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→Modern renderings: Add John Bull's ''Fantasia «Ut Re Mi Fa Sol La»'' (late 1500s/early 1600s, from ''Fitzwilliam Virginal Book Vol.1 No.51'') – rendered by Claudi Meneghin (2020) in a 24 note per octave well-tempered system that combines golden meantone with peppermint
 
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== Peppermint 24 ==
'''Peppermint 24''' is a [[scale]] first documented by [[Margo Schulter]] on the Yahoo tuning forum: [https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_38440.html#38440 M. Schulter (7/3/2002 3:51:43 AM)]


[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_38440.html#38440 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.
==Concept==
Peppermint 24 aims to map [[superparticular]] and other ratios within [[wikipedia:Harry_Partch|Harry Partch's]] larger [[17-limit]] set, to two conventional piano keyboards.


== Margo Schulter's article ==
It takes as its basis a [[Regular_Temperaments|regular temperament]] mentioned in [[Erv_Wilson|Ervin Wilson]]'s Scale Tree and described on the Tuning List by [[Keenan Pepper]], with a fifth of about 704.096 [[Cent|cents]], and a precise ratio of [[wikipedia:Golden_ratio|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.
An interesting feature of tuning systems, as implemented on keyboards (conventional or alternative), is the [[keyboard_mappings|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 [https://en.wikipedia.org/wiki/Harry_Partch 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_Temperaments|regular temperament]] mentioned in [[Erv_Wilson|Ervin Wilson]]'s Scale Tree and described on the Tuning List by [[Keenan_Pepper|Keenan Pepper]], with a fifth of about 704.096 cents, and a precise ratio of [https://en.wikipedia.org/wiki/Golden_ratio 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).
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:
Here is a 24-note keyboard arrangement, with an asterisk (*) showing a note on the upper keyboard:


<pre>     187.349  346.393              683.253    891.445 1050.488
<pre>
    187.349  346.393              683.253    891.445 1050.488
       C#*      Eb*                  F#*      G#*    Bb*
       C#*      Eb*                  F#*      G#*    Bb*
   C*        D*          E*      F*        G*        A*      B*    C*
   C*        D*          E*      F*        G*        A*      B*    C*
Line 24: Line 22:
   C        D          E      F          G        A        B    C
   C        D          E      F          G        A        B    C
   0      208.191    416.382 495.904    704.096  912.287 1120.478 1200
   0      208.191    416.382 495.904    704.096  912.287 1120.478 1200
 
</pre>
 </pre>


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).
== Intervals ==


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.
=== Single chain ===
Offset two of these by 58.680 cents.
<pre>
128.669
208.191
287.713
416.382
495.904
624.574
704.096
832.765
912.287
991.809
1120.478
1200.000
</pre>


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).
=== Combined (both chains) ===
<pre>
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
</pre>


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.
==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).


'''1. Multiplex (n:1) and [[superparticular|superparticular]] (n+1:n) intervals'''
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.
 
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)
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).


7:9 (435.08) -- Fourth - QD (e.g. G*4-C5, 437.22, +2.14)
Many ratios of 2-3-7-9-11-13 are represented quite accurately, with 14:17:21 and related ratios also closely approximated.


7:12 (933.13) -- Minor seventh - QD (e.g. G*3-F4), at a pure 7:12.
===Multiplex (n:1) and [[superparticular]] (n+1:n) intervals===


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


9:16 (996.09) -- Usual minor seventh (e.g. G4-F4, 991.81, -4.28)
===Other ratios===
Many of these fall within the [[17-odd-limit]].


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


7:11 (782.49) -- Usual minor sixth (e.g. A3-F4, 783.62, +1.13)
== Subsets ==
=== Diatonic and related scales ===
C Major
* 208.191
* 416.382
* 495.904
* 704.096
* 912.287
* 1120.478
* 1200.000


8:11 (551.32) -- Fourth + QD (e.g. G3-C*4, 554.58, +3.27)
=== Salt and pepper scale and its subsets ===
Salt and pepper{{idiosyncratic}}


9:11 (347.41) -- Minor third + QD (e.g. G3-Bb*3, 346.39, -1.02)
''A 12-tone subset of Peppermint-24 designed by [[Budjarn Lambeth]] to concentrate the most frequently used intervals on just one keyboard.''
* 128.669
* 187.349
* 208.191
* 266.871
* 287.713
* 495.904
* 704.096
* 832.765
* 891.445
* 970.967
* 1050.488
* 1200.000


8:13 (840.53) -- Minor sixth + QD (e.g. G3-Eb*3, 842.30, +1.77)
<small>
Evacuated planet{{idiosyncratic}} (approximated from [[66afdo|66]][[afdo]])
* 128.669
* 495.904
* 704.096
* 1050.488
* 1200.000


9:13 (636.62) -- Diminished fifth + QD (e.g. A3-Eb*4, 634.11, -2.51)
Flattened pseudo-[[equiheptatonic]]
* 128.669
* 266.871
* 495.904
* 704.096
* 832.765
* 1050.488
* 1200.000


11:13 (289.21) -- Usual minor third (e.g. D3-F3, 287.71, -1.50)
Geode{{idiosyncratic}} (approximated from [[6afdo]])
* 266.871
* 495.904
* 704.096
* 1050.488
* 1200.000


11:14 (417.51) -- Usual major third (e.g. D3-F#3, 416.38, -1.13)
Minor hexatonic (approximated from [[12edo]])
* 187.349
* 287.713
* 495.904
* 704.096
* 970.967
* 1200.000


11:16 (648.68) -- Fifth - QD (e.g. F*3-C4, 645.42, -3.27)
Pepperbass{{idiosyncratic}} (original/default tuning)


11:18 (852.59) -- Major sixth - QD (e.g. G*4-E5, 853.61, +1.02)
(''works well with jungle- or trap-style sub bass'')
* 208.191
* 704.096
* 891.445
* 1050.488
* 1200.000


11:21 (1119.46) -- Usual major seventh (e.g. F3-E4, 1120.48, +1.02)
Pseudo-[[6afdo]]
* 266.871
* 495.904
* 704.096
* 891.445
* 1050.488
* 1200.000


12:17 (603.00) -- Augmented third + QD (e.g. Eb4-G#*4, 603.73, +0.73)
Pseudo-akebono I (approximated from [[12edo]])
* 208.191
* 287.713
* 704.096
* 891.445
* 1200.000


13:16 (359.47) -- Major third - QD (e.g. C*4-E4, 357.70, -1.77)
Pseudo-akebono II (approximated from [[12edo]])
* 128.669
* 495.904
* 704.096
* 832.765
* 1200.000


13:18 (563.38) -- Augmented fourth - QD (e.g. C*4-F#4, 565.89, +2.51)
Pseudo-[[equipentatonic]]
* 266.871
* 495.904
* 704.096
* 970.967
* 1200.000


13:21 (830.25) -- Usual augmented fifth (e.g. C4-G#4, 832.76, +2.51)
Pseudo-hirajoshi (approximated from [[12edo]])
* 208.191
* 287.713
* 704.096
* 832.765
* 1200.000


13:22 (910.79) -- Usual major sixth (e.g. G3-E4, 912.29, +1.50)
Sharpened pseudo-[[pelog]]
* 128.669
* 287.713
* 704.096
* 832.765
* 1200.000
</small>


13:23 (987.75) -- Usual minor seventh (e.g. D4-C5, 991.81, +4.06)
=== Ketchup and mustard scale and its subsets ===
Ketchup and mustard{{idiosyncratic}}


13:24 (1061.43) -- Major seventh - QD (e.g. F*3-E4), 1061.80, +0.37)
''A 12-tone subset of Peppermint-24 designed by [[Budjarn Lambeth]] to map intervals which sound nice with an inharmonic [[gamelan]]-like timbre to a 12-key keyboard (e.g. [https://scaleworkshop.plainsound.org/scale/h2qwnm0-l this timbre in Scale Workshop]).''
* 58.680
* 128.669
* 187.349
* 266.871
* 475.062
* 683.253
* 762.775
* 832.765
* 912.287
* 970.967
* 1050.488
* 1200.000


14:17 (336.13) -- Usual augmented second (e.g. F4-G#4, 336.86, +0.73)
<small>
Inharmonic geode{{idiosyncratic}}
* 266.871
* 475.062
* 683.253
* 1050.488
* 1200.000


14:27 (1137.04) -- Octave - QD (e.g. F*4-F5, 1141.32, +4.28)
Inharmonic minor hexatonic
* 187.349
* 266.871
* 475.062
* 683.253
* 970.967
* 1200.000


15:17 (216.69) -- Diminished third + QD (e.g. C#4-Eb*4, 217.72, +1.04)
Inharmonic pepperbass{{idiosyncratic}}
* 187.349
* 683.253
* 762.775
* 1050.488
* 1200.000


16:21 (470.71) -- Major third + QD (e.g. C4-E*4, 475.06, +4.28)
Inharmonic pseudo-[[6afdo]]
* 266.871
* 475.062
* 683.253
* 832.765
* 1050.488
* 1200.000


16:23 (628.27) -- Usual augmented fourth (e.g. C4-F#4, 624.57, -3.70)
Inharmonic pseudo-akebono I
* 187.349
* 266.871
* 683.253
* 912.287
* 1200.000


18:23 (424.36) -- Diminished fourth + QD (e.g. B4-Eb*5, 425.91, +1.55)
Inharmonic pseudo-akebono II
* 58.680
* 475.062
* 683.253
* 762.775
* 1200.000


16:25 (772.63) -- Diminished fourth + QD (e.g. F#4-Bb*4, 774.09, +1.46)
Inharmonic pseudo-[[equipentatonic]]
* 266.871
* 475.062
* 704.096
* 970.967
* 1200.000


17:20 (281.36) -- Augmented second - QD (e.g. F*4-G#4, 278.18, -3.18)
Inharmonic pseudo-hirajoshi
* 187.349
* 266.871
* 683.253
* 832.765
* 1200.000


17:21 (365.83) -- Usual diminished fourth (e.g. F#4-Bb4, 367.24, +1.41)
Unsharpened pseudo-[[pelog]]
* 128.669
* 266.871
* 683.253
* 762.775
* 1200.000
</small>


17:28 (863.87) -- Usual diminished seventh (e.g. F#4-Eb4, 863.14, -0.73)
=== Miscellaneous ===
Undecimal picardy hexatonic{{idiosyncratic}} (original/default tuning)
* 58.680
* 266.871
* 346.393
* 704.096
* 970.967
* 1200.000


21:34 (834.17) -- Usual augmented fifth (e.g. F3-C#4, 832.76, +1.41)
Unflattened pseudo-[[equiheptatonic]]
* 187.349
* 346.393
* 495.904
* 704.096
* 832.765
* 1050.488
* 1200.000


28:51 (1038.08) -- Usual augmented sixth (e.g. Eb3-C#4, 1040.96, +2.87)
== Instruments ==
=== Lumatone ===
* [[:File:Peppermint-C62.ltn]] & [[:File:MillerPeppermintLumatone.jpeg]] — [[Herman Miller]]'s [[Lumatone]] mapping for peppermint-24.


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


21:26 (369.75) -- Usual diminished fourth (e.g. C#4-F4, 367.24, -1.51)
=== Modern Renderings ===
; {{W|John Bull (composer)|John Bull}}
* [https://www.youtube.com/watch?v=Ku32F-zEtmU ''Fantasia «Ut Re Mi Fa Sol La»''] (late 1500s/early 1600s, from ''Fitzwilliam Virginal Book Vol.1 No.51'') – rendered by Claudi Meneghin (2020) in a 24 note per octave well-tempered tuning system that uses both [[golden meantone]] fifths and peppermint fifths (tuning specification in video description).


23:27 (277.59) -- Augmented second - QD (e.g. Eb*4-F#4, 278.18, +0.59)
; [[wikipedia:Wolfgang Amadeus Mozart|Wolfgang Amadeus Mozart]]
* [https://www.youtube.com/watch?v=eRzdbzJah20 ''Mozart's Gigue KV 574, Arranged for Fortepiano (PEPPERMINT)''] (rendered in the 12 note subset by [[Claudi Meneghin]], 2025)
* [https://www.youtube.com/watch?v=2-4oaNq7jwo ''2025-05-24 CHACONNE «LES REGRETS» - PEPPERMINT''] (rendered in a 46EDO-related subset by [[Claudi Meneghin]], (2025) ([https://www.youtube.com/shorts/I8NbVZFsIh0 short version])


26:33 (412.75) -- Usual major third (e.g. F4-A4, 416.38, +3.63)
=== 21st Century ===
; [[Budjarn Lambeth]]
* [https://www.youtube.com/watch?v=g6e3zYlbsWc ''Microtonal Jungle-Inspired Track in the "Salt and Pepper Scale"''] (2025)


28:33 (284.45) -- Usual minor third (e.g. E4-G4, 287.71, +3.27)
; [[Claudi Meneghin]]
* [https://www.youtube.com/watch?v=5vPvI6MXWFM ''ST LOUIS FUGUE (Fugue on St Louis Blues), for Baroque Ensemble - (Microtonal, PEPPERMINT)''] (2025)
* [https://www.youtube.com/watch?v=iZlvKLg4CoM ''PEPPERMINT FUGUE in 5 parts «Les Regrets»''] (2025)


33:56 (915.55) -- Usual major sixth (e.g. G4-E5, 912.29, -3.27)
[[Category:24-tone scales]]
[[Category:24-tone scales]]
[[Category:Todo:cleanup]]
[[Category:Tempered scales]]
[[Category:Todo:clarify]]
[[Category:Todo:clarify]]
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