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.Introduction1. 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.mp3OutlineA 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.ExampleErlich, PaulResourcesThe Forms of TonalityRothenberg, DavidA Model for Pattern Perception with Musical ApplicationsMathematical Systems Theory vol. 11, 1978. Miller, GeorgeThe Magical Number Seven, Plus or Minus TwoThe Psychological Review vol. 63, 1956, pp. 81-97.