User:CompactStar/Ed9/n

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backing up the mmtm page just in the off chance there is any useful theory here

An ed9/n is an equal-step tuning devised by equally dividing an interval of the form 9/n where n is any integer.

Ed9/1

See Ed9.

Ed9/2

The equal division of 9/2 (ed9/2) is a tuning obtained by dividing two octaves and a Pythagorean major second (9/2) into a number of equal steps.

Properties

Division of 9/2 into equal parts does not necessarily imply directly using this interval as an equivalence. Many, though not all, ed9/2 scales have a perceptually important false octave, with various degrees of accuracy.

One approach to ed9/2 tunings is the use of the 9:10:14 chord as the fundamental complete sonority in a very similar way to the 4:5:6:(8) chord in meantone. Whereas in meantone it takes four 3/2 to get to 5/1, here it takes seven 14/9 to get to 10/9 (tempering out the comma 215233605/210827008 in the 9/2.5.7 fractional subgroup). This temperament yields 7-, 10-, 17-, and 27-note mos scales.

Ed9/4

The equal division of 9/4 (ed9/4) is a tuning obtained by dividing the Pythagorean ninth (9/4) in a certain number of equal steps. An ed9/4 can be generated by taking every other tone of an edf, so even-numbered ed9/4's are integer edfs.

Properties

Division of 9/4 into equal parts does not necessarily imply directly using this interval as an equivalence. Many, though not all, ed9/4 scales have a perceptually important false octave, with various degrees of accuracy.

The structural utility of 9/4 or another major ninth is apparent by being the standard replacement for the root in jazz piano voicings. Also, as a ninth is the double of a fifth, the fifth of normal root position triads will become the common suspension (5-4 or 5-6) of a ninth-based system.

Joseph Ruhf's ed9/4 theory

This article or section contains multiple idiosyncratic terms. Such terms are used by only a few people and are not regularly used within the community.

In ed9/4 systems, thirds and sixths are no longer inverses, and thus an octatonic scale (i. e. any of those of the proper Napoli temperament family, which are generated by a fourth optionally with a period equivalent to three or six macrotones, in particular ones at least as wide as 101.083 cents) takes 1-3-6, which is not equivalent to a tone cluster as it would be in an edf tuning, as the root position of its regular triad.

One way to approach some ed9/4 tunings is the use of the 5:6:8 chord as the fundamental complete sonority in a very similar way to the 4:5:6:(8) chord in meantone. Whereas in meantone it takes four 3/2 to get to 5/1, here it takes four 9/8 to get to 8/5 (tempering out the schisma). So, doing this yields 6-, 8-, 14- and 20- or 22-note 2mos. While the notes are rather farther apart, the scheme is superficially similar to certain versions of the regularly tempered approximate ("full"-status) shrutis. Joseph Ruhf proposes the name "macroshrutis" for this reason.

The branches of the Napoli family are named thus:

5&3: Grandfather

Bipentachordal:

  • 4&4: Macrodiminshed
  • 6&2: Macroshrutis

The temperament family in the Neapolitan temperament area which has an interlaced enneatonic scale is named for parts of Maryland further west of the Middletown Valley as its generator rises:

3&6: South Mountain Scale

4&5: Hagerstown (particularly in ~9/4)

2&7: Allegany

The temperament family in the Neapolitan temperament area which has an octatonic scale of seven generators and a remainder is named Fujiyama (i. e. the volcano viewable from practically anywhere in Japan due to the Japanese archipelago consisting of such flat islands).

Surprisingly, though sort of obviously, due to 9/4 being the primary attractor for Neapolitan temperaments, the golden and pyrite tunings of edIXs must be forced to turn out to divide a (nearly) pure 9:4 (in particular, using Aeolian mode gives the 2/7-comma meantone major ninth as almost exactly the pyrite tuning of the period, or (8φ+6)/(7φ+5).

Individual pages for ed9/4's

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21 23 25 27 29 31 33 35 37 39
41 43 45 47 49 51 53 55 57 59
61 63 65 67 69 71 73 75 77 79
81 83 85 87 89 91 93 95 97 99

Ed9/5

The equal division of 9/5 (ed9/5) is a tuning obtained by dividing the classic minor seventh (9/5) in a certain number of equal steps.

Properties

Division of 9/5 into equal parts does not necessarily imply directly using this interval as an equivalence. Many, though not all, ed9/5 scales have a perceptually important false octave, with various degrees of accuracy.

The structural importance of 9/5 is suggested by its being the most common width for a tetrad in Western harmony, though it could be argued that this distinction belongs instead to 7/4 or 16/9 depending how one converts 10\12 into JI.

One approach to some ed9/5 tunings is the use of the 5:6:7:(9) chord as the fundamental complete sonority in a very similar way to the 4:5:6:(8) chord in meantone. Whereas in meantone it takes four 3/2 to get to 5/1, here it takes four 7/5 to get to 6/5 (tempering out the comma 2430/2401). So, doing this yields 5-, 7-, and 12-note mos scales, just like meantone. While the notes are rather closer together, the scheme is exactly identical to meantone. Joseph Ruhf proposed the term "microdiatonic"[idiosyncratic term] for this because it uses a scheme that turns out exactly identical to meantone, though severely compressed.

However, a just or very slightly flat 9/5 leads to the just 7/5 generator converging to a scale where the '6/5' and '7/6' are only ~3.6 cents off of half a generator in opposite directions, which is acceptable with the harmonic entropy of a pelogic temperament, but this temperament is not intended to be understood as pelogic. It is rather intended to be understood as meantone, albeit severely compressed, but in theory still has plus or minus ~36.85 cents of harmonic entropy for '7/6' and '6/5', though in potential practice no more than ~36% of this should ever be used very seriously.

Individual pages for ed9/5's

0…99
0 1 2 3 4 5 6 7 8 9
10 11 12 13 14 15 16 17 18 19
20 21 22 23 24 25 26 27 28 29
30 31 32 33 34 35 36 37 38 39
40 41 42 43 44 45 46 47 48 49

Ed9/7

The equal division of 9/7 (ed9/7) is a tuning obtained by dividing the septimal supermajor third (9/7) in a certain number of equal steps.

ED9/7 tuning systems that accurately represent the intervals 9/8 and 8/7 include: 15ed9/7 (0.87 cent error), 17ed7/5 (0.84 cent error), and 32ed7/5 (0.04 cent error).

15ed9/7, 17ed9/7, and 32ed9/7 are to the division of 9/7 what:

Ed9/8

The equal division of 9/8 (ed9/8) is a tuning obtained by dividing the major whole tone (9/8) into a number of equal steps.

Properties

Division of 9/8 into equal parts does not necessarily imply directly using this interval as an equivalence. Many, though not all, ed9/8 scales have a perceptually important false octave, with various degrees of accuracy.

9/8 is notably the difference between two intervals sometimes considered equivalences: the triple octave (8/1), and the double tritave (9/1).

List of ed9/8s


Todo: review , cleanup, improve layout
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