Ratio space on keyboard

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Ratio Space on Keyboard:
The Peppermint 24 system
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An interesting feature of tuning systems, as implemented on keyboards
(conventional or alternative), is the mapping of pure or tempered
ratios to positions on the keyboard layout.

Here I shall explore the mapping of approximate ratios, and especially
of superparticular and other ratios within Harry Partch's larger
17-limit set, in the tuning system and keyboard arrangement I call
Peppermint 24.

Peppermint 24 takes as its basis a regular temperament mentioned in
Ervin Wilson's Scale Tree and described on the Tuning List by Keenan
Pepper, with a fifth of about 704.096 cents, and a precise ratio of
Phi, the Golden Section (~1.618) between the larger chromatic semitone
(e.g. C-C#) at about 128.669 cents and the smaller diatonic semitone
(e.g. C#-D) at about 79.522 cents.

In Peppermint 24, two regular 12-note chains of this temperament are
placed at a distance of approximately 58.680 cents, so as to yield
some pure ratios of 6:7 (~266.871 cents).

Here is a 24-note keyboard arrangement, with an asterisk (*) showing a
note on the upper keyboard:

187.349 346.393 683.253 891.445 1050.488
C#* Eb* F#* G#* Bb*
C* D* E* F* G* A* B* C*
58.680 266.871 475.062 554.584 762.775 970.967 1179.158 1258.680
7/6
-------------------------------------------------------------------------
128.669 287.713 624.574 832.765 991.809
C# Eb F# G# Bb
C D E F G A B C
0 208.191 416.382 495.904 704.096 912.287 1120.478 1200

In the following catalogue of some ratio equivalents and mappings, I
will focus on intervals no further from just than 8:9 or 9:16, which
vary from their pure sizes by about 4.282 cents (twice the tempering
of the fifth, at about 2.141 cents wide of 2:3).

Octave numbers appear in a MIDI-style notation, with C4 as middle C;
just ratios and tempered equivalents are given values in cents, shown
in parentheses, with tempered variations in cents also shown.

To describe the 58.68-cent interval between the two keyboards, whose
addition or subtraction plays a role in obtaining or approximating
many ratios, I shall the term "quasi-diesis," or QD for short. This
"artificial" diesis-like interval is actually somewhat larger than the
natural diesis in the regular Wilson/Pepper temperament at about 49.15
cents (12 tempered fifths less 7 pure octaves).

As this partial catalogue might suggest, many ratios of 2-3-7-9-11-13
are represented quite accurately, with 14:17:21 and related ratios
also closely approximated.

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1. Multiplex (n:1) and superparticular (n+1:n) intervals
--------------------------------------------------------

1:2 (1200) -- This is the usual octave (e.g. F3-F4), at a pure 1:2.

2:3 (701.96) -- This is the usual fifth (e.g. F3-C4, 704.10, +2.14).

3:4 (498.04) -- Usual fourth (e.g. C4-F4, 495.90, -2.14).

6:7 (266.87) -- Major second + QD (e.g. D4-E*4), at a pure 6:7.

7:8 (231.17) -- Minor third - QD (e.g. C*4-Eb4, 229.03, -2.14)

8:9 (203.91) -- Usual major second (e.g. C4-D4, 208.19, +4.28)

11:12 (150.64) -- Major second - QD (e.g. C*4-D4, 149.51, -1.13)

12:13 (138.57) -- Minor second + QD (e.g. E4-F*4, 138.20, -0.37)

13:14 (128.30) -- Usual apotome (e.g. C4-C#4, 128.67, +0.37)

17:18 (98.95) -- Diminished third - QD (e.g. G#*4-Bb4, 100.36, -1.41)

21:22 (80.54) -- Usual minor second (e.g. E4-F4, 79.52, -1.02)

24:25 (70.67) -- Apotome - QD (e.g. E*4-Eb4, 69.99, -0.68)

27:28 (62.96) -- QD (e.g. E4-E*4, 58.68, -4.28)

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2. Other ratios -- many within 17-odd limit
-------------------------------------------

4:7 (968.83) -- Major sixth + QD (e.g. G3-E*4, 970.97, +2.14)

7:9 (435.08) -- Fourth - QD (e.g. G*4-C5, 437.22, +2.14)

7:12 (933.13) -- Minor seventh - QD (e.g. G*3-F4), at a pure 7:12.

9:14 (764.92) -- Fifth + QD (e.g. G4-D*5, 762.78, -2.14)

9:16 (996.09) -- Usual minor seventh (e.g. G4-F4, 991.81, -4.28)

6:11 (1049.36) -- Minor seventh + QD (e.g. G3-F*4, 1050.49, +1.13)

7:11 (782.49) -- Usual minor sixth (e.g. A3-F4, 783.62, +1.13)

8:11 (551.32) -- Fourth + QD (e.g. G3-C*4, 554.58, +3.27)

9:11 (347.41) -- Minor third + QD (e.g. G3-Bb*3, 346.39, -1.02)

8:13 (840.53) -- Minor sixth + QD (e.g. G3-Eb*3, 842.30, +1.77)

9:13 (636.62) -- Diminished fifth + QD (e.g. A3-Eb*4, 634.11, -2.51)

11:13 (289.21) -- Usual minor third (e.g. D3-F3, 287.71, -1.50)

11:14 (417.51) -- Usual major third (e.g. D3-F#3, 416.38, -1.13)

11:16 (648.68) -- Fifth - QD (e.g. F*3-C4, 645.42, -3.27)

11:18 (852.59) -- Major sixth - QD (e.g. G*4-E5, 853.61, +1.02)

11:21 (1119.46) -- Usual major seventh (e.g. F3-E4, 1120.48, +1.02)

12:17 (603.00) -- Augmented third + QD (e.g. Eb4-G#*4, 603.73, +0.73)

13:16 (359.47) -- Major third - QD (e.g. C*4-E4, 357.70, -1.77)

13:18 (563.38) -- Augmented fourth - QD (e.g. C*4-F#4, 565.89, +2.51)

13:21 (830.25) -- Usual augmented fifth (e.g. C4-G#4, 832.76, +2.51)

13:22 (910.79) -- Usual major sixth (e.g. G3-E4, 912.29, +1.50)

13:23 (987.75) -- Usual minor seventh (e.g. D4-C5, 991.81, +4.06)

13:24 (1061.43) -- Major seventh - QD (e.g. F*3-E4), 1061.80, +0.37)

14:17 (336.13) -- Usual augmented second (e.g. F4-G#4, 336.86, +0.73)

14:27 (1137.04) -- Octave - QD (e.g. F*4-F5, 1141.32, +4.28)

15:17 (216.69) -- Diminished third + QD (e.g. C#4-Eb*4, 217.72, +1.04)

16:21 (470.71) -- Major third + QD (e.g. C4-E*4, 475.06, +4.28)

16:23 (628.27) -- Usual augmented fourth (e.g. C4-F#4, 624.57, -3.70)

18:23 (424.36) -- Diminished fourth + QD (e.g. B4-Eb*5, 425.91, +1.55)

16:25 (772.63) -- Diminished fourth + QD (e.g. F#4-Bb*4, 774.09, +1.46)

17:20 (281.36) -- Augmented second - QD (e.g. F*4-G#4, 278.18, -3.18)

17:21 (365.83) -- Usual diminished fourth (e.g. F#4-Bb4, 367.24, +1.41)

17:28 (863.87) -- Usual diminished seventh (e.g. F#4-Eb4, 863.14, -0.73)

21:34 (834.17) -- Usual augmented fifth (e.g. F3-C#4, 832.76, +1.41)

28:51 (1038.08) -- Usual augmented sixth (e.g. Eb3-C#4, 1040.96, +2.87)

21:23 (157.49) -- Usual diminished third (e.g. C#4-Eb4, 159.04, +1.55)

21:26 (369.75) -- Usual diminished fourth (e.g. C#4-F4, 367.24, -1.51)

23:27 (277.59) -- Augmented second - QD (e.g. Eb*4-F#4, 278.18, +0.59)

26:33 (412.75) -- Usual major third (e.g. F4-A4, 416.38, +3.63)

28:33 (284.45) -- Usual minor third (e.g. E4-G4, 287.71, +3.27)

33:56 (915.55) -- Usual major sixth (e.g. G4-E5, 912.29, -3.27)

Most appreciatively,

Margo Schulter
mschulter@value.net