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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 [http://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 [http://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:Tempered scales]]
[[Category:Todo:clarify]]

Latest revision as of 04:33, 15 May 2026

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)

Subsets

Diatonic and related scales

C Major

  • 208.191
  • 416.382
  • 495.904
  • 704.096
  • 912.287
  • 1120.478
  • 1200.000

Salt and pepper scale and its subsets

Salt and pepper[idiosyncratic term]

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

Evacuated planet[idiosyncratic term] (approximated from 66afdo)

  • 128.669
  • 495.904
  • 704.096
  • 1050.488
  • 1200.000

Flattened pseudo-equiheptatonic

  • 128.669
  • 266.871
  • 495.904
  • 704.096
  • 832.765
  • 1050.488
  • 1200.000

Geode[idiosyncratic term] (approximated from 6afdo)

  • 266.871
  • 495.904
  • 704.096
  • 1050.488
  • 1200.000

Minor hexatonic (approximated from 12edo)

  • 187.349
  • 287.713
  • 495.904
  • 704.096
  • 970.967
  • 1200.000

Pepperbass[idiosyncratic term] (original/default tuning)

(works well with jungle- or trap-style sub bass)

  • 208.191
  • 704.096
  • 891.445
  • 1050.488
  • 1200.000

Pseudo-6afdo

  • 266.871
  • 495.904
  • 704.096
  • 891.445
  • 1050.488
  • 1200.000

Pseudo-akebono I (approximated from 12edo)

  • 208.191
  • 287.713
  • 704.096
  • 891.445
  • 1200.000

Pseudo-akebono II (approximated from 12edo)

  • 128.669
  • 495.904
  • 704.096
  • 832.765
  • 1200.000

Pseudo-equipentatonic

  • 266.871
  • 495.904
  • 704.096
  • 970.967
  • 1200.000

Pseudo-hirajoshi (approximated from 12edo)

  • 208.191
  • 287.713
  • 704.096
  • 832.765
  • 1200.000

Sharpened pseudo-pelog

  • 128.669
  • 287.713
  • 704.096
  • 832.765
  • 1200.000

Ketchup and mustard scale and its subsets

Ketchup and mustard[idiosyncratic term]

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. 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

Inharmonic geode[idiosyncratic term]

  • 266.871
  • 475.062
  • 683.253
  • 1050.488
  • 1200.000

Inharmonic minor hexatonic

  • 187.349
  • 266.871
  • 475.062
  • 683.253
  • 970.967
  • 1200.000

Inharmonic pepperbass[idiosyncratic term]

  • 187.349
  • 683.253
  • 762.775
  • 1050.488
  • 1200.000

Inharmonic pseudo-6afdo

  • 266.871
  • 475.062
  • 683.253
  • 832.765
  • 1050.488
  • 1200.000

Inharmonic pseudo-akebono I

  • 187.349
  • 266.871
  • 683.253
  • 912.287
  • 1200.000

Inharmonic pseudo-akebono II

  • 58.680
  • 475.062
  • 683.253
  • 762.775
  • 1200.000

Inharmonic pseudo-equipentatonic

  • 266.871
  • 475.062
  • 704.096
  • 970.967
  • 1200.000

Inharmonic pseudo-hirajoshi

  • 187.349
  • 266.871
  • 683.253
  • 832.765
  • 1200.000

Unsharpened pseudo-pelog

  • 128.669
  • 266.871
  • 683.253
  • 762.775
  • 1200.000

Miscellaneous

Undecimal picardy hexatonic[idiosyncratic term] (original/default tuning)

  • 58.680
  • 266.871
  • 346.393
  • 704.096
  • 970.967
  • 1200.000

Unflattened pseudo-equiheptatonic

  • 187.349
  • 346.393
  • 495.904
  • 704.096
  • 832.765
  • 1050.488
  • 1200.000

Instruments

Lumatone

Music

Modern Renderings

John Bull
  • 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).
Wolfgang Amadeus Mozart

21st Century

Budjarn Lambeth
Claudi Meneghin
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