Peppermint-24

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Peppermint 24

Original article by Margo Schulter, on the Yahoo tuning forum, is quoted below. In addition to what it says, it may be noted that the Wilson/Pepper fifth it mentions, of size approximately 704.096 cents, has a precise value of (67 + √5)/118 octaves, which is (40200 + 600 √5)/59 cents.

Margo Schulter's article

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.

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)

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)