Peppermint-24

Peppermint 24 is a scale first documented by Margo Schulter on the Yahoo tuning forum: M. Schulter (7/3/2002 3:51:43 AM)

Concept

Peppermint 24 aims to map superparticular and other ratios within Harry Partch's larger 17-limit set, to two conventional piano keyboards.

It 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. Said fifth has a precise value of (67 + √5)/118 octaves, which is (40200 + 600 √5)/59 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).

Keyboard arrangement

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

Intervals

Single chain

Offset two of these by 58.680 cents.

128.669
208.191
287.713
416.382
495.904
624.574
704.096
832.765
912.287
991.809
1120.478
1200.000

Combined (both chains)

58.680
128.669
187.349
208.191
266.871
287.713
346.393
416.382
475.062
495.904
554.584
624.574
683.253
704.096
762.775
832.765
891.445
912.287
970.967
991.809
1050.488
1120.478
1179.157
1200.000

Catalogue of ratio equivalents

What follows is a catalogue of some ratio equivalents and mappings 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, the term "quasi-diesis" or "QD" is used. 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).

Many ratios of 2-3-7-9-11-13 are represented quite accurately, with 14:17:21 and related ratios also closely approximated.

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)

Other ratios

Many of these fall within the 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)