The Too-Condensed Tuning-Math Outline
Introduction
This document hopes to outline some of the stuff that happens on the
tuning-math mailing list...
http://groups.yahoo.com/group/tuning-math/
Thanks to Paul Erlich, Gene Ward Smith, Graham Breed, and Dave Keenan.
Carl Lumma
Berkeley, California.
Outline
1. Constraints.
1.1 Observation: Reducing the number of different intervals in
a scale makes melodies in that scale easier to sing.
1.1.1 The number of different intervals in a scale,
normalized to the number of notes in it, is Rothenberg's
*mean variety*.
1.2 The human hearing system is specially adapted to extract
information from a particular type of signal, typical of human
speech. By constraining music to a similar domain we approach
the channel capacity of human listeners.
1.2.1 This has traditionally been achieved by limiting
the set of musical intervals to the 'small whole number
ratios' of just intonation (5:4, 3:2, etc.) or some
approximation thereof.
1.3 Insofar as pitch memory is important in melody, the
*Miller limit* suggests scales consisting of no more than 9
octave-equivalent notes.
2. Repetition for reinforcement and fun (and profit).
2.1 Besides exact repetition, one or more parameters may be
changed between instances.
2.1.1 Repeating a theme in a different mode keeps
generic intervals (3rds, 5ths, etc.) the same while
changing their quality (major/minor, etc.).
2.1.1.1 *Rothenberg stability* is supposed to
measure the ease of accomplishing such 'modal
transposition' in a given scale.
2.1.2 A theme played in a different key keeps absolute
intervals the same while changing the pitches involved.
2.1.2.1 As goes *Rothenberg efficiency*, rules
such as those implied in Western tonal music
become more important in making key changes
recognizable.
3. This suggests we build scales by selecting a few concordant
intervals and stacking them. Such scales can be graphed on a *lattice*
with an axis for each of the selected intervals.
3.1 All intervals in the scale can be expressed as a path
through the lattice.
3.2 Scales which maximize the number of concordant intervals
per note will generally be compact and convex on the lattice.
3.3 Ignoring octaves (factors of 2), 3-limit just intonation
can be graphed on a linear chain (of "fifths"), 5-limit JI on a
planar lattice, 7-limit JI on the face-centered cubic lattice...
4. As we add notes to our scale by stacking, we will eventually find
some that are very near in pitch to ones we already have. The small
intervals (such as 81:80) between such pairs of notes (such as 9/8 and
10/9) are called *commas*.
4.1 By forbidding the occurrence of commas in our scale, we can
delimit a finite section of the lattice known as a "block" or
"periodicity block".
4.2 We create a *pun* if we use the same name ("Eb") for both
notes in such a pair.
4.3 We create a *comma pump* by writing a chord progression
whose starting and ending roots involve a pun. Every time the
progression is repeated, our pitch standard (concert pitch)
*drifts* sharp or flat by the size of the associated comma.
5. We can make a comma vanish by *tempering* the intervals in its
lattice path (in the case of 81:80, four 3:2s and one 5:4) so that its
size shrinks to zero.
5.1 If we do this uniformly across the lattice, the associated
comma pumps can all be performed without causing drift in our
pitch standard.
5.1.1 This effectively reduces the dimensionality of
the lattice.
6. Observation: The most popular scales across cultures and times
correspond to periodicity blocks or temperaments defined by *simple*
commas.
6.1 The complexity of an interval may be measured by the
distance it spans on the lattice. Thus, simpler commas tend to
delimit smaller periodicity blocks.
6.1.1 The size of the denominator of a comma provides a
good heuristic for its complexity.
6.2 The complexity of a scale or temperament may be defined in
terms of notes per concordant intervals, and thus is closely
related to the complexity of their defining comma(s).
7. To further evaluate temperaments, complexity may be balanced against
the error relative to just intonation. This error is bounded by the
size of the comma(s) being tempered out (trivial) and the number of
concordant intervals over which it/they must vanish (complexity again).
7.1 It's easy to find tiny commas with arbitrarily large
denominators, such as 1001:1000. We want temperaments based on
the simplest commas in a given size range.
7.1.1 As a matter of coincidence, the same math is
behind *harmonic entropy*...
7.2 *Badness* for temperaments is thus complexity*error, and
badness for commas is complexity*size.
8. To see a database of 5-limit linear temperaments, go to...
http://groups.yahoo.com/group/tuning/database/
Or try the Excel version.
Try sorting by comma denominator (the Excel version should be sorted
this way by default).
8.1 You can see that the syntonic comma (81:80), which defines
the meantone temperaments that have dominated Western music for
hundreds of years, is one of the simplest 5-limit commas and is
by far the smallest among the few most simple (on a list which
is itself the result of searching 5-limit ratio space for commas
with low badness).
8.2 Another comma that looks good is the major diesis
(648:625), which leads to the "diminished" temperament. If we
take 8 tones/octave of this temperament, we get the octatonic
scale of Stravinsky and Messiaen (not to mention jazz).
8.3 However, while diminished makes a good showing, you can
see that "porcupine" is better. Note, however...
8.3.1 Diminished, but not porcupine, is a subset of
12-tone equal temperament.
8.3.2 The octatonic scale was not used by composers
until after 12-tET had finally become widespread.
9. This suggests that porcupine is a potentially fertile direction for
new music. Indeed, the porcupine temperament is named after a piece by
composer Herman Miller, the Mizarian Porcupine Overture...
http://lumma.org/music/theory/tctmo/MizarianPorcupineOverture.mp3
Example
A periodicity block may be viewed as a tiling on the untempered
lattice...
Here the "unison vectors" (Fokker's term for commas) 81:80 and 32:25
delimit the tile shown in red, and the scale shown in green, on the
5-limit lattice. Neighboring tiles will contain copies of the scale,
transposed by either 81:80 or 32:35.
As mentioned in 5.1.1, tempering one of these commas out will give a
linear temperament (chain), and tempering both out an equal temperament.
In the 7-limit, three commas are required to delimit a block. Tempering
one out gives a planar temperament, two a linear temperament, three an
equal temperament.
In the 3-limit, the standard diatonic scale may be seen as a block
delimited by the apotome (2187:2048), not tempered out. In the 5-limit,
it may be seen as a linear temperament of the block delimited by 81:80
and 25:24, where 81:80 is tempered out and 25:24 is not.
Resources
Erlich, Paul
The Forms of Tonality
Rothenberg, David
A Model for Pattern Perception with Musical Applications
Mathematical Systems Theory vol. 11, 1978.
Miller, George
The Magical Number Seven, Plus or Minus Two
The Psychological Review vol. 63, 1956, pp. 81-97.