Why often 12? -- Moments of Symmetry
Margo Schulter
respond to a recent thread on the general topic of "Why 12 notes per
octave on a fixed-pitch instrument such as a keyboard?" -- which I'd
revise to, "Why often a number such as 5, 7, 12, 17, 19, 24, or 31,
for example?"
First of all, please let me emphasize that 12 is merely one possibility:
on a microtunable synthesizer with two standard 12-note keyboards, my most
common tuning sizes are 12, 17, or 24 notes per octave, with 24 usually
representing "two times 12," that is, two identically arranged 12-note
tunings at some convenient distance apart.
However, 19 and 31 are also important keyboard sizes for meantone systems
in the 16th-17th century era, and also to a degree in the 20th-century
tradition, with Joseph Yasser's advocacy of 19-tone equal temperament and
Adriaan Fokker's of 31-tone equal temperament for his famous organ, the
latter system the source of an interesting performance tradition in the
Netherlands.
In response to some comments in the earlier thread, I would respectfully
caution (as did some people there) that what the theorist Ervin Wilson has
developed as the "Moment of Symmetry" (MOS) concept does not necessarily
imply a circulating tuning, much less an equal temperament! It can apply
to Pythagorean intonation, or meantone, or some modern eventone system
with fifths a bit wider than pure. What it does tend to imply is a
"regular" tuning based on a single nonoctaval generating interval, usually
taken in the European tradition (and some others) to be what is termed in
Greek a diapente or diatessaron, and in English as a fifth or fourth, at
or rather close to the pure ratios of 3:2 and 4:3 respectively. By the
way, the Greek terms mean "through five" and "through four," showing that
some terminology at least has certain connections through lots of musical
and cultural changes.
The basic quality of a Moment of Symmetry or MOS scale or system, as
presented by Wilson, is that there are only two different sizes of
adjacent steps -- with these two sizes themselves identical in the special
case of an equal temperament. Let us consider a few examples.
Here I'll take Pythagorean intonation as a basis, since the pure fifths
and fourths permit the special case of an MOS system which is also a just
intonation system -- that is, with all intervals based on integer ratios,
and with the suggestion that at least some of those ratios are relatively
simple ones deemed "concordant" (here the 2:1 octave, 3:2 fifth, 4:3
fourth, and also to some degree, if we follow for example Jacobus of Liege
around 1325, the 9:8 whole-tone or major second).
If we tune a chain of five notes in pure fifths, making adjustments to
keep our notes within a given desired octave, and add the pure 2:1 octave
above the original note, we get a pentatonic system like this, with the
notes generated in the order F-C-G-D-A, and F an octave higher then added:
s s L s L
F G A C D F
1/1 9/8 81/64 3/2 27/16 2/1
0 204 408 702 906 1200
9:8 9:8 32:27 9:8 32:27
204 294 294 204 294
Here I've used one common MOS notation: since there are only two sizes of
adjacent steps, we can the small step "s" and the large step "L." Our
small step is the 9:8 whole tone at around 204 cents, and the large step
what we'd call in European terminology a minor third.
Suppose we add more notes? With our fifth generating interval of a pure
3:2 fifth, we get F-C-G-D-A-E, giving us the new type of step E-F, a
Pythagorean diatonic semitone or limma at 256:243. Adding one additional
note -- seven in all -- gives us a new kind of MOS:
L L L s L L s
F G A B C D E F
1/1 9/8 81/64 729/512 3/2 27/16 243/128 2/1
0 204 408 612 702 906 1110 1200
9:8 9:8 9:8 256:243 9:8 9:8 256:243
204 204 204 90 204 204 90
Here we again have two step sizes, an L of 9:8, and an s of 256:243, or a
rounded 204 cents and 90 cents respectively. As it happens, this makes an
excellent melodic scale, and also yields a fine set of vertical intervals
for lots of musical purposes. It is a usual medieval European Pythagorean
diatonic scale: the rather complex ratios for major and minor thirds
(81:64 and 32:27, for example) nicely resolve to stable intervals such as
fifths and fourths. There are also three-voice and four-voice sonorities
available like the pure 6:8:9 (e.g. C-F-G) or 4:6:9 (e.g. G-D-A) which
likewise provide a stimulating element of "imperfect concord" -- unstable
but relatively blending -- in sophisticated 13th-14th century technique.
It should be noted that a 7-note diatonic scale doesn't necessarily have
to be an MOS: we are talking about one common tendency in tunings, not
_the_ formula for an acceptable or useful scale in a given cultural
tradition. For example, consider a beautiful tuning of the Greek theorist
Archytas, which I'll give in a form I often use (sometimes with certain
temperings, I admit), a version of the medieval European Dorian mode on
D-D:
D E F G A B C D
1/1 9/8 7/6 4/3 3/2 27/16 7/4 2/1
0 204 267 498 702 906 969 1200
9:8 28:27 8:7 9:8 9:8 28:27 8:7
204 63 231 204 204 63 231
While Pythagorean intonation is one kind of just tuning, the term "just
intonation" most commonly tends to imply a system like this exquisite
example, with some characteristic traits. First, many or often (as here)
all of the steps of the system have superparticular (n+1:n) ratios, here
8:7, 9:8, or 28:27.
Secondly, there are typically pure vertical intervals of at least a few
different types with rather simple ratios, often superparticular, here 3:2
fifths, 4:3 fourths, 9:8 whole-tones, 7:6 minor thirds, 9:7 major thirds,
and 7:4 minor sevenths. While 3:2, 4:3, and 9:8 also occur in Pythagorean
tuning, the pure 7:6 and 9:7 thirds and 7:4 minor sevenths (along with the
8:7 whole-tones) are not obtainable from any chain of pure fifths alone
(although, curiously, a chain carried far enough can yield some very close
approximations).
Third, this mixture of different prime factors -- here 2-3-7 -- produces
some interesting variations and sometimes "complications" in a tuning
system, which can range from academic to charming to daunting depending on
the musical style in question.
Let us first consider the beauties of the Archytas diatonic tuning.
Melodically, the 28:27 semitones, sometimes described as "thirdtones"
since they are equal to something like a third of a 9:8 whole-tone
(actually a bit smaller), are even more economical or incisive than the
fine Pythagorean semitones at 256:243. Vertically, if one performs in this
scale above a drone (for example the fifth D-A at a pure 3:2), one gets
not only a pure 3:2 fifth, 4:3 fourth, and 9:8 whole-tone as in
Pythagorean, but also a 7:6 minor third and 7:4 minor seventh, often very
prominent notes in a 12th-13th century European Dorian melody, for
example. The scale thus seems to optimize certain melodic and vertical
elements.
Further, the contrast in the common figure B-C-D between the narrow 28:27
semitone B-C and the extra-wide 8:7 whole tone C-D, which I find pleasing
and might describe as "stately," can give a charming touch to an already
engaging modal character.
As long as we are playing or improvising melody above a drone, this scale
seems ideal -- and _is_ one ideal choice for those with my inclinations,
whatever one might say about the probability of anyone in 13th-century
Europe using or approximating such a tuning.
For discant or polyphony, using only these seven notes, we face the
complication that C-G is not a 3:2 fifth, but a "fifth" at 32:21, larger
by an interval which George Secor has termed the comma of Archytas at
64:63 (~27.26 cents). In usual European types of harmonic timbres, it is
unlikely that one would confuse a 32:21 with a pure 3:2 -- the former will
beat very heavily, and in short present a kind of "Wolf" fifth interval,
that is, an interval whose beating might suggest "the howling of wolves."
For a conventional polyphonic composition or improvisation in 13th-century
style using _only_ these seven notes, this is a significant navigational
hazard, since the fifth C-G is very likely to come up in lots of pieces.
In other settings, however, a "problem" can become a high musical
opportunity: a sonority like G-C-D-F, or 16:21:24:28 (~0-471-702-969
cents), can be heard as one of the most "way cool" things in music,
featuring a narrow 21:16 fourth (the octave complement of the 32:21
fifth). My most typical idiom for this sonority involves a resolution to
the fifth A-E, with the outer 7:4 minor seventh contracting to a fifth and
the upper 7:6 minor third to a unison. This is the result of mixing a
favorite sonority of the modern composer and performer LaMonte Young with
a bit of 13th-14th century voice-leading.
Anyway, let's note again that this is _not_ an MOS system, because we have
two sizes of whole-tones at 8:7 and 9:8, as well as the 28:27 semitone.
The difference in size between 9:8 and 8:7 is equal to the 64:63 comma,
and this difference applies to some other related sizes of intervals as
well: for example, the minor thirds at 7:6 (e.g. D-F) or 32:27 (e.g. E-G).
Some people consider different sizes for whole-tones an expressive
virtue, and others a less desirable complication -- so that the contrast
between an MOS and a complex just intonation or JI tuning like this one
can attract proponents of either "side" of the question.
While our Archytas tuning isn't an MOS, it does have an important element
of symmetry: the two symmetrical (identically arranged) tetrachords or
divisions of the 4:3 fourth at D-E-F-G and A-B-C-D, with 9:8-28:27-8:7
steps within each tetrachord, and a connecting 9:8 whole-tone at G-A (the
difference between the 3:2 fifth and 4:3 fourth).
This kind of non-MOS diatonic scale is especially popular with Ptolemy,
who seems generally to favor tuning systems with as many superparticular
steps as possible. Here's his "Equable Diatonic," based on ratios of
2-3-5-11, following the Greek convention of placing the smallest interval
of a tetrachord first:
E F* G A B C* D E
1/1 12/11 6/5 4/3 3/2 18/11 9/5 2/1
0 151 316 498 702 853 1018 1200
12:11 11:10 10:9 9:8 12:11 11:10 10:9
151 165 182 204 151 165 182
This scale has all superparticular steps, like the Archytas diatonic: here
what we might call neutral seconds at 12:11 and 11:10, or around 151 and
165 cents, with the notations F* and C* used to show steps such as E-F*
and B-C*, somewhere between a minor and major second, and also the small
10:9 whole-tone at around 182 cents, as well as the 9:8 step connecting
the two tetrachords. Each tetrachord is arranged 12:11:10:9.
This arrangement produces what we might call minor thirds at 6:5 or a
rounded 316 cents (12:11 + 11:10); neutral thirds at 11:9 or about 347
cents (11:10 + 10:9); and a 5:4 major third at around 386 cents (10:9 +
9:8). Some liturgical music in the Byzantine tradition is said to favor
this kind of tuning system, and neutral seconds and thirds are very common
in a medieval Near Eastern tradition which often draws on the Greek
theorists.
As it happens, neither Pythagorean tuning nor the two more complex JI
systems we have just looked at fit the specific musical agenda of the 16th
century in Europe, but another tuning system of Ptolemy, his "Syntonic
Diatonic," did, at least more or less:
E F G A B C D E
1/1 16/15 6/5 4/3 3/2 8/5 9/5 2/1
0 112 316 498 702 814 1018 1200
16:15 9:8 10:9 9:8 16:15 9:8 10:9
112 204 182 204 112 204 182
This could be a version of a Renaissance Phrygian mode (E-E) as it might
be played on a specially designed keyboard (likely to have extra notes to
solve certain problems in polyphonic music like the fifth D-A at 40:27 or
about 680 cents, narrow of a pure 2:1 by 81:80 or about 21.51 cents), or
possibly approximated by some singers (again, making adjustments to
maintain vertical concords such as the fifth D-A in polyphony). We have
all superparticular steps -- two tetrachords of 16:15-9:8-10:9, connected
by the usual 9:8 whole-tone. As it happens, this version agrees with
Zarlino's convention (1558) that the small 10:9 whole-tone should be
placed at D-E and G-A -- subject to modification when interval such a pure
fifth D-A are needed.
The special charm of this tuning, from a 16th-century perspective, is that
it yields pure thirds at the then favorite ratios of 5:4 and 6:5. In that
era, a system with factors of 2-3-5 neatly fit prevailing tastes, and for
many people following traditions based on Renaissance-Romantic
European music "just intonation" tends to imply specifically 2-3-5, rather
than the 2-3-7 of the Archytas diatonic or the 2-3-5-11 of the Ptolemy
equable diatonic, for example. From a 21st-century perspective, each
tuning has its own utility, beauty, and actual or potential musical
constituency.
While complex JI systems are indeed possible on keyboards, the
complications often tend to make musicians in the European tradition
prefer MOS keyboard tunings such as Pythagorean, meantone (fifths narrower
than pure), or eventone (like meantone, but with fifths wider than pure).
Such tunings, if sufficiently large, can _approximate_ complex JI systems
while maintaining a regular chain of fifths -- or possibly, for example,
two such chains at some convenient distance apart.
With any of these types of regular systems -- assuming that the fifths are
not too far from a pure 3:2 -- the next MOS after 5 and 7 is 12, a point
reached for medieval European keyboards (generally tuned in Pythagorean)
sometime around the early to middle 14th century. Around 1325, Jacobus of
Liege tells us that the whole-tones on keyboards are "almost everywhere"
divided into two unequal semitones, and the Robertsbridge Codex (dating
somewhere between 1325 and 1365, if we take different musicological
suggestions) has music using 12 steps and nicely fitting an Eb-G#
keyboard:
114 318 498 816 996
2187/2048 19683/16384 4/3 6561/4096 16/9
F# G# Bb C# Eb
F G A B C D E F
1/1 9/8 81/64 729/512 3/2 27/16 243/128 2/1
0 204 408 612 702 906 1110 1200
In this tuning _every_ diatonic whole-tone is divided into the two
semitones mentioned by Jacobus: e.g. F-F#-G, with a 2187:2048 chromatic
semitone or apotome F-F# at around 114 cents plus the usual 90-cent
diatonic semitone or limma F#-G. Indeed there are only two kind of
adjacent steps, a large or L at 2187:2048 and a small or s at 256:243. As
with the 5-note and 7-note Pythagorean tunings, we have a just system
which is also an MOS.
As it happens, in Renaissance meantone (with fifths tempered a bit
narrower than pure in order to achieve or approximate regular major and
minor thirds at 5:4 and 6:5), 12 is also an MOS, but here with the
diatonic semitone _larger_ than the chromatic semitone. This was
convenient for keyboard designers around the late 15th century, when the
Pythagorean to meantone transition likely took root: the same 12-note
arrangement more or less "standard" for the old kind of tuning system
could equally fit the new.
However, if we wish to go beyond 12 notes, then each system has its own
characteristic MOS patterns. For Pythagorean, the next stop after 12 is
17, where each diatonic whole-tone such as G-A gets divided into _three_
steps, e.g. G-Ab-G#-A, with G-Ab and G#-A as usual 90-cent limmas or
diatonic semitones, and Ab-G# a new kind of much smaller interval known as
the Pythagorean comma (531441:524288, ~23.46 cents).
In 1413, the Italian musician and theorist Prosdocimus advocated a 17-note
Pythagorean tuning (Gb-A#) of this kind, and noted that the accidental A#
did not seem to have come into use in the practical repertory. Pieces of
the period do use tuning sets of up to 15 notes (e.g. Gb-G#), and a
17-note Pythagorean keyboard would accommodate such pieces as well as
giving some creative intonational choices for realizing certain pieces in
an early 15th-century style (e.g. regular Pythagorean major thirds in a
standard cadential progression like M3-5, but possibly some diminished
fourths at 8192:6561 (~384.36 cents) very close to 5:4 (~386.31 cents),
for some prolonged noncadential sonorities).
Around 1425-1440, Ugolino of Orvieto specifically advocates a 17-note
Pythagorean organ, and recognizes what we might term the "MOS" properties
of this system with each whole-tone divided into limma-comma-limma.
For Renaissance meantone, however, the next MOS after 12 is 19, and we
find that 19 is a rather popular size for a harpsichord in Naples around
1600, the milieu of Gesualdo. Here the MOS mechanics are a bit different
because the diatonic semitone is _larger_ than the chromatic semitone.
Thus in addition to each of five whole-tones getting divided into three
steps, e.g. G-G#-Ab-A, here with the chromatic semitones G-G# and Ab-A
_smaller_ than the diatonic semitones G-Ab and G#-A, the two diatonic
semitones E-F and B-C also get subdivided E-E#-F and B-B#-C. The MOS steps
are here the _chromatic_ semitone (in meantone the _small_ semitone) and a
step corresponding to the Pythagorean comma but considerably larger in a
characteristic Renaissance meantone, e.g. G#-Ab, E#-F, or B#-C, most often
called a _diesis_.
In 1/4-comma temperament with pure 5:4 major thirds, these MOS steps are
about 76 cents (small or chromatic semitone, ~76.05 cents) and 41 cents
(the 128:125 diesis, ~41.06 cents). The latter is, from a receptive
melodic point of view such as Vicentino's, an intriguing "fifthtone" step
with certain affinities to Greek tunings of the "enharmonic" type dividing
a semitone into two more or less equal dieses. Thus each whole-tone of the
7-note MOS from which our 19-note system can be derived gets divided L-s-L
(e.g. G-G#-Ab-A), and each diatonic semitone of this same 7-note MOS L-s
(e.g. E-E#-F).
If we carry Pythagorean tuning further than 17, we get MOS conditions at
29, 41, and 53 -- the last, as recognized in ancient Chinese theory, an
almost mathematically perfect circle, since tuning 53 fifths up less 31
octaves yields a note _almost_ identical to the starting note. The
difference is a tiny comma of about 3.615 cents, later known in Europe as
the "comma of Mercator" after one theorist (related to the famous
cartographer, possibly his son?) of the 17th century interested in 53-note
systems.
Thus 53-note Pythagorean is a just intonation system and MOS which happens
also to be a very nice circulating tuning -- the minor differences of
3.615 cents between certain interval sizes should cause little
inconvenience either for a performance along traditional medieval European
lines, or for approximating some more complex JI systems such as the
Archytas diatonic (2-3-7) or Ptolemy's syntonic diatonic (2-3-5) shown
above.
With meantone, the next stop beyond 19 is 31, Vicentino's favorite, and in
1/4-comma a complete circulating system like 53-note Pythagorean. As
Vicentino puts it, every kind of interval is available from every step --
with the minor qualification, we might add, that some interval sizes vary
by roughly 6.07 cents, the difference between 31 fifths in 1/4-comma
meantone and 18 pure octaves.
Both Pythagorean 53 and 1/4-comma meantone 31 have attracted their share
of attention in the last century or a bit more, and one modification is to
use a precisely equal temperament with 53 or 31 identical steps per
octave. Some people prefer the mathematical symmetry of precisely
identical divisions of the octaves, while others relish the slight
asymmetries of the Pythagorean and 1/4-comma meantone versions (including,
in the 1/4-comma system, one "odd" fifth which is 6.07 cents wider than
the others at about 5.38 cents narrow, or in other words almost pure!).
In the 16th century, while the more modest (and technically less completed
on fixed-pitch keyboards) 19-note meantone MOS was not a circulating
system in 1/4-comma (this required 31, as favored by Vicentino or
Colonna), another tuning system did provide a circulating 19-note system:
1/3-comma meantone with pure 6:5 minor thirds, or the almost equivalent
19-tone equal temperament. The French composer and theorist Guillaume
Costeley described a division of the octave into 19 equal thirdtones in
1570, and wrote some chromatic music very artfully fitting this system,
while the Spanish theorist Francisco Salinas advocated 1/3-comma
temperament in 1577 as one choice even while recognizing the compromises
it involves for fifths and major thirds (both narrow by a bit more than 7
cents, more specifically about 7.17 cents in 1/3-comma and 7.22 cents in
19-equal, virtually identical systems).
While circulation does not generally become an important concern in
Western European music until around the late 17th century, when 12-note
well-temperaments come into vogue, it would seem that some 16th-century
musicians, as Mark Lindley concludes, may have favored 1/3-comma or
19-equal in order to get a smaller circulating system than the 31 notes
required in 1/4-comma.
However, a 19-tone tuning in 1/4-comma (Gb-E#) was also popular in the
Neapolitan milieu around 1600, where one composer gives the player a
choice between, if I recall, D#-F##-A# (available on a 31-note instrument,
but not a 19-note instrument in Gb-E#) or D#-F#-A# with the minor third
substituted for the major (accommodating a usual _cembalo cromatico_ or
"chromatic harpsichord" with 19 notes). On a 19-note instrument in
1/3-comma or 19-equal, F##=Gb and D#-F##-A# is equivalent to D#-Gb-A#, so
that one can play the major third without any problem. On a 19-note
keyboard in 1/4-comma, however, if I am correct, D#-Gb would actually be a
neutral third not too far from 11:9, interesting but not here the
composer's intention!
To conclude, MOS systems often seem to have a certain attraction, either
in the patterns of tuning systems or the design of fixed-pitch instruments
such as European keyboards -- 5, 7, 12, and larger sets such as 17, 29,
41, or 53 in Pythagorean (and also regular tunings with fifths a bit
larger than pure) or 19 and 31 in meantone.
Often I like the solution two identical 12-note MOS scales at some
convenient distance apart: this happens "naturally" in 24-note Pythagorean
intonation or 1/4-comma meantone, for example, where the interval between
the two identically patterned keyboards is the 531441:524888 comma or
128:125 diesis. Each keyboard provides a ready and familiar musical "map,"
while combining notes from the two keyboards permits one to do such things
as approximate complex JI systems or get a variety of special intervals
and effects.
Here my goal has been to offer a possible explanation, based on Erv
Wilson's concept of an MOS, as to why 12 is _one_ popular choice. I should
add that lots of non-MOS sets can also have great utility: for example,
the eight notes of the medieval European system of _musica recta_, Bb-B
(with B/Bb as a flexible step); and 13-16 note systems of a kind
implemented on some organs and harpsichord of the 15th-18th century era
with certain accidental keys "split" into two parts and notes (e.g. G# in
front, Ab in back).
Also, I hope that this article will invite more discussion on "Why 12 --
or 17 or 19 or 31, etc." and related questions such as different kinds of
JI systems or temperaments.
Most appreciatively,
Margo Schulter
msch...@value.net
Bob Pease
"Margo Schulter" <msch...@wopr.bestII.com> wrote in message
news:bhidtb$1a9k$1...@boredom.ennui.org...
> Hello, everyone, and now that I'm back on the Internet, I'd like to
> respond to a recent thread on the general topic of "Why 12 notes per
> octave on a fixed-pitch instrument such as a keyboard?" -- which I'd
> revise to, "Why often a number such as 5, 7, 12, 17, 19, 24, or 31,
> for example?"
Snip rest..
While you have done a nice job of telling about all of this,
I think you have made issue uncommonly complicated.
There are twelve notes in the scale because it forms a cycle ( slightly
compromised) on the cycle of natural overtones of natural instruments.
Bob Pease
J. B. Wood
<bobnosp...@concentric.net> wrote:
> There are twelve notes in the scale because it forms a cycle ( slightly
> compromised) on the cycle of natural overtones of natural instruments.
>
> Bob Pease
Hello, and just what is "the cycle of natural overtones of natural
instruments?" I'm lost. Didn't know that overtones came in cycles
although they do come in hertz ;-) Also, you might want to keep in mind
that when one sounds a C major triad (C-E-G), for example, on a polyphonic
musical instrument we are dealing with the aggregate effect of partials
(overtones) of all three notes, not just those partials represented by the
harmonic series of the root/tonic. IOW, in the real world we have to be
concerned about beats occurring among the partials of all three notes lest
the triad be unacceptable harmonically. Complicating this (and a big
reason why piano tuning is an art) are acoustic musical instruments that
generate partials that are not quite at integer multiples of the
fundamental pitch frequency. It is the trained ear of the tuner that can
bring all those beating partials into a range that results in consonance.
Granted, the ancient Greeks were melody oriented but if certain intervals
like just perfect fifths, fourths, and thirds sound pleasing harmonically,
they also represent pleasing melodic transitions (and probably
vice-versa).
As for Margo, I think she has provided the detailed discussion that many
of us have come to expect and respect. Sincerely
John Wood (Code 5550) e-mail: wo...@itd.nrl.navy.mil
Naval Research Laboratory
4555 Overlook Avenue, SW
Washington, DC 20375-5337
Margo Schulter
> While you have done a nice job of telling about all of this,
> I think you have made issue uncommonly complicated.
Hello, there, Bob, and please let me frankly acknowledge that your
response might reflect the reaction of lots of other readers.
Possibly it's a bit like physics. If one is in a situation where the
relevant velocities are taken as of insignificant magnitude in comparison
to the speed of light, then Newtonian mechanics is a simple and adequate
approximation. If those velocities become a substantial fraction of the
speed of light, however, then the more complicated Einsteinian relatively
becomes a necessity rather than a luxury if we wish valid results.
Thus in many 20th-century settings, taking an octave as divided into 12
equal semitones and counting intervals accordingly is a highly elegant and
useful approach. Similarly, in many Renaissance or "Xeno-Renaissance"
settings, taking an octave as divided into 31 more or less equal
fifthtones and counting accordingly is very helpful: Vicentino does it in
1555, and I do it now (e.g. neutral third in 31 equals 9/5-tone or 9
fifthtone steps).
In 17-tone equal temperament or well-temperament, similarly, one counts in
thirdtones: 5 for a neutral third for example. I'd say that 17 is my
favorite size for a circulating temperament, with George Secor's
well-temperament (1978) as a masterpiece. This reflects, of course, my
taste in thirds: I tend often to relate to 6:7:9, for example, where
others might prefer 4:5:6 or possibly 16:19:24. In a Renaissance or
Manneristic setting where I really do want 4:5:6, I tend to go with some
kind of meantone.
This is the musical universe I inhabit, and I try to make my theory --
actually borrowed from lots of other people, possibly with a few original
touches -- somewhat reflective of that reality. A given counting system
of the kind I've discussed above can be very useful for some tuning which
seems to be made up of more or less equal "units" of convenience: I'd be
less likely to use it for a just tuning (e.g. Pythagorean or the Archytas
diatonic I discussed in my post), or for some regular tunings where the
set of smaller steps seems a bit more complicated.
> There are twelve notes in the scale because it forms a cycle ( slightly
> compromised) on the cycle of natural overtones of natural instruments.
Here I would say that this is statement is a fair and reasonable statement
about why 12-tone well-temperament or equal-temperament are attractive
options for lots of music based on certain cyclical identities (e.g.
G#=Ab), and relying on more or less accurate approximations of _some_
interval types or sizes which can be taken as representing simple integer
ratios (certainly the 3:2 fifth and 4:3 fourth, and major and minor thirds
at ratios such as 4:5:6 or 16:19:24, the latter a sonority represented
quite accurately (0-300-700 cents, close to a just 0-298-702 cents).
However, what about the simple and very useful ratios of 7:6 (~266.87
cents) and 7:4 (~968.83 cents)? If that's what I'm looking for, or 9:7
major thirds (~435.05 cents), then a 12-note circle isn't a very likely
choice. Interestingly, if one really wants the "isotropic" qualities of
12-equal, then an excellent solution is a cycle of 36 notes, or three
12-equal circles spaced in intervals of 1/6-tone (33-1/3 cents). This
gives great ratios of 2-3-7, including a virtually just 7:6 and 12:7, two
of my favorite intervals.
Certainly I agree that 12-equal is a fine compromise for lots of purpose,
and not so much a compromise as the ideal and intended tuning for lots of
music involving 12-note serialism, for example, or late Romantic music
where the uniform color of all the keys is an important premise.
The way I'd describe this tuning is as a kind of compromise between
medieval Pythagorean and Renaissance meantone -- and a very useful one for
lots of purposes.
Possibly, also, we are addressing two different questions in this
dialogue. One of them might be, "Why is 12-equal or a 12-note
well-temperament a very attractive option for lots of relevant music, and
a more or less passable option for even more?"
The other question I might put this way, "Why are keyboard tunings of 12
notes, with each whole-tone among the diatonic keys divided into two
semitones of whatever size, such a popular option for European and related
musics, whether the tuning system is Pythagorean, meantone, or some kind
of circulating temperament?"
To put some dates on this: 12-note keyboards seem to have become standard
by around 1325-1365 or so, let's say (with G# possibly as the last note to
become routine), but circulating 12-note well-temperaments only seem to
have caught on by around 1700, with 12-note meantone keyboards remaining
quite popular for organs, for example, in some localities through the
mid-19th century. Today, a 12-note instrument might be tuned to any of
these systems, or also to an "eventone" system with fifths a bit wider
than pure (I often use 12-note tunings with fifths of this kind).
My point, historical and of practical utility today, is that there are
lots of useful 12-note keyboard tunings which are not "cycles" -- that is,
do not form a circulating system. In Pythagorean, meantone, or a
21st-century eventone, G#-Eb will get you something very different than
G#-D# or Ab-Eb, and I can tell you that the difference is quite audible.
Of course, I have an implicit agenda in my explanation: to let people know
that there are lots of keyboard tuning systems out there besides 12-equal,
and that I like them and consider them worth celebrating and
understanding.
This is not merely exposition, but indeed advocacy: I would emphasize that
my purpose is not to denigrate the value of 12-equal, but simply to say
that it is one option among many others.
Bob Pease
"Margo Schulter" <msch...@web1.calweb.com> wrote in message
news:3f3d6308$0$218$d36...@news.newshosting.com...
Thank you for a clear and polite response to my attempt to express a
concentrated explanation.
RJ P
Bob Pease
"J. B. Wood" <wo...@itd.nrl.navy.mil> wrote in message
news:wood-15080...@jbw-mac.itd.nrl.navy.mil...
> In article <bhiqqt$b...@dispatch.concentric.net>, "Bob Pease"
> <bobnosp...@concentric.net> wrote:
>
> > There are twelve notes in the scale because it forms a cycle ( slightly
> > compromised) on the cycle of natural overtones of natural instruments.
> >
> > Bob Pease
>
> Hello, and just what is "the cycle of natural overtones of natural
> instruments?" I'm lost. Didn't know that overtones came in cycles
> although they do come in hertz ;-)
Snip rest
The OP has dealt with these issues in a polite tone.
The original statement was carefully constructed.
Basically a cycle is anything that repeats itself.
This applies to names given to the overtones of NATURAL
instruments,
Which may or may not be integral ratios of some fundamental.
But the point of my post was to answer the issue that the OP's answer was
much too involved for the way the Question was phrased.
Mine was too terse, but does not imply ignorance of Basic Acoustics, as you
imply.
RJ Pease
Dan Williams
to our posts. Her articles are so thorough that I admit I fail to respond in
kind. However, I think your compromised cycle of natural overtones is
oversimplified. Since intervals are composed of prime numbers, by definition
there can never be a point at which say, the series of just fifths (powers
of 3) will merge with the series of fifths (powers of five). In even
temperament, the thirds are especially compromised and the JI sevenths are
not even close, so there is no uncomplicated answer that explains twelve
tones. Some gamelan instruments are tuned to 7 equal intervals between the
octave. There are no half steps or whole steps, so why don't we use 7 equal
steps instead of 12?
The 12-note equal temperament was not at all accepted by many composers and
music theorists for centuries and was a topic for virulent debate (i.e.
Zarlino vs. Galilei [Galileo's father]). It is something that was finally
settled upon as a convenient compromise among the schemes.
Dan Williams
> "Margo Schulter" <msch...@wopr.bestII.com> wrote in message
> news:bhidtb$1a9k$1...@boredom.ennui.org...
> > Hello, everyone, and now that I'm back on the Internet, I'd like to
> > respond to a recent thread on the general topic of "Why 12 notes per
> > octave on a fixed-pitch instrument such as a keyboard?" -- which I'd
> > revise to, "Why often a number such as 5, 7, 12, 17, 19, 24, or 31,
> > for example?"
> "Bob Pease" <bobnosp...@concentric.net> wrote in message
news:bhiqqt$b...@dispatch.concentric.net...
Bob Pease
"Dan Williams" <danmxwm...@bellsouth.net> wrote in message
news:DBs%a.1067$mE5...@fe05.atl2.webusenet.com...
> I think we owe a lot to Margo for her thoughtful and informative responses
> to our posts. Her articles are so thorough that I admit I fail to respond
in
> kind. However, I think your compromised cycle of natural overtones is
> oversimplified.
It isn't even well defined,
Sort of a skeleton guide to bonier things
> Since intervals are composed of prime numbers, by definition
HUH??
Several Music Dictionaries give
An Interval is the "Distance between two pitches"
There are several different definitions of the term "distance" in this
context
Please demonstrate what you mean.
> there can never be a point at which say, the series of just fifths (powers
> of 3) will merge with the series of fifths (powers of five).
Hmmm...
Are you saying that There are no integers p and q such that p^3 = q ^ 5 ??
This is true for primes. But the definition of Interval as a prime is
uncommon.
> In even
> temperament, the thirds are especially compromised and the JI sevenths are
> not even close, so there is no uncomplicated answer that explains twelve
> tones. Some gamelan instruments are tuned to 7 equal intervals between the
> octave. There are no half steps or whole steps, so why don't we use 7
equal
> steps instead of 12?
I have , at times.
> The 12-note equal temperament was not at all accepted by many composers
and
> music theorists for centuries and was a topic for virulent debate (i.e.
> Zarlino vs. Galilei [Galileo's father]). It is something that was finally
> settled upon as a convenient compromise among the schemes.
Yup
> Dan Williams
>
>
>
Dan Williams
HUH??
Sorry meant to say: JUST intervals.
John Chalmers
Dan wrote:
>Some gamelan instruments are tuned to 7 equal intervals between the
>octave. There are no half steps or whole steps, so why don't we use 7 equal
>steps instead of 12?
I don't know of any gamelan-type instruments in Indonesia tuned to anything
close to 7-tone equal temperament, though 9-tone has been tried and one finds 23
discussed theoretically. I'm told that some 12-tone gamelans were built for
Christian church music, but I don't have a reference.
Traditionally, each gamelan was tuned uniquely, but some tunings have become
common, i.e., that of Radio Jakarta (IIRC). In general there are two tunings: a
nearly equal 5-tone system called Slendro and an unequal 7, termed Pelog.
However, there are regional variations-- there is a Sundanese 9-tone unequal
system called Surupan Melog from which 5 and 7 tone scales are taken.
Thai, Cambodian (Khmer) and, I believe, Burmese (Myanmar) music uses a nearly
equal 7-tone scale on fixed pitch instruments, but vocal scales are quite
different. Nearly equal 7-tone is also found in Sub-Saharan Africa.
--John
Dan Williams
Ellis studied, although the Javanese 5-note scale equal scale is another
good example.
"John Chalmers" <jhcha...@ucsd.edu> wrote in message
news:3F3E7400...@ucsd.edu...
Dan Williams
??
> This is true for primes. But the definition of Interval as a prime is
> uncommon.
BTW, I am saying that there are no prime numbers raised to any (non-zero)
power will equal any other prime raised to any (non-zero) power.
In this case, for any positive integers x and y, 3^x will never = 5^y.
This is only to demonstrate that a succession of JI fifths can never derive
the same (octave-adjusted) pitch as a a succession of JI thirds.
Dr.Matt
Bob Pease <bobnosp...@concentric.net> wrote:
>
>"J. B. Wood" <wo...@itd.nrl.navy.mil> wrote in message
>news:wood-15080...@jbw-mac.itd.nrl.navy.mil...
>> In article <bhiqqt$b...@dispatch.concentric.net>, "Bob Pease"
>> <bobnosp...@concentric.net> wrote:
>>
>> > There are twelve notes in the scale because it forms a cycle ( slightly
>> > compromised) on the cycle of natural overtones of natural instruments.
>> >
>> > Bob Pease
>>
>> Hello, and just what is "the cycle of natural overtones of natural
>> instruments?" I'm lost. Didn't know that overtones came in cycles
>> although they do come in hertz ;-)
>
>Snip rest
>
>The OP has dealt with these issues in a polite tone.
>
>The original statement was carefully constructed.
>Basically a cycle is anything that repeats itself.
>This applies to names given to the overtones of NATURAL
>instruments,
>Which may or may not be integral ratios of some fundamental.
Natural instruments mostly make white noise (bang two rocks together...).
Instruments which make nearly-harmonic partials are all artifices--except
for voice.
>But the point of my post was to answer the issue that the OP's answer was
>much too involved for the way the Question was phrased.
>Mine was too terse, but does not imply ignorance of Basic Acoustics, as you
>imply.
As I'm sure you know, harmonics don't form closed cycles at all, but
you may quit adding harmonics when you get the set of sounds you
want-- and optionally, you may include near-circulation and some
compromises in your set of choices. This distinction is rather
important to make: cycles of 12 notes are artifices based on choices
and compromises, not something arising naturally without human
intervention. A lot of bogus philosophy is based on the mistaken
assumption that conventions like 12 notes per octave are handed us by
Mother Nature or The Gods--when really, as a species and a culture, we
deserve to pat ourselves on the back a whole lot more for having invented
them.
>RJ Pease
>
>
>
--
Matthew H. Fields http://personal.www.umich.edu/~fields
Music: Splendor in Sound
Brights have a naturalistic world-view. http://www.the-brights.net/
Dr.Matt
Bob Pease <bobnosp...@concentric.net> wrote:
>
>"Dan Williams" <danmxwm...@bellsouth.net> wrote in message
>news:DBs%a.1067$mE5...@fe05.atl2.webusenet.com...
>> I think we owe a lot to Margo for her thoughtful and informative responses
>> to our posts. Her articles are so thorough that I admit I fail to respond
>in
>> kind. However, I think your compromised cycle of natural overtones is
>> oversimplified.
>
>
>It isn't even well defined,
>Sort of a skeleton guide to bonier things
>
>
>> Since intervals are composed of prime numbers, by definition
>
>HUH??
>
>Several Music Dictionaries give
>An Interval is the "Distance between two pitches"
>
>There are several different definitions of the term "distance" in this
>context
> Please demonstrate what you mean.
He's referring to the frequency relationships involved in just
intonation using RATIOS of "small" integers. Most of Western harmony
is nominally 5-limit, meaning that consonances are built from ratios
of numbers with the factors 2, 3, and 5 only (in practice, the ratio
19:10 may fleetingly arise, but usually the ratio ends up being
15:8). Much of medieval music is based on 3-limit harmony, where
factors of 2 and 3 are the only factors allowed. Margo favors 7-limit
harmony without (or with only very rare) factors of 5, as a rich way
of making medieval music with an expanded pallette, somewhat supported
by some medieval sources.
>> there can never be a point at which say, the series of just fifths (powers
>> of 3) will merge with the series of fifths (powers of five).
>
>Hmmm...
>Are you saying that There are no integers p and q such that p^3 = q ^ 5 ??
> This is true for primes. But the definition of Interval as a prime is
>uncommon.
Ummm, there was a misunderstanding--see my previous paragraph. The definition
of just intervals as small ratios of a limited set of prime numbers is
actually quite common.
>> In even
>> temperament, the thirds are especially compromised and the JI sevenths are
>> not even close, so there is no uncomplicated answer that explains twelve
>> tones. Some gamelan instruments are tuned to 7 equal intervals between the
>> octave. There are no half steps or whole steps, so why don't we use 7
>equal
>> steps instead of 12?
>
>
>I have , at times.
Indeed, there's lots of great ways to go.
>
>> The 12-note equal temperament was not at all accepted by many composers
>and
>> music theorists for centuries and was a topic for virulent debate (i.e.
>> Zarlino vs. Galilei [Galileo's father]). It is something that was finally
>> settled upon as a convenient compromise among the schemes.
>
>
>Yup
Yup :)
Bob Pease
"Dr.Matt" <fie...@rygar.gpcc.itd.umich.edu> wrote in message
news:L4x%a.1782$H91....@news.itd.umich.edu...
Thanks for clarifying the "Ratio" business.
I suspected as much, as it didn't make much sense to define an interval as a
Prime Number.
Bob Pease
paramucho
God invented the octave, all else is the work of man.
(Well, not quite)
--
Ian
Impressive If Haughty - Q Magazine
Dan Williams
> > intonation using RATIOS of "small" integers. Most of Western harmony
> > is nominally 5-limit, meaning that consonances are built from ratios
> > of numbers with the factors 2, 3, and 5 only (in practice, the ratio
> > 19:10 may fleetingly arise, but usually the ratio ends up being
> > 15:8). Much of medieval music is based on 3-limit harmony, where
> > factors of 2 and 3 are the only factors allowed. Margo favors 7-limit
> > harmony without (or with only very rare) factors of 5, as a rich way
> > of making medieval music with an expanded pallette, somewhat supported
> > by some medieval sources.
> Thanks for clarifying the "Ratio" business.
> I suspected as much, as it didn't make much sense to define an interval as
a
> Prime Number.
>
> Bob Pease
Well, since Margo's whole post was about ratios, I kinda figured we knew
what we were talking about. I was referring to the building blocks. A
complete trip through the circle of just fifths (controlled by 3 or in this
case 3^12) won't come around again to the same tonic or octave (controlled
by 2). The ratio is 3^2 to 2^18 is 531441:262144 or about 2.027. And
stacking three just M3rds comes out way short of an octave. 5^3 to 2^6 is
125/64 or about 1.953. Since pitch degrees are adjusted freely at the
octave, 2 can be raised to any convenient power.
Interestingly, in 1519 Willaert wrote a motet that modulates wildly through
the 12 tones, expressly to demonstrate the problem of JI octaves, thirds,
and fifths. One in-tune interval would morph into another, leaving a singer
hanging on a sour note.
Thank you Dr. Matt for your usual candor.
Dan Williams
Jo Totland
out of curiosity:
How do you solve the practical problems of working with weird tunings, do
you simply remap your standard keyboard (I'm assuming electronic here, as I
guess acoustic would be even more trouble), or do you have special keyboards
with more regular (and probaby more compact) layout that would make more
sense in non-12-tone tunings? Or do you simply work with other instruments,
such as unfretted string instruments?
I assume some tunings are more "mapped" than others. Most people understand
12-tet pretty well. There are probably quite a few others that are well
understood, from folk-music and other compositions. But when working with a
new random tuning, how does one start to make sense of it? What kind of
theory would be useful in understanding them? Or do you simply work it all
out by ear? Do you choose a tuning to fit your musical idea, or choose a
neat tuning and try to come up with one?
Most microtonal music I've heard in say 17-tet or 31-tet have been pretty
experimental. While they can certainly sound pleasant, they aren't exactly
something you would whistle while walking down the street. Is this because
nobody is interested in writing simple catchy melodies in weird tunings, or
is it because they are incapable of it, or is it because I've just listened
to the weird stuff, or is it because my ear is not accustomed to it so I'm
incapable of hearing it properly?
Margo Schulter
> Thanks, very interesting stuff, both posts actually. Just a few questions
> out of curiosity:
Hello, there, and thank you for practical questions of a kind which can
make this kind of discussion a bit more concrete. I've found that a lot of
my theory has developed through practice: I run into some "surprise," and
than figure out how to describe it and possibly account for it or let it
take me in turn to another "discovery."
> How do you solve the practical problems of working with weird tunings, do
> you simply remap your standard keyboard (I'm assuming electronic here, as I
> guess acoustic would be even more trouble), or do you have special keyboards
> with more regular (and probaby more compact) layout that would make more
> sense in non-12-tone tunings? Or do you simply work with other instruments,
> such as unfretted string instruments?
This depends on the tuning system. I use a synthesizer with two
standard 12-note keyboards which can be "part-tuned" -- to get up to 24
notes per octave with each keyboard in a usual repeating 12-note octave
arrangement. This isn't necessarily ideal for everything: a 17-note
keyboard with split accidentals might sometimes be more convenient, and
with two of those I could have up to 34 notes per octave in regular
arrangements.
However, two regular 12-note keyboards are reasonably convenient for lots
of things. Often I tend to use a 24-note "regularized keyboard" with the
same pattern of steps and intervals on each keyboard, with a convenient
distance between the manuals. Also, I often use 17-note systems with the
more "usual" accidentals (Eb, Bb, F#, C#, G#) on the lower keyboard, and
the others (Db, D#, Gb, Ab, A#) on the upper keyboard.
With a more "unusual" tuning system like 13-tET, I come up with curious
solutions -- another story. However, for regular tunings of a more or less
"conventional" kind -- Pythagorean, meantone, or eventone -- it's usually
straightforward.
> I assume some tunings are more "mapped" than others. Most people understand
> 12-tet pretty well. There are probably quite a few others that are well
> understood, from folk-music and other compositions. But when working with a
> new random tuning, how does one start to make sense of it? What kind of
> theory would be useful in understanding them? Or do you simply work it all
> out by ear? Do you choose a tuning to fit your musical idea, or choose a
> neat tuning and try to come up with one?
The problem of a really "random" tuning is an interesting one, but I'd say
that for the most part, I have certain ready "maps" to start with on a
given tuning --and then find out possibilities to add to the map, and
apply to other similar or possibly "not so similar" tunings that I might
meet later.
For example, one of the first eventones I experimented with (regular
tunings with fifths typically around two cents _wider_ than pure) was what
I call the "e-based tuning," with a ratio between the large or chromatic
and small or diatonic semitone equal to Euler's e (~2.71828). That was
just a mathematical intuition: it was somewhere between 29-tET and 17-tET,
and might be fun.
Having tuned it up, I could play traditional progressions (for me, from
Perotin to Machaut or Solage is "classical," a usual and accustomed kind
of stylistic orientation) much as if the keyboard were in the
"historically correct" Pythagorean tuning -- no problem. The major thirds
are wider, of course, very close to 14:11 (~417.51 cents), and actually
not quite a cent wider, and the diatonic semitones rather narrower (~76.97
cents) -- so we have a kind of "accentuated Pythagorean."
Then came my big "discovery": I hit on a three-voice progression like
this, with C4 showing middle C:
Bb3 B3
F#3 E3
Eb3 E3
We might call this a "supraminor third" (around 341 cents) contracting to
a unison, and a "submajor third" (around 363 cents) expanding to a fifth,
with motion of the two outer voices by chromatic semitones of about 132.25
cents. A bit like Machaut -- and rather different. I had come up a neat
family of "supraminor/submajor" or "semineutral" thirds. Here they're
closer to neutral, and with fifths a bit more mildly tempered they are
closer to ratios of 17:14 and 21:17, which I've decided might be ideal to
define the central area of this general category.
Anyway, I played around with this tuning in 12 notes, but only at the end
of that summer realized something exciting: in a 24-note tuning, I'd get
excellent approximations of ratios of 2-3-7, with the 7:4 (15 fifths up,
fifths ~704.61 cents) virtually pure. I tried this, and found myself
playing a kind of progression I called "metachromatic."
A neat thing about this temperament is that it gives very good
approximations of 14:11 and 13:11, plus some supraminor/submajor thirds,
plus some near-2-3-7 thirds and sevenths, in a _regular_ 24-note tuning.
That means, for purposes of this new group, that I have a tuning with a
true E# and B# -- which I tend to call F* and C* (the * showing a note
raised by an enharmonic diesis or extra-small semitone of about 55.28
cents). This makes a neat cadential semitone, also.
Maybe for this question about more "weird" tunings, 13-tET would be more
like it. There, I started out by finding some timbres where the equivalent
of a "fifth," 8/13 octave or about 738 cents, could sound like a stable
interval. This is the kind of thing that John Chowning, Wendy Carlos, and
Bill Sethares have done, for example.
Then I was in for some surprises. The 1-step of about 92 cents makes a
fine semitone, just a tad larger than Pythagorean, and one "surprise" is
that the 5-step (about 462 cents) can represent either a stable interval
analogous to the fourth, or a cadential interval equivalent to a very
large major third! It's a fascinating "neo-Gothic" tuning because the
interval arithmetic "adds up differently" than in European conventions: I
had fun improvising the equivalent of a conductus around 1200, and finding
that "third-to-fifth" progressions were somehow similar, but different!
Anyway, often I tend to have a set of usual progressions, say
Renaissance-style counterpoint for meantone or Perotin-Solage for
Pythagorean or eventone -- but can modify or add things as I go along. As
a tuning gets more "unconventional," I must admit to often "translating"
familiar models to the new systems, which can mean some timbral
adjustments.
What I will admit is that two standard 12-note keyboards are not
necessarily ideal for a 17-note or 24-note tuning if you want quickly and
gracefully to negotiate scales mixing notes from the two keyboards. I
guess that I tend to take the regular chain of fifths on each keyboard as
the "norm," with the other possibilities as enriching.
In other words, if I tune two keyboards in a regular tuning at 704 cents
at some convenient distance apart, it probably means that I like the
regular intervals and progressions available on either keyboard as well as
the new categories of intervals available by mixing notes. For some people
who use certain kinds of tuning systems, the situation is a bit less
convenient -- another discussion.
> Most microtonal music I've heard in say 17-tet or 31-tet have been pretty
> experimental. While they can certainly sound pleasant, they aren't exactly
> something you would whistle while walking down the street. Is this because
> nobody is interested in writing simple catchy melodies in weird tunings, or
> is it because they are incapable of it, or is it because I've just listened
> to the weird stuff, or is it because my ear is not accustomed to it so I'm
> incapable of hearing it properly?
That's a good question. In a 1/4-comma meantone tuning with 24 notes
(almost identical to 24-out-of-31-tET), I tend to get "Renaissancy," with
some diesis shifts and neutral thirds and the like mixed in, as people
like Vicentino and Colonna do in their treatises. With 17, equal-tempered
or well-tempered, I often tend to follow a more-or-less-Gothic-like style,
but again taking advantage of things like neutral thirds or sevenths, or
possibly drawing on some medieval Near Eastern scales (which often map
reasonably well in some kind of 17).
In part, I suspect, it could be a matter of getting accustomed to an
unfamiliar tuning or style; in part, it could also be a matter of taste. I
must admit to having my own favorite "cliches," or riffs, and tending to
carry them into a lot of different tuning systems.
In a 17-note circle, for example, I very much like this kind of
progression:
C#4 D4
G3 A3
Eb3 D3
The augmented sixth, here a neutral seventh, expands to an octave with
each voice moving by a diatonic semitone -- about 70.59 cents in 17-equal,
and varying a bit in a well-temperament. To me it's a sort of cross
between Perotin -- or Machaut? -- and some curious French Impressionist
organ school for an alternative universe.
Anyway, this is just a first quick response -- and people who do use
generalized keyboards and the like could give another side of this
question.
Margo Schulter
> The 12-note equal temperament was not at all accepted by many composers and
> music theorists for centuries and was a topic for virulent debate (i.e.
> Zarlino vs. Galilei [Galileo's father]). It is something that was finally
> settled upon as a convenient compromise among the schemes.
>
> Dan Williams
Hello, there, and the experiments of Vincenzo Galilei with 12-note equal
temperament raise an interesting point sometimes discussed in this
newsgroup: the interplay between tuning and timbre.
In theory, Galilei indeed described 12-tET as the "perfect" tuning, and
regarded the use of this tuning on the lute as one of the perfections of
the instrument.
In practice, however, he concluded that the result was less ideal on a
harpsichord, where the difference in the material of the strings and the
more "vehement" sound production were possible factors. He concluded that
a meantone temperament of 2/7-comma, the system advocated by his former
teacher Zarlino, was best for the harpsichord, although he considered the
nonequivalence between notes such as C# and Db as an imperfection of this
solution.
In modern terms, we might say that the more prominent fifth partials of
the harpsichord made a major third at 400 cents more tense than on the
lute, and thus less suitable for a style where 5:4 (~386 cents) is the
ideal ratio.
Margo Schulter
> 15:8). Much of medieval music is based on 3-limit harmony, where
> factors of 2 and 3 are the only factors allowed. Margo favors 7-limit
> harmony without (or with only very rare) factors of 5, as a rich way
> of making medieval music with an expanded pallette, somewhat supported
> by some medieval sources.
Hello, there, Dr. Matt, and maybe I should explain why "somewhat
supported" is one way to put it -- it's a question of certain medieval
sources reporting that singers sometimes vary the sizes of their diatonic
semitone steps, for example, or giving some clues about the direction of
variation, but leaving open the precise ratios desired for intervals like
major thirds and sixths in directed or cadential progressions.
In 1318, Marchettus of Padua tells us that major thirds and sixths
involving sharps in directed progressions (major third to fifth, major
sixth to octave, with the voices proceeding by stepwise contrary motion)
should be adjusted so as "more closely to approach" the sizes of the
intervals toward which they seek resolution. This means intervals somewhat
wider than in a standard Pythagorean tuning -- but exactly how wide is a
matter of interpretation.
Personally I'd say that around 9:7 for a cadential major third, or 12:7
for a cadential major sixth, is "about right": this gives pure or
near-pure ratios like 7:9:12 for the common sonority with outer major
sixth, lower major third, and upper fourth. A small 7:6 minor third before
a unison is also neat, and I love this for some music in an Italian
trecento style, for example.
People reading the original text, however, might argue that Marchettus
could at least as easily be read to advocate something a bit more radical:
singing cadential major thirds and sixths at around 450 and 950 cents, or
about 13:10 and 26:15 if one wants to use integer ratios. It depends on
how, and how literally, one takes his division of the tone into "five
parts" -- are they geometrically equal (as in 29-note equal temperament?),
or an arithmetic division one could find on a monochord, or more of an
impressionistic guide for singers?
Anyway, if music history had developed a bit differently in Europe, a
theorist sometime around 1500 might have explained how something like
6:7:9 or 7:9:12 fits both vertical simplicity (relatively low integer
ratios for the concords) and directed efficiency (for 7:9:12 expanding to
2:3:4, for example, fine semitones around 28:27 or 63 cents, and also the
"closest approach" of Marchettus with a 9:7 third very nicely resolving to
3:2 or a 12:7 sixth to 2:1).
It's a bit of medieval theory reporting on one kind of vocal intonational
practice, a bit of interpretation, and a bit of making an alternative
history happen.
By the way, Christopher Page advocates this general kind of intonation for
some beautiful French and Italian music of the era, and urges that singers
be given an opportunity to perform without intervention of fixed-pitch
instruments so that they can free make these kinds of adjustments to
taste.
On a fixed-pitch keyboard, picking something near 9:7 or 12:7 or 7:6 is
one possible solution.
In fairness, I should also emphasize that ratios such as 7:6 are not a
standard part of period theory, except in certain more "academic" or
philosophical discussions. To Jacobus of Liege, a superparticular (n+1:n)
ratio like 7:6, or _sesquisexta_ in Latin ("again a sixth part," i.e.
7/6) might well have an agreeable effect on an instrument built to support
this ratio. However, he considers it impractical because it cannot be
derived from proper Pythagorean steps.
This is possibly not so different from the kind of comment one sometimes
reads in 20th-century sources: "The seventh partial, a kind of small minor
seventh, might be musically useful, but it is not supported by our system
of temperament dividing the octave into 12 equal parts."
Buster Mudd
>
> Natural instruments mostly make white noise (bang two rocks together...).
More often "clangorous timbres" (non-periodic waveforms w/
non-harmonically related overtones) than genuine "white noise" (equal
energy per frequency division). But your point is still valid.
> Instruments which make nearly-harmonic partials are all artifices--except
> for voice.
So true.
Graham Breed
>
>>Thanks, very interesting stuff, both posts actually. Just a few questions
>>out of curiosity:
As Margo's asked for a generalized keyboard, perspective, I'll supply one.
>>How do you solve the practical problems of working with weird tunings, do
>>you simply remap your standard keyboard (I'm assuming electronic here, as I
>>guess acoustic would be even more trouble), or do you have special keyboards
>>with more regular (and probaby more compact) layout that would make more
>>sense in non-12-tone tunings? Or do you simply work with other instruments,
>>such as unfretted string instruments?
As far as keyboards are concerned, I'm fully electronic. There's a lot
you can do with a conventional, re-mapped keyboard. Even a synthesizer
that only supports 12 note octave based tunings still has a lot of
flexibibility, the trick being knowing which 12 notes you want.
There are two extended mappings I've found particularly helpful.
Firstly, the schismic fourth, explained here
http://www.microtonal.co.uk/schv12.htm
It works with similar tunings to the eventones Margo typically works
with. You end up with a lot of notes (29 to the octave) but it has room
for a lot of subtlety. The main feature is that it fits the white/black
pattern of the keyboard.
More recently, I've been working with Miracle temperament, as explained at
http://www.microtonal.co.uk/miracle.htm
and it lends itself to a 24 note mapping
http://www.microtonal.co.uk/miracle/keyboard.html
As that only has 24 notes to an octave, it's less of a stretch than the
29 note one. There's still good scope for expression, and it's rairly
easy to remember the pattern of intervals. Compared to a 12 note
meantone, it is harder to play, but not that daunting. You have to give
up on the standard polyphonic technique because chords and melodies need
to be shared between both hands. But if you've either got friends or a
multitrack recorder, it's easier to play than most instruments.
I do now have a special keyboard, the Starr Labs Z6. See this link
under "ZTars"
It's designed as a MIDI guitar, but works well for microtonality because
it has a regular array of 144 small keys. It lends itself well to a
Miracle layout with 60 notes to the octave, but I've found it preferable
to squash that down to 30, like this (if you can follow my notation):
1^ 3^ 5^ 7^ 9^ 1^ 3^ 5^ 7^ 9^ 1^ ...
1 3 5 7 9 1 3 5 7 9 1 ...
1v 3v 5v 7v 9v 1v 3v 5v 7v 9v 1v ...
0^ 2^ 4^ 6^ 8^ 0^ 2^ 4^ 6^ 8^ 0^ ...
0 2 4 6 8 0 2 4 6 8 0 ...
0v 2v 4v 6v 8v 0v 2v 4v 6v 8v 0v ...
for a total of 4 and a bit octaves. It's good for experimentation, but
I still haven't done as much with it as a normal keyboard. So it isn't
like you have any excuse for not getting results because you don't have
fancy hardware.
Another mapping I've played with is for Mystery temperament, which I
don't think I've written up yet. For that, the tuning imitates 29
equally tempered frets to the octave, with strings tuned in alternating
neutral thirds of around 347 and 356 cents. It's a good approximation
to 15-limit harmony, and works well with a guitar layout, so I can play
it with triggers as if strumming a guitar. It would probably work well
for a real guitar if it were fretted correctly -- maybe even better
because the stretch won't be as great.
The idea is that three consecutive strings stopped on the same fret will
give a triad composed of two neutral thirds. If you sharpen the middle
note by one fret, you get some kind of major triad. There's a 50%
chance this will be an approximation to 4:5:6. If not, you can sharpen
the middle note again and get an approximation to a "car horn triad"
(Margo has a latin name for this) where the major third is 7:9.
Flattening the middle note of a neutral triad may give you a 5-limit
minor triad, and if not flattening it again will give an approximation
to 6:7:9.
Depending on which three strings you chose, you either get "utonal" or
"otonal" chords. For otonal, the neutral triad has lower neutral third
11:9, the major triad approximates 4:5:6 and the subminor triad 6:7:9.
For utonal, the lower neutral third approximates 16:13, the minor triad
approximates 10:12:15 and the supermajor triad is as I described above.
If you tune to 58-equal, the otonal/utonal distinction vanishes, and
everything's still in tune to the nearest 8.3 cents.
Although it's easy to find approximations to simple chords on this
layout, it's still cumbersome to work with. But 15-limit harmony in all
its majesty is like that, and there's no simpler way of getting it.
With serious study you could probably get the hang of it, but I'm
concentrating on Miracle for now.
>>I assume some tunings are more "mapped" than others. Most people understand
>>12-tet pretty well. There are probably quite a few others that are well
>>understood, from folk-music and other compositions. But when working with a
>>new random tuning, how does one start to make sense of it? What kind of
>>theory would be useful in understanding them? Or do you simply work it all
>>out by ear? Do you choose a tuning to fit your musical idea, or choose a
>>neat tuning and try to come up with one?
I have experimented with random tunings. There are always good melodies
to be found, but harmony is a problem (not only finding "nice chords"
but satisfying progressions) so I didn't pursue them. The best results
were random perturbations relative to equal temperament rather than
choosing a random set of pitches within the octave. In the latter case,
notes tend to cluster together.
Usually, I construct tunings around theoretical ideas like approximating
consonant intervals. Usually these are linear temperaments, so there's
an obvious scale that may work as a diatonic, and a way of mapping it to
a two dimensional keyboard. Then it's a question of finding musical
ideas within that scale. In the and, I may find I only need a subset
for a particular piece, and it can be tuned to a 12 note keyboard.
Also, as linear temperaments have a free parameter (the size of the
fifth for a meantone) is possible to control that manually and so find
the setting that sounds best.
>>Most microtonal music I've heard in say 17-tet or 31-tet have been pretty
>>experimental. While they can certainly sound pleasant, they aren't exactly
>>something you would whistle while walking down the street. Is this because
>>nobody is interested in writing simple catchy melodies in weird tunings, or
>>is it because they are incapable of it, or is it because I've just listened
>>to the weird stuff, or is it because my ear is not accustomed to it so I'm
>>incapable of hearing it properly?
Probably a bit of all four. I'm sure it's possible to write whistlable
microtonal melodies. I've found myself writing natural sounding
melodies in Miracle, where it is possible to get usual diatonic scales.
Then, when checking how the look in a diatonic context, I found a much
weaker fit than I expected. A naive listener wouldn't have so much
exposure to the tuning, and so would find them stranger, but less so on
repeated listening.
For a proof ad hominem, Natacha Atlas makes conscious use of
quartertones, but perhaps this isn't a weird enough tuning for you.
Graham
Margo Schulter
>
> As Margo's asked for a generalized keyboard, perspective, I'll supply one.
>
Thank you, Graham, for just the kind of contribution I wanted to invite;
it's a neat summary of lots of tuning systems, and how you've gone about
finding solutions for using them with different kinds of keyboards. I'd
say that my range of experience in this is much more modest.
> It's designed as a MIDI guitar, but works well for microtonality because
> it has a regular array of 144 small keys. It lends itself well to a
> Miracle layout with 60 notes to the octave, but I've found it preferable
> to squash that down to 30, like this (if you can follow my notation):
>
> 1^ 3^ 5^ 7^ 9^ 1^ 3^ 5^ 7^ 9^ 1^ ...
> 1 3 5 7 9 1 3 5 7 9 1 ...
> 1v 3v 5v 7v 9v 1v 3v 5v 7v 9v 1v ...
> 0^ 2^ 4^ 6^ 8^ 0^ 2^ 4^ 6^ 8^ 0^ ...
> 0 2 4 6 8 0 2 4 6 8 0 ...
> 0v 2v 4v 6v 8v 0v 2v 4v 6v 8v 0v ...
>
> for a total of 4 and a bit octaves. It's good for experimentation, but
> I still haven't done as much with it as a normal keyboard. So it isn't
> like you have any excuse for not getting results because you don't have
> fancy hardware.
A quick aside for the curious: the Miracle temperament, originally
proposed by George Secor (a very important person in the history of the
generalized keyboard, as well as a microtonal theorist and musician) in
1975 (_Xenharmonikon_ 3), was independently "rediscovered" by a group of
people including David Keenan and Paul Erlich on the Tuning List (hosted
by Yahoogroups) in early 2001. The temperament uses a generator of
approximately 7/72 octave, which is one of the possible realizations (in
72-equal); the optimal generator as calculated by Secor for minimizing
variations of certain ratios from just is very slightly larger than 7/72
octave (116-2/3 cents), at about 116.716 cents.
Anyway, the Miracle Temperament in either an equal-tempered form (31, 41,
or 72) or this precisely optimized form, tends to offer lots of near-just
ratios but musical structures often quite different than those of
traditional European modality or tonality.
An interesting feature of Secor's optimal, non-equal version (close to
72-equal, but a bit different) is that it forms a 72-note cycle with some
variation in certain interval sizes. I wonder if anyone has tried tuning
this on a 72-note keyboard, if there are generalized keyboards which
support such a large number of notes per octave.
Anyway, I hope that this bit of background might get people interested in
Graham's Web page on Miracle, which has a lot more information on the
theory and keyboard practice of this tuning system.
> The idea is that three consecutive strings stopped on the same fret will
> give a triad composed of two neutral thirds. If you sharpen the middle
> note by one fret, you get some kind of major triad. There's a 50%
> chance this will be an approximation to 4:5:6. If not, you can sharpen
> the middle note again and get an approximation to a "car horn triad"
> (Margo has a latin name for this) where the major third is 7:9.
> Flattening the middle note of a neutral triad may give you a 5-limit
> minor triad, and if not flattening it again will give an approximation
> to 6:7:9.
This Mystery sounds like a very interesting tuning system, and I'll
explain that I refer to the 7:9 as the _tertia automotiva_, or "automative
third," since it does sometimes get compared to a car horn. In lots of
timbres and registers, it tends to be an active interval, and I often use
it in cadences where it expands by stepwise motion to a fifth.
It can be more mellow in a sonority like 6:7:9 (with 6:7 minor third
below) or 7:9:12, where it can also invite excellent cadences involving
its expansion to a fifth as one of the two-voice resolutions.
Another name I have for the 7:9 is _tertia clarionis_, or "clarion third,"
which expresses the often rather strident qualities of this third and also
its frequent "clarion call" to a directed cadence. A nice feature is how
this interval can participate in sonorities I often find very rich and
euphonious like 7:9:12 or 14:18:21:24, and at the same time can resolve in
very effective cadences to 2:3:4. Some people might describe these
sonorities as inversions of 6:7:9 or 12:14:18:21, with the 6:7 below the
7:9, which are often used as preferred concords or even stable sonorities,
for example by George Secor and Easley Blackwood.
I've found that sometimes the effect of 7:9 can vary depending on the
register: high enough up, it can be quite "sweet" even as a simple
interval or in 14:18:21 (7:9 below 6:7), Graham's "utonal" form or
"supramajor triad."
> Depending on which three strings you chose, you either get "utonal" or
> "otonal" chords. For otonal, the neutral triad has lower neutral third
> 11:9, the major triad approximates 4:5:6 and the subminor triad 6:7:9.
> For utonal, the lower neutral third approximates 16:13, the minor triad
> approximates 10:12:15 and the supermajor triad is as I described above.
> If you tune to 58-equal, the otonal/utonal distinction vanishes, and
> everything's still in tune to the nearest 8.3 cents.
>
This sounds like a really neat tuning system! I'll just mention that
either in 58-equal, or 29-equal (with which I'm familiar), there are neat
ratios like 15:13 or around 248 cents, a very small minor third or
possibly a very large major second.
Anyway, the practical keyboard information as well as the survey of tuning
systems (with much more on your Web site, of course!) helps to really make
this thread.