consonance, than lining up the overtones is what you want to do. But
it's not my goal. A lot of expression lies in the way we derivate
from consonance into dissonance. There is no "perfect" scale because
it depends on the exact sound you wish in your music. Of course, a
scale with no consonance would find itself lacking this fundamental
contrast leaving us with noise. My guess is that exactly which tones
we decide to use in a scale is largely cultural preference for the
sound qualities generated by those notes due to the harmonic series.
Does this make sense to anyone? As I said, I am currenly trying to
figure all this out. A lot of this is speculation, so take it with
a grain of salt. I am quite interested in other people's opinions
on this.
--
--------------------------------------------------------------------
This mailbox is UBE filtered. Junk automatically sent to /dev/null.
<mi...@bulwark1.ic.gc.ca>
Matt
In article <6i2vfa$b...@necco.harvard.edu>, ghost <j...@deas.harvard.edu> wrote:
>In article <6i2e2r$npi$1...@news.eecs.umich.edu> fie...@zip.eecs.umich.edu (Matt) writes:
>
>>In article <6i2bsh$5...@necco.harvard.edu>, ghost <j...@deas.harvard.edu> wrote:
>
>>>In article <6i1qii$fop$1...@news.eecs.umich.edu> fie...@zip.eecs.umich.edu (Matt) writes:
>
>>>>Yes, it can't be done, and that's a rudimentary theorem of number theory
>>>>(uniqueness of prime factorization).
>
>
>>JMF is clearly trolling here,
>
>No, I'm not, & keep your interpetations of my questions free of
>of that kind of perjorative.
Hi, JMF,
Brian Mcgillen wrote:
>Is this the best historical answer? Wouldn't the idea of 12 tones in an octave
>have been worked out by Greek theorists who, for example, starting with the
>perfect fifth, would have found themselves at the same (theoretical) pitch 11
>tones later?
>
To which I rhetorically wrote:
No, they would have found themselves a Pythagorian comma away from the
same pitch. In classical thought, the octave is a 2:1 tuning ratio and
the fifth is a 3:2 tuning ratio. To what integer power greater than
zero can you raise 3 to get an integer power of 2?
To which Tritone King wrote:
>
>That is a question that I've been grappling with for quite some time now,
>but have never found an answer.
>
>Do you have the answer?
You have jumped into a thread in the middle.
>>has deleted the yes/no question and is
>>now going to go on a straw-man rampage. Bulk of straw-man-pushing
>>deleted below.
>
>I'd like to hear how any "theory of music" accounts for all music,
>not just specially-manicured music, designed to fit into a particular
>theory.
Yes, I'd like to see that too. And I have some interesting wide-ranging
examples that I'd like to throw at any such theory. But in the meanwhile,
in anticipation of the grand millenial unified music theory, there's
lots of observations to be drawn about music as it is. One of the things
we notice is that
> In other words, the theory, if its really going to explain
>"how music works", should fit any music, rather than having a theory that
>only fits specific music, music which has been pruned to fit the theory.
What about theories based on a pre-existing body of music, that speak
about how that body of music works for its particular audience, and
from a multiplicity of which you might eventually hope to form a GUMT
of all musics and all audiences? Gotta start somewhere.
>>>*Integers* have a bunch to do with the physics of sound, because a
>>>vibrating string produces harmonics according to integer divisions of
>
>>A *driven* vibrating string, yes. But not a plucked string ringing freely.
>>The modes of vibration of a freely ringing string are measurably wide of
>>integer ratios.
>Oh really? Not the way I've read it. As your vibration dies down you'll
>possibly stray wide of the original series, but most players damp out their
>strings long before then.
Okay, I suggest you try reading Backus on this, and actually doing
some of the physics experiments yourself.
>>>Seriously: If anyone has any physiology articles to cite that say
>>>that human hearing responds differently to something having to do with
>>>prime numbers than it does to something having to do with non-primes,
>>>please cite it.
>
>>That is a different topic.
>
>Sure is. Has nothing to do with "prime number theory".
>Has lots to do with "how & what humans hear".
>Have anything informative to say on it?
Yes. Humans hear with a combination of hardwiring and learned responses.
So there's a whole lot more complexity to be found in hearing than
we can account for by harmonic series.
My comment on prime factorization from number theory was completely
on-topic. If you have a difficulty with that, then let's please
take it to e-mail.
Matt
In article <6i2vp6$ibe$2...@news.crc.ca>,
Mike Dusseault <mi...@strategis.ic.gc.ca> wrote:
>The fourth makes perfect sense to me when you consider the harmonic
>series. The tonic is part of the fourth's harmonic series, rather
>than the other way around. That's how I hear it anyways (I think).
>
>Let's see...
>
In the case of classical music, it works ---in conjunction with an
important extra-harmonic use, the rhetorical function of calling into
question the role of the tonic. This is why you see cycles of the form
I-IV-V-I in classical music: the tonic is stated, then called into
question by motion I-IV, which can be interpreted as V-I in the
subdominant key, and just when you start wondering what's going on,
you get to V, which then can be held out almost indefinitely, so long
as it is followed by I. The corresponding retrograde cycle found in
much Rock, I-V-IV-I, is usually not extended beyond a few seconds, and
if you try extending the IV chord much longer, you may well find--as I
do-- that you've been conditioned to accept that as V-II-I---(what's
this added V on the end?) in the key of the subdominant.
So, I'm saying that harmonic relationships play a role in this way of
hearing things, but conventions and interdisciplinary combinations
like harmony with rhetoric, harmony with drama, and harmony with
semiotics provide important clues to what's going on that you can't
get just looking at the harmonic series.
>I think the harmonic series does not dictate the scale to us, but
>rather dictates how intervals will sound.
Up to a point. Does a pile of perfect 16/15 minor seconds sound restful
to you? Is there a context in which it does? or 10/9 or 9/8 whole-steps:
when you introduce those between the fourth and the fifth of a scale,
does V7 lose its sense of forward momentum? I don't think we can
afford to overlook the learned component of how intervals sound.
There's a long tradition of passing off consonance on harmonic alignment...
yet I know that I can enjoy music in major modes on carillon bells
(they have minor-mode partials!) and music in minor modes on flutes
(they come about as close as any natural instrument to pure harmonic spectra).
John Sheehy
k...@hpsl.demon.co.uk (K C Moore) writes:
>Scots bagpipes have a half flattened 4th, like an 11th harmonic. I'm
>sure that happens in other ethnic musics.
The 11th partial is a half-flatted #4th, or half-sharped 4th.
M. Schulter
John Sheehy <jsh...@ix.netcom.com> wrote:
: k...@hpsl.demon.co.uk (K C Moore) writes:
: >Scots bagpipes have a half flattened 4th, like an 11th harmonic. I'm
: >sure that happens in other ethnic musics.
: The 11th partial is a half-flatted #4th, or half-sharped 4th.
Hello, there, and let's see -- that would be something like 555 cents, if
I figure a perfect fourth at 498 cents, and a tritone at 612 cents.
BTW, I have an idea for a new tuning unit -- the "bit."
If things worked out simply, there would be eight bits to a 12tet
semitone, or 12.5 cents per bit -- analogous to eight bits in a byte (if
we take a semitone in a certain context as the smallest discrete unit of
musical information), or eight bits to a dollar U.S.A.
However, if "getting one's two bits in" means using an interval with a
Pythagorean comma, then we need two sizes of bits: a "major bit" equal to
half of the 24-cent comma defining the difference between 12 fifths and an
even octave -- or is that closer to 23 cents?; and a "minor bit" equal to
half of the 22-cent comma by which a major third (81:64) exceeds a 5:4
ratio.
In order to "get our two cents in," as you suggested in an earlier thread,
on each perfect fifth (702 cents vs. 700 for 12tet), and also to "get our
two bits in" on the various thirds and sixths, as well as the "Wolf" fifth
and fourth, we need to do a bit of creative rounding -- but it has a
certain appeal.
Most appreciatively,
Margo Schulter
msch...@value.net
K C Moore
> In article <6i2e2r$npi$1...@news.eecs.umich.edu> fie...@zip.eecs.umich.edu (Matt)
> writes:
>
> >In article <6i2bsh$5...@necco.harvard.edu>, ghost <j...@deas.harvard.edu> wrote:>
> >>*Integers* have a bunch to do with the physics of sound, because a
> >>vibrating string produces harmonics according to integer divisions of
>
> >A *driven* vibrating string, yes. But not a plucked string ringing freely.
> >The modes of vibration of a freely ringing string are measurably wide of
> >integer ratios.
>
> Oh really? Not the way I've read it. As your vibration dies down you'll
> possibly stray wide of the original series, but most players damp out their
> strings long before then.
Consensus of those who sound as though they understand mechanics of
vibrations agrees with Matt. Have you read Benade? He treats this
subject and makes the distinction between forced (harmonic) and free
(potentially inharmonic) vibrations.
Nowadays strings are pretty good though, with high "harmonics" pretty
well in tune to my ear. Not so during the time of Spohr, whose violin
method warns players to ensure that the taper on all the strings goes
in the same direction.
--
Ken Moore
k...@hpsl.demon.co.uk
K C Moore
In article <6i2vp6$ibe$2...@news.crc.ca>
mi...@strategis.ic.gc.ca "Mike Dusseault" writes:
> In article <6hvesr$ceo$1...@news.eecs.umich.edu>,
> fie...@zip.eecs.umich.edu (Matt) writes:
> > So the real curiousity is not that the harmonic series plays a role in
> > the structure of western music, but rather that it plays so small a
> > role. We've become so accustomed to our artifices that we tend to
> > imagine they're perfectly natural and universal, just like FORTRAN!
>
> I think the harmonic series does not dictate the scale to us, but
> rather dictates how intervals will sound.
According to the psycho-acousticians (Plomp & Levelt, Pierce,
Slaymaker) the influence is indirect. Partials determine whether an
interval sounds consonant or dissonant, but partials of most of the
instruments used in Western music are harmonic. Slaymaker points out
that there is a positive feed-back here:
1) We started music with a number of instruments (including the human
voice) with harmonic partials.
2) We developed a harmonic system in which consonant intervals have
frequency ratios derived from small integers.
3) The acceptability of new instruments was judged by how well they
sounded in this harmonic system: consequently most new instruments have
harmonic partials.
> If your goal is maxiumum
> consonance, than lining up the overtones is what you want to do.
^^^^^^^^^
Precisely
> But
> it's not my goal. A lot of expression lies in the way we derivate
> from consonance into dissonance. There is no "perfect" scale because
> it depends on the exact sound you wish in your music.
There is also no perfect scale of harmonic timbres because of annoying
arithmetic like (4/3)*(5/4) <> (9/8)*(5/4) and there is no perfect
equal tempered scale of harmonic timbres because 2^(p/q) <> r/s, where
p, q, r and s are integers, and p<q. However, see J R Pierce,
"Attaining consonance in arbitrary scales", Journal of the Acoustical
Society of America 40 (1968), for perfect scales in synthesized
non-harmonic timbres.
> [ ... ] As I said, I am currenly trying to
> figure all this out. A lot of this is speculation, so take it with
> a grain of salt. I am quite interested in other people's opinions
> on this.
There is a lot of good work on this subject reported in the literature,
and most of it hangs together pretty well, though an expanding perimeter
continues to show more interesting areas not yet occupied.
--
Ken Moore
k...@hpsl.demon.co.uk
K C Moore
In article <6i33r5$3rt$1...@news.eecs.umich.edu>
fie...@zip.eecs.umich.edu "Matt" writes:
> In article <6i2vfa$b...@necco.harvard.edu>, ghost
<j...@deas.harvard.edu> wrote:
> >I'd like to hear how any "theory of music" accounts for all music,
> >not just specially-manicured music, designed to fit into a particular
> >theory.
>
> Yes, I'd like to see that too.
A good place (not necessarily the only one) to start would be the last
6 million years of evolution and what else makes homo sapiens different
from chimpanzees and bonobos. I was pleased to hear recently that I
was preceded in my idea that man's musical ability evolved to meet the
need for increased social cohesion by no less an authority than E O
Wilson (who wrote "Sociobiology", but the comment on music is more
recent).
--
Ken Moore
k...@hpsl.demon.co.uk
Matt
It just occured to me to consider the minor triad in terms of
the harmonic series of the subdominant. If you take 6th, 7th, and 9th
partials together, you get a triad with a Pythagorian 5th, and the
oft-overlooked ratio 7/6 for the third (we tend to use 6/5 for the minor
third, i.e. to truly invert the just major triad). This whets my appetite
to try that out in Csound and see whether my ears will accept 7/6 as any
form of minor third at all. At the moment I don't have access to any
platform on which I can run Csound. Anybody out there can synthesize
a nice 6:7:9 triad and a 6:7 interval for us and put it in web space
in audio/basic (au) format?
David F. Place
ghost wrote:
>
[...]
> I say that taking a scale from the overtone series is exactly what all
> forms of traditional music do. They just don't all take the *same* scale.
> There's lots of overtones to choose from (there's an infinite amount).
> You then get to choose to form your scale out of groups of notes that
> compliment each other in some way (such as by sharing members of their
> own harmonic series with each other).
>
The existence of an infinite number of overtones is only mathematical.
For musical purposes, can't we say that inaudible equals nonexistence?
Also, only bowed strings and wind instruments (including voice)
vibrate harmonically. Many cultures use plucked stringed instruments or
various kind of idiophones. Saying that people pick the scales from
mathematical abstractions which are not present to their ears seems
unlikely to be true.
Also, in my experience playing hindustani music and central javanese
gamelan,
intonation is always a matter of lively debate. Exactly how small or
large
an interval should be is often decided by the player on the spot.
--
David F. Place
President -- Enhanced Intuition, mailto:dpl...@entuit.com
http://www.shore.net/~dplace/
Albert Aribaud
Hi Matt and all of you people out (or is it in ?) there,
On 27 Apr 1998 17:08:43 GMT, fie...@zip.eecs.umich.edu (Matt) wrote:
>The question was what integer power of 3 is an integer power of 2 and
>the answer is none. Corrolary: no matter how many Pythagorian-tuned
>fifths you stack, you'll never exactly match frequency ratios with
>compounded octaves, though you may get as close as you like.
I take this opportunity to ask a side question about that.
I'm perfectly convinced that stacks of fifths (3:2 ratios) and octaves
(2:1) will never have any common frequencies (except, of course, the
fundamental).
What I'm wondering about is: why would one want them to have any? Why
is it desirable that stacked fifth and stacked octaves share a common
tone?
Albert Aribaud.
Albert Aribaud
On 27 Apr 1998 22:10:46 GMT, mi...@strategis.ic.gc.ca (Mike Dusseault)
wrote:
>The fourth makes perfect sense to me when you consider the harmonic
>series. The tonic is part of the fourth's harmonic series, rather
>than the other way around. That's how I hear it anyways (I think).
>
>Let's see...
>
>Take a C major scale. The tonic, C0, would contain as it's first harmonics
>C1, G1, C2, E2, G2 or the octave, fifth, another octave, third and fifth.
>The fourth would be F0. That note's first few harmonics would be
>F1, C1, F2, A3, C3. Note that C figures as F's 5th. I think that is
>the fundamental relationship of a tonic to it's fourth.
>
>We could do the same for the sixth. The sixth of C is A. Let's see
>what the harmonic series is for A: A1, E1, A2, C#2, E2, G2, A3... still
>no C, only a C sharp. Not only do the harmonics not coincide, at least
>early on, but now we have a C#2 in the series for F that would create
>a physiologically dissonant interval of a minor second with C. To me,
>that covers a lot of the character of the sixth interval.
>
>I don't think we always have to get everything to "line up" perfectly.
>Sometimes it's the imperfections that make it interesting. The
>interactions between the overtones of two notes affects greatly the
>sound and color of the interval.
>
>So in response, I don't see why we can't consider the fourth's harmonic
>series to learn why it works as it does, at least when considering the
>fourth and how it sounds against the tonic.
BTW, it looks like the harmonic series approach would consist in
constructing the scale from *one tone only* and its harmonics... But
when considering the scale from a harmonic series viewpoint, shouldn't
we consider harmonic series of *all* tones in the scale, rather than
only one?
Not even two $.02, I guess.
Albert Aribaud.
Matt
In article <3546da09...@news.easynet.co.uk>,
Albert Aribaud <ari...@easynet.fr> wrote:
>What I'm wondering about is: why would one want them to have any? Why
>is it desirable that stacked fifth and stacked octaves share a common
>tone?
I guess, were it true, it would lend the support of mathematics and
physics to enharmonicism, which provides some neat alternative paths
through western classical music, especially since the time of Chopin.
ghost
In article <3548dbb9...@news.easynet.co.uk> ari...@easynet.fr (Albert Aribaud) writes:
>BTW, it looks like the harmonic series approach would consist in
>constructing the scale from *one tone only* and its harmonics... But
>when considering the scale from a harmonic series viewpoint, shouldn't
>we consider harmonic series of *all* tones in the scale, rather than
>only one?
You *are* doing that when you decide that some tones in a scale
"go togther" better than a differently-chosen subset, whether you've
consciously analyzed the higher harmonics of that subset of tones or not.
On some level, you'll made preference-choices for grouping tones
with lots of shared harmonics; you'll also make personal preferences
known by *which* higher harmonics you like your grouped-tones to share.
(Very few subsets are likely to be entirely consonant througout their
harmonic series.)
ghost
>The existence of an infinite number of overtones is only mathematical.
>For musical purposes, can't we say that inaudible equals nonexistence?
Nah; if you happen on one of these inaudible higher-series harmonics
& *make* it audible (by playing it), you'll recognize it belongs in the
group.
>Also, only bowed strings and wind instruments (including voice)
>vibrate harmonically. Many cultures use plucked stringed instruments or
You can get a plucked instrument to vibrate pretty harmonically
(if you want to).
>various kind of idiophones. Saying that people pick the scales from
>mathematical abstractions which are not present to their ears seems
>unlikely to be true.
Like I said above, they pick the scales from things that become present
to their ears as the result of experimenting in general with sound,
not just from causing a single note to vibrate & trying to hear the harmonics
in it. They don't have to know anything about the math to be able to hear
what fits in when it presents itself, & what *doesn't* fit in when *it*
presents itself.
>Also, in my experience playing hindustani music and central javanese
>gamelan,
>intonation is always a matter of lively debate. Exactly how small or
>large
>an interval should be is often decided by the player on the spot.
Yeah, but there are things they would *never* choose in that type of music.
And "exactly how small or large an interval should be is often a subject of
lively debate, & decided on the spot by the player" in various forms of
more-western-sounding music, too.
That's my point; an infinite number of choices that fit, but also an
infinite number of choices that *don't* fit. You have to have some
pattern in your head to go by to make a choice. You *don't* need to be
doing mathematical calculations; you need to hear it.
I have to say I'm very wary of reports of "playing music like hindustani music
& central Javanese gamelan" being given by people who didn't grow up immersed
in the cultures that play these things. You tend to focus
on the wrong issues.
Marc Sabatella
>I'm perfectly convinced that stacks of fifths (3:2 ratios) and octaves
>(2:1) will never have any common frequencies (except, of course, the
>fundamental).
>
>What I'm wondering about is: why would one want them to have any? Why
>is it desirable that stacked fifth and stacked octaves share a common
>tone?
If they had a common tone - perhaps, after "N" cycles of fifths - then
we would know once and for all that "N" was the "proper" number of
divisions to split an octave into. Or, stated another way, equal
"temperament" would rule the world in peace and, yes, harmony.
--------------
Marc Sabatella
ma...@outsideshore.com
"The Outside Shore"
A Jazz Improvisation Primer, Scores, Sounds, & More:
http://www.outsideshore.com/
John Chalmers
For a thorough discussion of the relation of timbre to scale, see William
Sethares's new book from Springer called something like Scale,
Spectrum,Timbre. It builds on the J.R. Pierce study mentioned and uses
Plomp and Levelt's theory to compute dissonance of arbitrary intervals
with arbitrary spectra. A CD comes with it too.
--John
Mike Dusseault
In article <6i34mo$44g$1...@news.eecs.umich.edu>,
fie...@zip.eecs.umich.edu (Matt) writes:
> In article <6i2vp6$ibe$2...@news.crc.ca>,
> Mike Dusseault <mi...@strategis.ic.gc.ca> wrote:
>
>>series. The tonic is part of the fourth's harmonic series, rather
>>than the other way around. That's how I hear it anyways (I think).
>>
>>Let's see...
>>
>
> important extra-harmonic use, the rhetorical function of calling into
> question the role of the tonic. This is why you see cycles of the form
> I-IV-V-I in classical music: the tonic is stated, then called into
> question by motion I-IV, which can be interpreted as V-I in the
> subdominant key, and just when you start wondering what's going on,
> you get to V, which then can be held out almost indefinitely, so long
> as it is followed by I. The corresponding retrograde cycle found in
> much Rock, I-V-IV-I, is usually not extended beyond a few seconds, and
> if you try extending the IV chord much longer, you may well find--as I
> do-- that you've been conditioned to accept that as V-II-I---(what's
> this added V on the end?) in the key of the subdominant.
I hadn't looked at it that way. That makes it much clearer for me.
> So, I'm saying that harmonic relationships play a role in this way of
> hearing things, but conventions and interdisciplinary combinations
> like harmony with rhetoric, harmony with drama, and harmony with
> semiotics provide important clues to what's going on that you can't
> get just looking at the harmonic series.
I completely agree. That's one of the things I love about music.
>>I think the harmonic series does not dictate the scale to us, but
>>rather dictates how intervals will sound.
>
> Up to a point. Does a pile of perfect 16/15 minor seconds sound restful
> to you? Is there a context in which it does?
Making the assumption that a physiologically dissonant interval cannot be
restful, I would say no. You would always get two tones within critical
bandwidths of each other that would both interfere with each other.
As for harmonic movement, I'm not sure.
> or 10/9 or 9/8 whole-steps:
> when you introduce those between the fourth and the fifth of a scale,
> does V7 lose its sense of forward momentum? I don't think we can
> afford to overlook the learned component of how intervals sound.
True. As with most things in music, the higher levels of the brain
interpret just about everything, which could color one's perception
of an interval sound, I think. For example, it helps explain why we
seem to have more tolerance for dissonance today than before.
> There's a long tradition of passing off consonance on harmonic alignment...
> yet I know that I can enjoy music in major modes on carillon bells
> (they have minor-mode partials!) and music in minor modes on flutes
> (they come about as close as any natural instrument to pure harmonic spectra).
Interesting thoughts. There certainly is a lot to think about. Thanks.
M. Schulter
K C Moore <k...@hpsl.demon.co.uk> wrote:
: There is also no perfect scale of harmonic timbres because of annoying
: arithmetic like (4/3)*(5/4) <> (9/8)*(5/4) and there is no perfect
: equal tempered scale of harmonic timbres because 2^(p/q) <> r/s, where
: p, q, r and s are integers, and p<q. However, see J R Pierce,
: "Attaining consonance in arbitrary scales", Journal of the Acoustical
: Society of America 40 (1968), for perfect scales in synthesized
: non-harmonic timbres.
Hello, there.
Query: might the intended example be that 4:3 * 5:4 (4 + M3, 498 + 386
cents, or an M6 of 5:3 or 884 cents) is not equal to 3:2 * 9:8 (5 + M2,
702 + 204 cents, or a Pythagorean M6 of 27:16 or 906 cents)?
(This problem doesn't seem to me to apply to medieval Pythagorean tuning,
since then our first alternative becomes 4:3 * 81:64 (4 + Pythagorean M3),
which nicely yields 324:192 = 27:16, the same result as 3:2 * 9:8. In
fact, M6 is in this period often called _tonus cum diapente_ or
"whole-tone with fifth." However, if we are going after just thirds and
sixths, as your example suggests, then we do, of course, have an
"anomaly" here.)
One solution for this specific case, if our goal is tertian just
intonation, is the 16th-century one (at least in theory) of substituting a
10:9 M2 in the second case, yielding 3:2 * 10:9 = 30:18 = 5:3 (702 + 182,
or 884 cents).
I realize that this is a fine point in the more general discussion, of
course.
Most respectfully,
Margo Schulter
msch...@value.net
M. Schulter
Matt <fie...@zip.eecs.umich.edu> wrote:
: It just occured to me to consider the minor triad in terms of
: the harmonic series of the subdominant. If you take 6th, 7th, and 9th
: partials together, you get a triad with a Pythagorian 5th, and the
: oft-overlooked ratio 7/6 for the third (we tend to use 6/5 for the minor
: third, i.e. to truly invert the just major triad). This whets my appetite
: to try that out in Csound and see whether my ears will accept 7/6 as any
: form of minor third at all.
Hello, there, and following the Pythagorean stereotype, my reaction to
this was not to pick up an instrument, but to try GNU Emacs and calculate
my cents <grin>.
Anyway, I get your 7:6 "m3" at about 267 cents, and your 9:7 "M3" at about
435 cents. By comparison, the Pythagorean versions of these intervals
(which nonmedievalists sometimes find less than ideal) are 32:27 (294
cents) and 81:64 (408 cents) respectively.
Thus I'm not sure if the intervals you suggest would be convincing
"allophones" of m3 and M3, but they're respectively 49 cents (a tidge more
than two 22-cent commas) smaller than 6:5 (318 cents) and 49 cents larger
than 5:4 (386 cents).
What I'd suspect is that "7-limit Just Intonation" fans might like these
sonorities a great deal, but that they would be "different," just the
"neutral third" at somewhere near 350 cents is also different.