Peppermint-24: Difference between revisions

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~~<h2>IMPORTED REVISION FROM WIKISPACES</h2>~~

'''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)]

~~This~~ is ~~an imported revision from Wikispaces. The revision metadata is included below for reference:<br>~~

~~: This revision was~~ by ~~author~~ [[~~User:xenjacob|xenjacob~~]] ~~and made~~ on ~~<tt>2007-06-21 00~~:24:~~54 UTC<~~/~~tt>~~.~~<br>~~

~~: The original revision id was <tt>5371407<~~/~~tt>~~.~~<br>~~

~~: The revision comment was: <tt>formatted, hyperlinked~~.</~~tt><br>~~

~~The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>~~

~~<h4>Original Wikitext content:<~~/~~h4>~~

~~<div style="width~~:~~100%; max-height~~:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">=Peppermint 24=

[[~~http~~:~~//launch.groups.yahoo.com/group/tuning/message/40057~~|~~Original article~~]] ~~by Margo Schulter~~, ~~on the Yahoo tuning forum~~.

==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.

[[~~code~~]]

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~~

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).

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 Temperaments|regular temperament]] mentioned in~~

==Keyboard arrangement ==

~~[[Erv Wilson|Ervin Wilson]]'s Scale Tree and described on the Tuning List by [[Keenan Pepper]],~~

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

~~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:

<pre>

187.349 346.393 683.253 891.445 1050.488

C#* Eb* F#* G#* Bb*

Line 44:

Line 22:

C D E F G A B C

0 208.191 416.382 495.904 704.096 912.287 1120.478 1200

</pre>

== Intervals ==

~~In the following catalogue~~ of ~~some ratio equivalents and mappings, I~~

=== Single chain ===

~~will focus on intervals no further from just than 8:9 or 9:16, which~~

Offset two of these by 58.680 cents.

~~vary from their pure sizes~~ by ~~about 4.282 cents (twice the tempering~~

<pre>

~~of the fifth, at about 2.141 cents wide of 2:3).~~

128.669

208.191

~~Octave numbers appear in a MIDI-style notation, with C4 as middle C;~~

287.713

~~just ratios and tempered equivalents are given values in cents, shown~~

416.382

~~in parentheses, with tempered variations in cents also shown.~~

495.904

624.574

~~To describe the~~ 58.~~68-cent interval between the two keyboards, whose~~

704.096

~~addition or subtraction plays a role in obtaining or approximating~~

832.765

~~many ratios, I shall the term "quasi-diesis," or QD for short. This~~

912.287

~~"artificial" diesis-like interval is actually somewhat larger than the~~

991.809

~~natural diesis in the regular Wilson/Pepper temperament at about 49.15~~

1120.478

cents ~~(12 tempered fifths less 7 pure octaves)~~.

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~~.

~~----~~

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)~~

=== 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>

7:9 ~~(435~~.~~08) -- Fourth - QD~~ (~~e.g. G*4-C5~~, ~~437~~.~~22, +~~2.~~14)~~

==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).

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

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.

~~9:14 (764~~.~~92)~~ -- ~~Fifth +~~ QD ~~(e.g~~. G4-~~D*5, 762~~.~~78, -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).

9~~:16 (996.09)~~ -- ~~Usual minor seventh (e.g. G4-F4~~, ~~991.81, -4~~.~~28)~~

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

6:~~11 (1049.36~~) ~~-- Minor seventh + QD~~ (~~e.g. G3-F*4, 1050.49,~~ +1~~.13~~)

===Multiplex (n:1) and [[superparticular]] (n+1:n) intervals===

7:11 (~~782~~.49) -- Usual minor ~~sixth~~ (e.g. A3-F4, ~~783~~.62, +1.13)

* 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)

~~8:11 (551.32)~~ -- ~~Fourth + QD (e.g. G3-C*4, 554.58, +3~~.~~27)~~

===Other ratios===

Many of these fall within the [[17-odd-limit]].

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

* 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)

~~8:13 (840~~.~~53) -- Minor sixth + QD (e~~.g. ~~G3-Eb~~*~~3, 842~~.~~30, +1~~.~~77)~~

== Subsets ==

=== Diatonic and related scales ===

C Major

* 208.191

* 416.382

* 495.904

* 704.096

* 912.287

* 1120.478

* 1200.000

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

=== Salt and pepper scale and its subsets ===

Salt and pepper{{idiosyncratic}}

~~11:13 (289.21)~~ -- ~~Usual minor third (e~~.g. ~~D3-F3,~~ 287.~~71, -1~~.~~50)~~

''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

~~11:14~~ (~~417~~.~~51) -- Usual major third (e~~.g. ~~D3-F#3, 416~~.~~38, -1~~.~~13)~~

<small>

Evacuated planet{{idiosyncratic}} (approximated from [[66afdo|66]][[afdo]])

* 128.669

* 495.904

* 704.096

* 1050.488

* 1200.000

~~11:16 (648~~.~~68) -- Fifth - QD (e~~.g. F*~~3-C4, 645~~.~~42, -3~~.~~27)~~

Flattened pseudo-[[equiheptatonic]]

* 128.669

* 266.871

* 495.904

* 704.096

* 832.765

* 1050.488

* 1200.000

~~11:18~~ (~~852~~.~~59) -- Major sixth - QD (e~~.g. G*~~4-E5, 853~~.~~61, +1~~.~~02)~~

Geode{{idiosyncratic}} (approximated from [[6afdo]])

* 266.871

* 495.904

* 704.096

* 1050.488

* 1200.000

~~11:21~~ (~~1119~~.~~46) -- Usual major seventh (e~~.g. ~~F3-E4, 1120~~.~~48, +1~~.~~02)~~

Minor hexatonic (approximated from [[12edo]])

* 187.349

* 287.713

* 495.904

* 704.096

* 970.967

* 1200.000

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

Pepperbass{{idiosyncratic}} (original/default tuning)

~~13:16~~ (~~359.47)~~ -- ~~Major third - QD (e~~.g. C*~~4-E4, 357~~.~~70, -1~~.~~77)~~

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

* 208.191

* 704.096

* 891.445

* 1050.488

* 1200.000

~~13:18 (563~~.~~38) -- Augmented fourth - QD (e~~.g. C*~~4-F#4, 565~~.~~89, +2~~.~~51)~~

Pseudo-[[6afdo]]

* 266.871

* 495.904

* 704.096

* 891.445

* 1050.488

* 1200.000

~~13:21~~ (~~830~~.~~25) -- Usual augmented fifth (e~~.g. ~~C4-G#4, 832~~.~~76, +2~~.~~51)~~

Pseudo-akebono I (approximated from [[12edo]])

* 208.191

* 287.713

* 704.096

* 891.445

* 1200.000

~~13:22~~ (~~910~~.~~79) -- Usual major sixth (e~~.g. ~~G3-E4, 912~~.~~29, +1~~.~~50)~~

Pseudo-akebono II (approximated from [[12edo]])

* 128.669

* 495.904

* 704.096

* 832.765

* 1200.000

~~13:23 (987~~.~~75) -- Usual minor seventh (e~~.g. ~~D4-C5, 991~~.~~81, +4~~.~~06)~~

Pseudo-[[equipentatonic]]

* 266.871

* 495.904

* 704.096

* 970.967

* 1200.000

~~13:24~~ (~~1061~~.~~43) -- Major seventh - QD (e~~.g. F*~~3-E4), 1061~~.~~80, +0~~.~~37)~~

Pseudo-hirajoshi (approximated from [[12edo]])

* 208.191

* 287.713

* 704.096

* 832.765

* 1200.000

~~14:17 (336~~.~~13) -- Usual augmented second (e~~.g. ~~F4-G#4, 336~~.~~86, +0~~.~~73)~~

Sharpened pseudo-[[pelog]]

* 128.669

* 287.713

* 704.096

* 832.765

* 1200.000

</small>

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

=== Ketchup and mustard scale and its subsets ===

Ketchup and mustard{{idiosyncratic}}

~~15:17 (216.69)~~ -- ~~Diminished third + QD~~ (e.g. ~~C#4~~-Eb*~~4, 217~~.~~72, +1~~.~~04)~~

''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

~~16:21 (470~~.~~71) -- Major third + QD (e~~.g. ~~C4-E~~*~~4, 475~~.~~06, +4~~.~~28)~~

<small>

Inharmonic geode{{idiosyncratic}}

* 266.871

* 475.062

* 683.253

* 1050.488

* 1200.000

~~16:23 (628~~.~~27) -- Usual augmented fourth (e~~.g. ~~C4-F#4, 624~~.~~57, -3~~.~~70)~~

Inharmonic minor hexatonic

* 187.349

* 266.871

* 475.062

* 683.253

* 970.967

* 1200.000

~~18:23 (424~~.~~36) -- Diminished fourth + QD (e~~.g. ~~B4-Eb~~*~~5, 425~~.~~91, +1~~.~~55)~~

Inharmonic pepperbass{{idiosyncratic}}

* 187.349

* 683.253

* 762.775

* 1050.488

* 1200.000

~~16:25 (772~~.~~63) -- Diminished fourth + QD (e~~.g. ~~F#4-Bb~~*~~4, 774~~.~~09, +1~~.~~46)~~

Inharmonic pseudo-[[6afdo]]

* 266.871

* 475.062

* 683.253

* 832.765

* 1050.488

* 1200.000

~~17:20 (281~~.~~36) -- Augmented second - QD (e~~.g. F*~~4-G#4, 278~~.~~18, -3~~.~~18)~~

Inharmonic pseudo-akebono I

* 187.349

* 266.871

* 683.253

* 912.287

* 1200.000

~~17:21 (365~~.~~83) -- Usual diminished fourth (e~~.g. ~~F#4-Bb4, 367~~.~~24, +1~~.~~41)~~

Inharmonic pseudo-akebono II

* 58.680

* 475.062

* 683.253

* 762.775

* 1200.000

~~17:28 (863~~.~~87) -- Usual diminished seventh (e~~.g. ~~F#4-Eb4, 863~~.~~14, -0~~.~~73)~~

Inharmonic pseudo-[[equipentatonic]]

* 266.871

* 475.062

* 704.096

* 970.967

* 1200.000

~~21:34 (834~~.~~17) -- Usual augmented fifth (e~~.g. ~~F3-C#4,~~ 832.~~76, +1~~.~~41)~~

Inharmonic pseudo-hirajoshi

* 187.349

* 266.871

* 683.253

* 832.765

* 1200.000

~~28:51 (1038~~.~~08) -- Usual augmented sixth (e~~.g. ~~Eb3-C#4, 1040~~.~~96, +2~~.~~87)~~

Unsharpened pseudo-[[pelog]]

* 128.669

* 266.871

* 683.253

* 762.775

* 1200.000

</small>

~~21:23~~ (~~157~~.~~49) -- Usual diminished third (e~~.g. ~~C#4-Eb4, 159~~.~~04, +1~~.~~55)~~

=== Miscellaneous ===

Undecimal picardy hexatonic{{idiosyncratic}} (original/default tuning)

* 58.680

* 266.871

* 346.393

* 704.096

* 970.967

* 1200.000

~~21:26 (369~~.~~75) -- Usual diminished fourth (e~~.g. ~~C#4-F4, 367~~.~~24, -1~~.~~51)~~

Unflattened pseudo-[[equiheptatonic]]

* 187.349

* 346.393

* 495.904

* 704.096

* 832.765

* 1050.488

* 1200.000

23:~~27 (277.59)~~ -~~- Augmented second - QD (e~~.g. ~~Eb*4~~-~~F#4, 278.18, +0~~.~~59)~~

== Instruments ==

=== Lumatone ===

* [[:File:Peppermint-C62.ltn]] & [[:File:MillerPeppermintLumatone.jpeg]] — [[Herman Miller]]'s [[Lumatone]] mapping for peppermint-24.

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

== Music ==

~~28:33~~ (~~284.45~~) ~~-- Usual minor third (e~~.g. E4-G4, ~~287~~.~~71, +3~~.27)

=== 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).

33:56 (~~915~~.~~55)~~ -- ~~Usual major sixth~~ (~~e.g. G4~~-E5, ~~912~~.~~29, -3~~.27)

; [[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])

~~[[code]]</pre></div>~~

=== 21st Century ===

~~<h4>Original HTML content:</h4>~~

; [[Budjarn Lambeth]]

~~<div style~~=~~"width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style~~=~~"margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class~~=~~"old-revision-html"><html><head><title>Peppermint-24</title></head><body><h1 id~~=~~"toc0"><a name="Peppermint 24"></a>Peppermint 24</h1>~~

* [https://www.youtube.com/watch?v=g6e3zYlbsWc ''Microtonal Jungle-Inspired Track in the "Salt and Pepper Scale"''] (2025)

~~<br />~~

~~<a class~~=~~"wiki_link_ext" href~~=~~"http://launch.groups.yahoo.com/group/tuning/message/40057" rel="nofollow">Original article</a> by Margo Schulter, on the Yahoo tuning forum.<br />~~

~~&lt~~;~~br />~~

~~<!-- ws:start:WikiTextCodeRule:0:~~

~~&lt;pre class=&quot;text&quot;&gt; An interesting feature of tuning systems, as implemented on keyboards&lt;br/&gt; (conventional or alternative), is the~~ [[~~keyboard mappings|mapping~~]] of pure or tempered&lt;br/&gt; ratios to positions on the keyboard layout.&lt;br/&gt;&lt;br/&gt; Here I shall explore the mapping of approximate ratios, and especially&lt;br/&gt; of superparticular and other ratios within [[Harry Partch]]'s larger&lt;br/&gt; 17-limit set, in the tuning system and keyboard arrangement I call&lt;br/&gt; Peppermint 24.&lt;br/&gt;&lt;br/&gt; Peppermint 24 takes as its basis a [[Regular Temperaments|regular temperament]] mentioned in&lt;br/&gt; [[Erv Wilson|Ervin Wilson]]'s Scale Tree and described on the Tuning List by [[Keenan Pepper]],&lt;br/&gt; with a fifth of about 704.096 cents, and a precise ratio of&lt;br/&gt; [[Phi]], the Golden Section (~1.618) between the larger chromatic semitone&lt;br/&gt; (e.g. C-C#) at about 128.669 cents and the smaller diatonic semitone&lt;br/&gt; (e.g. C#-D) at about 79.522 cents.&lt;br/&gt;&lt;br/&gt; In Peppermint 24, two regular 12-note chains of this temperament are&lt;br/&gt; placed at a distance of approximately 58.680 cents, so as to yield&lt;br/&gt; some pure ratios of 6:7 (~266.871 cents).&lt;br/&gt;&lt;br/&gt; Here is a 24-note keyboard arrangement, with an asterisk (*) showing a&lt;br/&gt; note on the upper keyboard:~~&lt;br~~/~~&gt;&lt;br~~/~~&gt; 187~~.~~349 346~~.~~393 683.253 891.445 1050.488&lt;br~~/&gt; C#* Eb* F#* G#* Bb*&lt;br/&gt; C* D* E* F* G* A* B* C*&lt;br/&gt;58.680 266.871 475.062 554.584 762.775 970.967 1179.158 1258.680&lt;br/&gt; 7/6&lt;br/&gt;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;-&lt;br/&gt; 128.669 287.713 624.574 832.765 991.809&lt;br/&gt; C# Eb F# G# Bb&lt;br/&gt; C D E F G A B C&lt;br/&gt; 0 208.191 416.382 495.904 704.096 912.287 1120.478 1200&lt;br/&gt;&lt;br/&gt;&lt;br/&gt; In the following catalogue of some ratio equivalents and mappings, I&lt;br/&gt; will focus on intervals no further from just than 8:9 or 9:16, which&lt;br/&gt; vary from their pure sizes by about 4.282 cents (twice the tempering&lt;br/&gt; of the fifth, at about 2.141 cents wide of 2:3).&lt;br/&gt;&lt;br/&gt; Octave numbers appear in a MIDI-style notation, with C4 as middle C;&lt;br/&gt; just ratios and tempered equivalents are given values in cents, shown&lt;br/&gt; in parentheses, with tempered variations in cents also shown.&lt;br/&gt;&lt;br/&gt; To describe the 58.68-cent interval between the two keyboards, whose&lt;br/&gt; addition or subtraction plays a role in obtaining or approximating&lt;br/&gt; many ratios, I shall the term &amp;quot;quasi-diesis,&amp;quot; or QD for short. This&lt;br/&gt; &amp;quot;artificial&amp;quot; diesis-like interval is actually somewhat larger than the&lt;br/&gt; natural diesis in the regular Wilson/Pepper temperament at about 49.15&lt;br/&gt; cents (12 tempered fifths less 7 pure octaves).&lt;br/&gt;&lt;br/&gt; As this partial catalogue might suggest, many ratios of 2-3-7-9-11-13&lt;br/&gt; are represented quite accurately, with 14:17:21 and related ratios&lt;br/&gt; also closely approximated.&lt;br/&gt;&lt;br/&gt;&lt;br/&gt;&amp;#45;&amp;#45;&amp;#45;-&lt;br/&gt; 1. Multiplex (n:1) and [[superparticular]] (n+1:n) intervals&lt;br/&gt;&amp;#45;&amp;#45;&amp;#45;-&lt;br/&gt;&lt;br/&gt; 1:2 (1200) &amp;#45;- This is the usual octave (e.g. F3-F4), at a pure 1:2.&lt;br/&gt;&lt;br/&gt; 2:3 (701.96) &amp;#45;- This is the usual fifth (e.g. F3-C4, 704.10, +2.14).&lt;br/&gt;&lt;br/&gt; 3:4 (498.04) &amp;#45;- Usual fourth (e.g. C4-F4, 495.90, -2.14).&lt;br/&gt;&lt;br/&gt; 6:7 (266.87) &amp;#45;- Major second + QD (e.g. D4-E*4), at a pure 6:7.&lt;br/&gt;&lt;br/&gt; 7:8 (231.17) &amp;#45;- Minor third - QD (e.g. C*4-Eb4, 229.03, -2.14)&lt;br/&gt;&lt;br/&gt; 8:9 (203.91) &amp;#45;- Usual major second (e.g. C4-D4, 208.19, +4.28)&lt;br/&gt;&lt;br/&gt; 11:12 (150.64) &amp;#45;- Major second - QD (e.g. C*4-D4, 149.51, -1.13)&lt;br/&gt;&lt;br/&gt; 12:13 (138.57) &amp;#45;- Minor second + QD (e.g. E4-F*4, 138.20, -0.37)&lt;br/&gt;&lt;br/&gt; 13:14 (128.30) &amp;#45;- Usual apotome (e.g. C4-C#4, 128.67, +0.37)&lt;br/&gt;&lt;br/&gt; 17:18 (98.95) &amp;#45;- Diminished third - QD (e.g. G#*4-Bb4, 100.36, -1.41)&lt;br/&gt;&lt;br/&gt; 21:22 (80.54) &amp;#45;- Usual minor second (e.g. E4-F4, 79.52, -1.02)&lt;br/&gt;&lt;br/&gt; 24:25 (70.67) &amp;#45;- Apotome - QD (e.g. E*4-Eb4, 69.99, -0.68)&lt;br/&gt;&lt;br/&gt; 27:28 (62.96) &amp;#45;- QD (e.g. E4-E*4, 58.68, -4.28)&lt;br/&gt;&lt;br/&gt;&lt;br/&gt;&amp;#45;&amp;#45;&amp;#45;-&lt;br/&gt; 2. Other ratios &amp;#45;- many within 17-odd limit&lt;br/&gt;&amp;#45;&amp;#45;&amp;#45;-&lt;br/&gt;&lt;br/&gt; 4:7 (968.83) &amp;#45;- Major sixth + QD (e.g. G3-E*4, 970.97, +2.14)&lt;br/&gt;&lt;br/&gt; 7:9 (435.08) &amp;#45;- Fourth - QD (e.g. G*4-C5, 437.22, +2.14)&lt;br/&gt;&lt;br/&gt; 7:12 (933.13) &amp;#45;- Minor seventh - QD (e.g. G*3-F4), at a pure 7:12.&lt;br/&gt;&lt;br/&gt; 9:14 (764.92) &amp;#45;- Fifth + QD (e.g. G4-D*5, 762.78, -2.14)&lt;br/&gt;&lt;br/&gt; 9:16 (996.09) &amp;#45;- Usual minor seventh (e.g. G4-F4, 991.81, -4.28)&lt;br/&gt;&lt;br/&gt; 6:11 (1049.36) &amp;#45;- Minor seventh + QD (e.g. G3-F*4, 1050.49, +1.13)&lt;br/&gt;&lt;br/&gt; 7:11 (782.49) &amp;#45;- Usual minor sixth (e.g. A3-F4, 783.62, +1.13)&lt;br/&gt;&lt;br/&gt; 8:11 (551.32) &amp;#45;- Fourth + QD (e.g. G3-C*4, 554.58, +3.27)&lt;br/&gt;&lt;br/&gt; 9:11 (347.41) &amp;#45;- Minor third + QD (e.g. G3-Bb*3, 346.39, -1.02)&lt;br/&gt;&lt;br/&gt; 8:13 (840.53) &amp;#45;- Minor sixth + QD (e.g. G3-Eb*3, 842.30, +1.77)&lt;br/&gt;&lt;br/&gt; 9:13 (636.62) &amp;#45;- Diminished fifth + QD (e.g. A3-Eb*4, 634.11, -2.51)&lt;br/&gt;&lt;br/&gt; 11:13 (289.21) &amp;#45;- Usual minor third (e.g. D3-F3, 287.71, -1.50)&lt;br/&gt;&lt;br/&gt; 11:14 (417.51) &amp;#45;- Usual major third (e.g. D3-F#3, 416.38, -1.13)&lt;br/&gt;&lt;br/&gt; 11:16 (648.68) &amp;#45;- Fifth - QD (e.g. F*3-C4, 645.42, -3.27)&lt;br/&gt;&lt;br/&gt; 11:18 (852.59) &amp;#45;- Major sixth - QD (e.g. G*4-E5, 853.61, +1.02)&lt;br/&gt;&lt;br/&gt; 11:21 (1119.46) &amp;#45;- Usual major seventh (e.g. F3-E4, 1120.48, +1.02)&lt;br/&gt;&lt;br/&gt; 12:17 (603.00) &amp;#45;- Augmented third + QD (e.g. Eb4-G#*4, 603.73, +0.73)&lt;br/&gt;&lt;br/&gt; 13:16 (359.47) &amp;#45;- Major third - QD (e.g. C*4-E4, 357.70, -1.77)&lt;br/&gt;&lt;br/&gt; 13:18 (563.38) &amp;#45;- Augmented fourth - QD (e.g. C*4-F#4, 565.89, +2.51)&lt;br/&gt;&lt;br/&gt; 13:21 (830.25) &amp;#45;- Usual augmented fifth (e.g. C4-G#4, 832.76, +2.51)&lt;br/&gt;&lt;br/&gt; 13:22 (910.79) &amp;#45;- Usual major sixth (e.g. G3-E4, 912.29, +1.50)&lt;br/&gt;&lt;br/&gt; 13:23 (987.75) &amp;#45;- Usual minor seventh (e.g. D4-C5, 991.81, +4.06)&lt;br/&gt;&lt;br/&gt; 13:24 (1061.43) &amp;#45;- Major seventh - QD (e.g. F*3-E4), 1061.80, +0.37)&lt;br/&gt;&lt;br/&gt; 14:17 (336.13) &amp;#45;- Usual augmented second (e.g. F4-G#4, 336.86, +0.73)&lt;br/&gt;&lt;br/&gt; 14:27 (1137.04) &amp;#45;- Octave - QD (e.g. F*4-F5, 1141.32, +4.28)&lt;br/&gt;&lt;br/&gt; 15:17 (216.69) &amp;#45;- Diminished third + QD (e.g. C#4-Eb*4, 217.72, +1.04)&lt;br/&gt;&lt;br/&gt; 16:21 (470.71) &amp;#45;- Major third + QD (e.g. C4-E*4, 475.06, +4.28)&lt;br/&gt;&lt;br/&gt; 16:23 (628.27) &amp;#45;- Usual augmented fourth (e.g. C4-F#4, 624.57, -3.70)&lt;br/&gt;&lt;br/&gt; 18:23 (424.36) &amp;#45;- Diminished fourth + QD (e.g. B4-Eb*5, 425.91, +1.55)&lt;br/&gt;&lt;br/&gt; 16:25 (772.63) &amp;#45;- Diminished fourth + QD (e.g. F#4-Bb*4, 774.09, +1.46)&lt;br/&gt;&lt;br/&gt; 17:20 (281.36) &amp;#45;- Augmented second - QD (e.g. F*4-G#4, 278.18, -3.18)&lt;br/&gt;&lt;br/&gt; 17:21 (365.83) &amp;#45;- Usual diminished fourth (e.g. F#4-Bb4, 367.24, +1.41)&lt;br/&gt;&lt;br/&gt; 17:28 (863.87) &amp;#45;- Usual diminished seventh (e.g. F#4-Eb4, 863.14, -0.73)&lt;br/&gt;&lt;br/&gt; 21:34 (834.17) &amp;#45;- Usual augmented fifth (e.g. F3-C#4, 832.76, +1.41)&lt;br/&gt;&lt;br/&gt; 28:51 (1038.08) &amp;#45;- Usual augmented sixth (e.g. Eb3-C#4, 1040.96, +2.87)&lt;br/&gt;&lt;br/&gt; 21:23 (157.49) &amp;#45;- Usual diminished third (e.g. C#4-Eb4, 159.04, +1.55)&lt;br/&gt;&lt;br/&gt; 21:26 (369.75) &amp;#45;- Usual diminished fourth (e.g. C#4-F4, 367.24, -1.51)&lt;br/&gt;&lt;br/&gt; 23:27 (277.59) &amp;#45;- Augmented second - QD (e.g. Eb*4-F#4, 278.18, +0.59)&lt;br/&gt;&lt;br/&gt; 26:33 (412.75) &amp;#45;- Usual major third (e.g. F4-A4, 416.38, +3.63)&lt;br/&gt;&lt;br/&gt; 28:33 (284.45) &amp;#45;- Usual minor third (e.g. E4-G4, 287.71, +3.27)&lt;br/&gt;&lt;br/&gt; 33:56 (915.55) &amp;#45;- Usual major sixth (e.g. G4-E5, 912.29, -3.27)&lt;br/&gt;&lt;/pre&gt;

~~-->~~

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; [[Claudi Meneghin]]

~~</style><pre class="text"&gt~~; ~~An interesting feature of tuning systems, as implemented on keyboards~~

* [https://www.youtube.com/watch?v=5vPvI6MXWFM ''ST LOUIS FUGUE (Fugue on St Louis Blues), for Baroque Ensemble - (Microtonal, PEPPERMINT)''] (2025)

~~(conventional or alternative), is the~~ [[~~keyboard mappings|mapping~~]] ~~of pure or tempered~~

* [https://www.youtube.com/watch?v=iZlvKLg4CoM ''PEPPERMINT FUGUE in 5 parts «Les Regrets»''] (2025)

~~ratios to positions on the keyboard layout.~~

~~&nbsp;~~

~~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.~~

~~&nbsp;~~

~~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]],~~

~~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.~~

~~&nbsp;~~

~~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)~~.

~~&nbsp;~~

~~Here is a 24-note keyboard arrangement, with an asterisk~~ (*) showing a

~~note on the upper keyboard:~~

~~&nbsp;~~

~~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~~

~~&nbsp;~~

~~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~~).

~~&nbsp;~~

~~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.~~

~~&nbsp;~~

~~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 &quot;quasi-diesis,&quot; or QD~~ for ~~short. This~~

~~&quot;artificial&quot; 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).~~

~~&nbsp;~~

~~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.~~

~~&nbsp;~~

~~----~~

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

~~----~~

~~&nbsp;~~

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

~~&nbsp;~~

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

~~&nbsp;~~

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

~~&nbsp;~~

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

~~&nbsp;~~

~~7:8 (231~~.~~17) -- Minor third - QD (e.g. C*4-Eb4, 229.03, -2.14)~~

~~&nbsp;~~

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

~~&nbsp;~~

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

~~&nbsp;~~

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

~~&nbsp;~~

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

~~&nbsp;~~

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

~~&nbsp;~~

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

~~&nbsp;~~

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

~~&nbsp;~~

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

~~&nbsp;~~

~~----~~

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

~~----~~

~~&nbsp;~~

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

~~&nbsp;~~

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

~~&nbsp;~~

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

~~&nbsp;~~

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

~~&nbsp;~~

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

~~&nbsp;~~

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

~~&nbsp;~~

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

~~&nbsp;~~

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

~~&nbsp;~~

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

~~&nbsp;~~

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

~~&nbsp;~~

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

~~&nbsp;~~

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

~~&nbsp;~~

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

~~&nbsp;~~

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

~~&nbsp;~~

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

~~&nbsp;~~

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

~~&nbsp;~~

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

~~&nbsp;~~

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

~~&nbsp;~~

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

~~&nbsp;~~

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

~~&nbsp;~~

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

~~&nbsp;~~

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

~~&nbsp;~~

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

~~&nbsp;~~

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

~~&nbsp;~~

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

~~&nbsp;~~

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

~~&nbsp;~~

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

~~&nbsp;~~

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

~~&nbsp;~~

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

~~&nbsp;~~

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

~~&nbsp;~~

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

~~&nbsp;~~

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

~~&nbsp;~~

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

~~&nbsp;~~

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

~~&nbsp;~~

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

~~&nbsp;~~

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

~~&nbsp;~~

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

~~&nbsp;~~

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

~~&nbsp;~~

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

~~&nbsp;~~

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

~~&nbsp;~~

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

~~&nbsp;</pre>~~

~~<!-~~- ws:~~end~~:~~WikiTextCodeRule~~:~~0 --></body></html></pre></div>~~

[[Category:24-tone scales]]

[[Category:Tempered scales]]

[[Category:Todo:clarify]]

@@ Line 1: / Line 1: @@
-<h2>IMPORTED REVISION FROM WIKISPACES</h2>
+'''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)]
-This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
-: This revision was by author [[User:xenjacob|xenjacob]] and made on <tt>2007-06-21 00:24:54 UTC</tt>.<br>
-: The original revision id was <tt>5371407</tt>.<br>
-: The revision comment was: <tt>formatted, hyperlinked.</tt><br>
-The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
-<h4>Original Wikitext content:</h4>
-<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">=Peppermint 24=
-[[http://launch.groups.yahoo.com/group/tuning/message/40057|Original article]] by Margo Schulter, on the Yahoo tuning forum.
+==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.
-[[code]]
+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
+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).
- of superparticular and other ratios within [[Harry Partch]]'s larger
--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
+==Keyboard arrangement ==
- [[Erv Wilson|Ervin Wilson]]'s Scale Tree and described on the Tuning List by [[Keenan Pepper]],
+Here is a 24-note keyboard arrangement, with an asterisk (*) showing a note on the upper keyboard:
- 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:
+<pre>
 .349  346.393              683.253    891.445 1050.488
         C#*      Eb*                   F#*       G#*     Bb*
@@ Line 44: / Line 22: @@
    C         D           E       F          G         A        B     C
        208.191    416.382 495.904    704.096   912.287 1120.478 1200
+</pre>
+== Intervals ==
- In the following catalogue of some ratio equivalents and mappings, I
+=== Single chain ===
- will focus on intervals no further from just than 8:9 or 9:16, which
+Offset two of these by 58.680 cents.
- vary from their pure sizes by about 4.282 cents (twice the tempering
+<pre>
- of the fifth, at about 2.141 cents wide of 2:3).
+.669
+.191
- Octave numbers appear in a MIDI-style notation, with C4 as middle C;
+.713
- just ratios and tempered equivalents are given values in cents, shown
+.382
- in parentheses, with tempered variations in cents also shown.
+.904
+.574
- To describe the 58.68-cent interval between the two keyboards, whose
+.096
- addition or subtraction plays a role in obtaining or approximating
+.765
- many ratios, I shall the term "quasi-diesis," or QD for short. This
+.287
- "artificial" diesis-like interval is actually somewhat larger than the
+.809
- natural diesis in the regular Wilson/Pepper temperament at about 49.15
+.478
- cents (12 tempered fifths less 7 pure octaves).
+.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.
-----
-. Multiplex (n:1) and [[superparticular]] (n+1:n) intervals
-----
-:2 (1200) -- This is the usual octave (e.g. F3-F4), at a pure 1:2.
-:3 (701.96) -- This is the usual fifth (e.g. F3-C4, 704.10, +2.14).
-:4 (498.04) -- Usual fourth (e.g. C4-F4, 495.90, -2.14).
-:7 (266.87) -- Major second + QD (e.g. D4-E*4), at a pure 6:7.
-:8 (231.17) -- Minor third - QD (e.g. C*4-Eb4, 229.03, -2.14)
-:9 (203.91) -- Usual major second (e.g. C4-D4, 208.19, +4.28)
-:12 (150.64) -- Major second - QD (e.g. C*4-D4, 149.51, -1.13)
-:13 (138.57) -- Minor second + QD (e.g. E4-F*4, 138.20, -0.37)
-:14 (128.30) -- Usual apotome (e.g. C4-C#4, 128.67, +0.37)
-:18 (98.95) -- Diminished third - QD (e.g. G#*4-Bb4, 100.36, -1.41)
-:22 (80.54) -- Usual minor second (e.g. E4-F4, 79.52, -1.02)
-:25 (70.67) -- Apotome - QD (e.g. E*4-Eb4, 69.99, -0.68)
-:28 (62.96) -- QD (e.g. E4-E*4, 58.68, -4.28)
-----
-. Other ratios -- many within 17-odd limit
-----
-:7 (968.83) -- Major sixth + QD (e.g. G3-E*4, 970.97, +2.14)
+=== Combined (both chains) ===
+<pre>
+.680
+.669
+.349
+.191
+.871
+.713
+.393
+.382
+.062
+.904
+.584
+.574
+.253
+.096
+.775
+.765
+.445
+.287
+.967
+.809
+.488
+.478
+.157
+.000
+</pre>
-:9 (435.08) -- Fourth - QD (e.g. G*4-C5, 437.22, +2.14)
+==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).
-:12 (933.13) -- Minor seventh - QD (e.g. G*3-F4), at a pure 7:12.
+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.
-:14 (764.92) -- Fifth + QD (e.g. G4-D*5, 762.78, -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).
-:16 (996.09) -- Usual minor seventh (e.g. G4-F4, 991.81, -4.28)
+Many ratios of 2-3-7-9-11-13 are represented quite accurately, with 14:17:21 and related ratios also closely approximated.
-:11 (1049.36) -- Minor seventh + QD (e.g. G3-F*4, 1050.49, +1.13)
+===Multiplex (n:1) and [[superparticular]] (n+1:n) intervals===
-:11 (782.49) -- Usual minor sixth (e.g. A3-F4, 783.62, +1.13)
+* 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)
-:11 (551.32) -- Fourth + QD (e.g. G3-C*4, 554.58, +3.27)
+===Other ratios===
+Many of these fall within the [[17-odd-limit]].
-:11 (347.41) -- Minor third + QD (e.g. G3-Bb*3, 346.39, -1.02)
+* 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)
-:13 (840.53) -- Minor sixth + QD (e.g. G3-Eb*3, 842.30, +1.77)
+== Subsets ==
+=== Diatonic and related scales ===
+C Major
+* 208.191
+* 416.382
+* 495.904
+* 704.096
+* 912.287
+* 1120.478
+* 1200.000
-:13 (636.62) -- Diminished fifth + QD (e.g. A3-Eb*4, 634.11, -2.51)
+=== Salt and pepper scale and its subsets ===
+Salt and pepper{{idiosyncratic}}
-:13 (289.21) -- Usual minor third (e.g. D3-F3, 287.71, -1.50)
+''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
-:14 (417.51) -- Usual major third (e.g. D3-F#3, 416.38, -1.13)
+<small>
+Evacuated planet{{idiosyncratic}} (approximated from [[66afdo|66]][[afdo]])
+* 128.669
+* 495.904
+* 704.096
+* 1050.488
+* 1200.000
-:16 (648.68) -- Fifth - QD (e.g. F*3-C4, 645.42, -3.27)
+Flattened pseudo-[[equiheptatonic]]
+* 128.669
+* 266.871
+* 495.904
+* 704.096
+* 832.765
+* 1050.488
+* 1200.000
-:18 (852.59) -- Major sixth - QD (e.g. G*4-E5, 853.61, +1.02)
+Geode{{idiosyncratic}} (approximated from [[6afdo]])
+* 266.871
+* 495.904
+* 704.096
+* 1050.488
+* 1200.000
-:21 (1119.46) -- Usual major seventh (e.g. F3-E4, 1120.48, +1.02)
+Minor hexatonic (approximated from [[12edo]])
+* 187.349
+* 287.713
+* 495.904
+* 704.096
+* 970.967
+* 1200.000
-:17 (603.00) -- Augmented third + QD (e.g. Eb4-G#*4, 603.73, +0.73)
+Pepperbass{{idiosyncratic}} (original/default tuning)
-:16 (359.47) -- Major third - QD (e.g. C*4-E4, 357.70, -1.77)
+(''works well with jungle- or trap-style sub bass'')
+* 208.191
+* 704.096
+* 891.445
+* 1050.488
+* 1200.000
-:18 (563.38) -- Augmented fourth - QD (e.g. C*4-F#4, 565.89, +2.51)
+Pseudo-[[6afdo]]
+* 266.871
+* 495.904
+* 704.096
+* 891.445
+* 1050.488
+* 1200.000
-:21 (830.25) -- Usual augmented fifth (e.g. C4-G#4, 832.76, +2.51)
+Pseudo-akebono I (approximated from [[12edo]])
+* 208.191
+* 287.713
+* 704.096
+* 891.445
+* 1200.000
-:22 (910.79) -- Usual major sixth (e.g. G3-E4, 912.29, +1.50)
+Pseudo-akebono II (approximated from [[12edo]])
+* 128.669
+* 495.904
+* 704.096
+* 832.765
+* 1200.000
-:23 (987.75) -- Usual minor seventh (e.g. D4-C5, 991.81, +4.06)
+Pseudo-[[equipentatonic]]
+* 266.871
+* 495.904
+* 704.096
+* 970.967
+* 1200.000
-:24 (1061.43) -- Major seventh - QD (e.g. F*3-E4), 1061.80, +0.37)
+Pseudo-hirajoshi (approximated from [[12edo]])
+* 208.191
+* 287.713
+* 704.096
+* 832.765
+* 1200.000
-:17 (336.13) -- Usual augmented second (e.g. F4-G#4, 336.86, +0.73)
+Sharpened pseudo-[[pelog]]
+* 128.669
+* 287.713
+* 704.096
+* 832.765
+* 1200.000
+</small>
-:27 (1137.04) -- Octave - QD (e.g. F*4-F5, 1141.32, +4.28)
+=== Ketchup and mustard scale and its subsets ===
+Ketchup and mustard{{idiosyncratic}}
-:17 (216.69) -- Diminished third + QD (e.g. C#4-Eb*4, 217.72, +1.04)
+''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
-:21 (470.71) -- Major third + QD (e.g. C4-E*4, 475.06, +4.28)
+<small>
+Inharmonic geode{{idiosyncratic}}
+* 266.871
+* 475.062
+* 683.253
+* 1050.488
+* 1200.000
-:23 (628.27) -- Usual augmented fourth (e.g. C4-F#4, 624.57, -3.70)
+Inharmonic minor hexatonic
+* 187.349
+* 266.871
+* 475.062
+* 683.253
+* 970.967
+* 1200.000
-:23 (424.36) -- Diminished fourth + QD (e.g. B4-Eb*5, 425.91, +1.55)
+Inharmonic pepperbass{{idiosyncratic}}
+* 187.349
+* 683.253
+* 762.775
+* 1050.488
+* 1200.000
-:25 (772.63) -- Diminished fourth + QD (e.g. F#4-Bb*4, 774.09, +1.46)
+Inharmonic pseudo-[[6afdo]]
+* 266.871
+* 475.062
+* 683.253
+* 832.765
+* 1050.488
+* 1200.000
-:20 (281.36) -- Augmented second - QD (e.g. F*4-G#4, 278.18, -3.18)
+Inharmonic pseudo-akebono I
+* 187.349
+* 266.871
+* 683.253
+* 912.287
+* 1200.000
-:21 (365.83) -- Usual diminished fourth (e.g. F#4-Bb4, 367.24, +1.41)
+Inharmonic pseudo-akebono II
+* 58.680
+* 475.062
+* 683.253
+* 762.775
+* 1200.000
-:28 (863.87) -- Usual diminished seventh (e.g. F#4-Eb4, 863.14, -0.73)
+Inharmonic pseudo-[[equipentatonic]]
+* 266.871
+* 475.062
+* 704.096
+* 970.967
+* 1200.000
-:34 (834.17) -- Usual augmented fifth (e.g. F3-C#4, 832.76, +1.41)
+Inharmonic pseudo-hirajoshi
+* 187.349
+* 266.871
+* 683.253
+* 832.765
+* 1200.000
-:51 (1038.08) -- Usual augmented sixth (e.g. Eb3-C#4, 1040.96, +2.87)
+Unsharpened pseudo-[[pelog]]
+* 128.669
+* 266.871
+* 683.253
+* 762.775
+* 1200.000
+</small>
-:23 (157.49) -- Usual diminished third (e.g. C#4-Eb4, 159.04, +1.55)
+=== Miscellaneous ===
+Undecimal picardy hexatonic{{idiosyncratic}} (original/default tuning)
+* 58.680
+* 266.871
+* 346.393
+* 704.096
+* 970.967
+* 1200.000
-:26 (369.75) -- Usual diminished fourth (e.g. C#4-F4, 367.24, -1.51)
+Unflattened pseudo-[[equiheptatonic]]
+* 187.349
+* 346.393
+* 495.904
+* 704.096
+* 832.765
+* 1050.488
+* 1200.000
-:27 (277.59) -- Augmented second - QD (e.g. Eb*4-F#4, 278.18, +0.59)
+== Instruments ==
+=== Lumatone ===
+* [[:File:Peppermint-C62.ltn]] & [[:File:MillerPeppermintLumatone.jpeg]] — [[Herman Miller]]'s [[Lumatone]] mapping for peppermint-24.
-:33 (412.75) -- Usual major third (e.g. F4-A4, 416.38, +3.63)
+== Music ==
-:33 (284.45) -- Usual minor third (e.g. E4-G4, 287.71, +3.27)
+=== 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).
-:56 (915.55) -- Usual major sixth (e.g. G4-E5, 912.29, -3.27)
+; [[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])
-[[code]]</pre></div>
+=== 21st Century ===
-<h4>Original HTML content:</h4>
+; [[Budjarn Lambeth]]
-<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;Peppermint-24&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextHeadingRule:1:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Peppermint 24"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:1 --&gt;Peppermint 24&lt;/h1&gt;
+* [https://www.youtube.com/watch?v=g6e3zYlbsWc ''Microtonal Jungle-Inspired Track in the "Salt and Pepper Scale"''] (2025)
- &lt;br /&gt;
-&lt;a class="wiki_link_ext" href="http://launch.groups.yahoo.com/group/tuning/message/40057" rel="nofollow"&gt;Original article&lt;/a&gt; by Margo Schulter, on the Yahoo tuning forum.&lt;br /&gt;
-&lt;br /&gt;
-&lt;!-- ws:start:WikiTextCodeRule:0:
-&amp;lt;pre class=&amp;quot;text&amp;quot;&amp;gt; An interesting feature of tuning systems, as implemented on keyboards&amp;lt;br/&amp;gt; (conventional or alternative), is the [[keyboard mappings|mapping]] of pure or tempered&amp;lt;br/&amp;gt; ratios to positions on the keyboard layout.&amp;lt;br/&amp;gt;&amp;lt;br/&amp;gt; Here I shall explore the mapping of approximate ratios, and especially&amp;lt;br/&amp;gt; of superparticular and other ratios within [[Harry Partch]]'s larger&amp;lt;br/&amp;gt; 17-limit set, in the tuning system and keyboard arrangement I call&amp;lt;br/&amp;gt; Peppermint 24.&amp;lt;br/&amp;gt;&amp;lt;br/&amp;gt; Peppermint 24 takes as its basis a [[Regular Temperaments|regular temperament]] mentioned in&amp;lt;br/&amp;gt; [[Erv Wilson|Ervin Wilson]]'s Scale Tree and described on the Tuning List by [[Keenan Pepper]],&amp;lt;br/&amp;gt; with a fifth of about 704.096 cents, and a precise ratio of&amp;lt;br/&amp;gt; [[Phi]], the Golden Section (~1.618) between the larger chromatic semitone&amp;lt;br/&amp;gt; (e.g. C-C#) at about 128.669 cents and the smaller diatonic semitone&amp;lt;br/&amp;gt; (e.g. C#-D) at about 79.522 cents.&amp;lt;br/&amp;gt;&amp;lt;br/&amp;gt; In Peppermint 24, two regular 12-note chains of this temperament are&amp;lt;br/&amp;gt; placed at a distance of approximately 58.680 cents, so as to yield&amp;lt;br/&amp;gt; some pure ratios of 6:7 (~266.871 cents).&amp;lt;br/&amp;gt;&amp;lt;br/&amp;gt; Here is a 24-note keyboard arrangement, with an asterisk (*) showing a&amp;lt;br/&amp;gt; note on the upper keyboard:&amp;lt;br/&amp;gt;&amp;lt;br/&amp;gt;     187.349  346.393              683.253    891.445 1050.488&amp;lt;br/&amp;gt;       C#*      Eb*                   F#*       G#*     Bb*&amp;lt;br/&amp;gt;  C*        D*          E*      F*        G*        A*       B*     C*&amp;lt;br/&amp;gt;58.680   266.871    475.062  554.584   762.775  970.967  1179.158 1258.680&amp;lt;br/&amp;gt;           7/6&amp;lt;br/&amp;gt;&amp;amp;#45;&amp;amp;#45;&amp;amp;#45;&amp;amp;#45;&amp;amp;#45;&amp;amp;#45;&amp;amp;#45;&amp;amp;#45;&amp;amp;#45;&amp;amp;#45;&amp;amp;#45;&amp;amp;#45;&amp;amp;#45;&amp;amp;#45;&amp;amp;#45;&amp;amp;#45;&amp;amp;#45;&amp;amp;#45;&amp;amp;#45;&amp;amp;#45;&amp;amp;#45;&amp;amp;#45;&amp;amp;#45;&amp;amp;#45;&amp;amp;#45;&amp;amp;#45;&amp;amp;#45;&amp;amp;#45;&amp;amp;#45;&amp;amp;#45;&amp;amp;#45;&amp;amp;#45;&amp;amp;#45;&amp;amp;#45;&amp;amp;#45;&amp;amp;#45;&amp;amp;#45;&amp;amp;#45;&amp;amp;#45;&amp;amp;#45;&amp;amp;#45;&amp;amp;#45;&amp;amp;#45;&amp;amp;#45;&amp;amp;#45;&amp;amp;#45;&amp;amp;#45;&amp;amp;#45;&amp;amp;#45;&amp;amp;#45;&amp;amp;#45;&amp;amp;#45;&amp;amp;#45;&amp;amp;#45;&amp;amp;#45;&amp;amp;#45;&amp;amp;#45;&amp;amp;#45;&amp;amp;#45;&amp;amp;#45;&amp;amp;#45;&amp;amp;#45;&amp;amp;#45;&amp;amp;#45;&amp;amp;#45;&amp;amp;#45;&amp;amp;#45;&amp;amp;#45;&amp;amp;#45;&amp;amp;#45;&amp;amp;#45;&amp;amp;#45;-&amp;lt;br/&amp;gt;     128.669  287.713              624.574    832.765 991.809&amp;lt;br/&amp;gt;        C#      Eb                    F#         G#     Bb&amp;lt;br/&amp;gt;  C         D           E       F          G         A        B     C&amp;lt;br/&amp;gt;  0       208.191    416.382 495.904    704.096   912.287 1120.478 1200&amp;lt;br/&amp;gt;&amp;lt;br/&amp;gt;&amp;lt;br/&amp;gt; In the following catalogue of some ratio equivalents and mappings, I&amp;lt;br/&amp;gt; will focus on intervals no further from just than 8:9 or 9:16, which&amp;lt;br/&amp;gt; vary from their pure sizes by about 4.282 cents (twice the tempering&amp;lt;br/&amp;gt; of the fifth, at about 2.141 cents wide of 2:3).&amp;lt;br/&amp;gt;&amp;lt;br/&amp;gt; Octave numbers appear in a MIDI-style notation, with C4 as middle C;&amp;lt;br/&amp;gt; just ratios and tempered equivalents are given values in cents, shown&amp;lt;br/&amp;gt; in parentheses, with tempered variations in cents also shown.&amp;lt;br/&amp;gt;&amp;lt;br/&amp;gt; To describe the 58.68-cent interval between the two keyboards, whose&amp;lt;br/&amp;gt; addition or subtraction plays a role in obtaining or approximating&amp;lt;br/&amp;gt; many ratios, I shall the term &amp;amp;quot;quasi-diesis,&amp;amp;quot; or QD for short. This&amp;lt;br/&amp;gt; &amp;amp;quot;artificial&amp;amp;quot; diesis-like interval is actually somewhat larger than the&amp;lt;br/&amp;gt; natural diesis in the regular Wilson/Pepper temperament at about 49.15&amp;lt;br/&amp;gt; cents (12 tempered fifths less 7 pure octaves).&amp;lt;br/&amp;gt;&amp;lt;br/&amp;gt; As this partial catalogue might suggest, many ratios of 2-3-7-9-11-13&amp;lt;br/&amp;gt; are represented quite accurately, with 14:17:21 and related ratios&amp;lt;br/&amp;gt; also closely approximated.&amp;lt;br/&amp;gt;&amp;lt;br/&amp;gt;&amp;lt;br/&amp;gt;&amp;amp;#45;&amp;amp;#45;&amp;amp;#45;-&amp;lt;br/&amp;gt; 1. Multiplex (n:1) and [[superparticular]] (n+1:n) intervals&amp;lt;br/&amp;gt;&amp;amp;#45;&amp;amp;#45;&amp;amp;#45;-&amp;lt;br/&amp;gt;&amp;lt;br/&amp;gt; 1:2 (1200) &amp;amp;#45;- This is the usual octave (e.g. F3-F4), at a pure 1:2.&amp;lt;br/&amp;gt;&amp;lt;br/&amp;gt; 2:3 (701.96) &amp;amp;#45;- This is the usual fifth (e.g. F3-C4, 704.10, +2.14).&amp;lt;br/&amp;gt;&amp;lt;br/&amp;gt; 3:4 (498.04) &amp;amp;#45;- Usual fourth (e.g. C4-F4, 495.90, -2.14).&amp;lt;br/&amp;gt;&amp;lt;br/&amp;gt; 6:7 (266.87) &amp;amp;#45;- Major second + QD (e.g. D4-E*4), at a pure 6:7.&amp;lt;br/&amp;gt;&amp;lt;br/&amp;gt; 7:8 (231.17) &amp;amp;#45;- Minor third - QD (e.g. C*4-Eb4, 229.03, -2.14)&amp;lt;br/&amp;gt;&amp;lt;br/&amp;gt; 8:9 (203.91) &amp;amp;#45;- Usual major second (e.g. C4-D4, 208.19, +4.28)&amp;lt;br/&amp;gt;&amp;lt;br/&amp;gt; 11:12 (150.64) &amp;amp;#45;- Major second - QD (e.g. C*4-D4, 149.51, -1.13)&amp;lt;br/&amp;gt;&amp;lt;br/&amp;gt; 12:13 (138.57) &amp;amp;#45;- Minor second + QD (e.g. E4-F*4, 138.20, -0.37)&amp;lt;br/&amp;gt;&amp;lt;br/&amp;gt; 13:14 (128.30) &amp;amp;#45;- Usual apotome (e.g. C4-C#4, 128.67, +0.37)&amp;lt;br/&amp;gt;&amp;lt;br/&amp;gt; 17:18 (98.95) &amp;amp;#45;- Diminished third - QD (e.g. G#*4-Bb4, 100.36, -1.41)&amp;lt;br/&amp;gt;&amp;lt;br/&amp;gt; 21:22 (80.54) &amp;amp;#45;- Usual minor second (e.g. E4-F4, 79.52, -1.02)&amp;lt;br/&amp;gt;&amp;lt;br/&amp;gt; 24:25 (70.67) &amp;amp;#45;- Apotome - QD (e.g. E*4-Eb4, 69.99, -0.68)&amp;lt;br/&amp;gt;&amp;lt;br/&amp;gt; 27:28 (62.96) &amp;amp;#45;- QD (e.g. E4-E*4, 58.68, -4.28)&amp;lt;br/&amp;gt;&amp;lt;br/&amp;gt;&amp;lt;br/&amp;gt;&amp;amp;#45;&amp;amp;#45;&amp;amp;#45;-&amp;lt;br/&amp;gt; 2. Other ratios &amp;amp;#45;- many within 17-odd limit&amp;lt;br/&amp;gt;&amp;amp;#45;&amp;amp;#45;&amp;amp;#45;-&amp;lt;br/&amp;gt;&amp;lt;br/&amp;gt; 4:7 (968.83) &amp;amp;#45;- Major sixth + QD (e.g. G3-E*4, 970.97, +2.14)&amp;lt;br/&amp;gt;&amp;lt;br/&amp;gt; 7:9 (435.08) &amp;amp;#45;- Fourth - QD (e.g. G*4-C5, 437.22, +2.14)&amp;lt;br/&amp;gt;&amp;lt;br/&amp;gt; 7:12 (933.13) &amp;amp;#45;- Minor seventh - QD (e.g. G*3-F4), at a pure 7:12.&amp;lt;br/&amp;gt;&amp;lt;br/&amp;gt; 9:14 (764.92) &amp;amp;#45;- Fifth + QD (e.g. G4-D*5, 762.78, -2.14)&amp;lt;br/&amp;gt;&amp;lt;br/&amp;gt; 9:16 (996.09) &amp;amp;#45;- Usual minor seventh (e.g. G4-F4, 991.81, -4.28)&amp;lt;br/&amp;gt;&amp;lt;br/&amp;gt; 6:11 (1049.36) &amp;amp;#45;- Minor seventh + QD (e.g. G3-F*4, 1050.49, +1.13)&amp;lt;br/&amp;gt;&amp;lt;br/&amp;gt; 7:11 (782.49) &amp;amp;#45;- Usual minor sixth (e.g. A3-F4, 783.62, +1.13)&amp;lt;br/&amp;gt;&amp;lt;br/&amp;gt; 8:11 (551.32) &amp;amp;#45;- Fourth + QD (e.g. G3-C*4, 554.58, +3.27)&amp;lt;br/&amp;gt;&amp;lt;br/&amp;gt; 9:11 (347.41) &amp;amp;#45;- Minor third + QD (e.g. G3-Bb*3, 346.39, -1.02)&amp;lt;br/&amp;gt;&amp;lt;br/&amp;gt; 8:13 (840.53) &amp;amp;#45;- Minor sixth + QD (e.g. G3-Eb*3, 842.30, +1.77)&amp;lt;br/&amp;gt;&amp;lt;br/&amp;gt; 9:13 (636.62) &amp;amp;#45;- Diminished fifth + QD (e.g. A3-Eb*4, 634.11, -2.51)&amp;lt;br/&amp;gt;&amp;lt;br/&amp;gt; 11:13 (289.21) &amp;amp;#45;- Usual minor third (e.g. D3-F3, 287.71, -1.50)&amp;lt;br/&amp;gt;&amp;lt;br/&amp;gt; 11:14 (417.51) &amp;amp;#45;- Usual major third (e.g. D3-F#3, 416.38, -1.13)&amp;lt;br/&amp;gt;&amp;lt;br/&amp;gt; 11:16 (648.68) &amp;amp;#45;- Fifth - QD (e.g. F*3-C4, 645.42, -3.27)&amp;lt;br/&amp;gt;&amp;lt;br/&amp;gt; 11:18 (852.59) &amp;amp;#45;- Major sixth - QD (e.g. G*4-E5, 853.61, +1.02)&amp;lt;br/&amp;gt;&amp;lt;br/&amp;gt; 11:21 (1119.46) &amp;amp;#45;- Usual major seventh (e.g. F3-E4, 1120.48, +1.02)&amp;lt;br/&amp;gt;&amp;lt;br/&amp;gt; 12:17 (603.00) &amp;amp;#45;- Augmented third + QD (e.g. Eb4-G#*4, 603.73, +0.73)&amp;lt;br/&amp;gt;&amp;lt;br/&amp;gt; 13:16 (359.47) &amp;amp;#45;- Major third - QD (e.g. C*4-E4, 357.70, -1.77)&amp;lt;br/&amp;gt;&amp;lt;br/&amp;gt; 13:18 (563.38) &amp;amp;#45;- Augmented fourth - QD (e.g. C*4-F#4, 565.89, +2.51)&amp;lt;br/&amp;gt;&amp;lt;br/&amp;gt; 13:21 (830.25) &amp;amp;#45;- Usual augmented fifth (e.g. C4-G#4, 832.76, +2.51)&amp;lt;br/&amp;gt;&amp;lt;br/&amp;gt; 13:22 (910.79) &amp;amp;#45;- Usual major sixth (e.g. G3-E4, 912.29, +1.50)&amp;lt;br/&amp;gt;&amp;lt;br/&amp;gt; 13:23 (987.75) &amp;amp;#45;- Usual minor seventh (e.g. D4-C5, 991.81, +4.06)&amp;lt;br/&amp;gt;&amp;lt;br/&amp;gt; 13:24 (1061.43) &amp;amp;#45;- Major seventh - QD (e.g. F*3-E4), 1061.80, +0.37)&amp;lt;br/&amp;gt;&amp;lt;br/&amp;gt; 14:17 (336.13) &amp;amp;#45;- Usual augmented second (e.g. F4-G#4, 336.86, +0.73)&amp;lt;br/&amp;gt;&amp;lt;br/&amp;gt; 14:27 (1137.04) &amp;amp;#45;- Octave - QD (e.g. F*4-F5, 1141.32, +4.28)&amp;lt;br/&amp;gt;&amp;lt;br/&amp;gt; 15:17 (216.69) &amp;amp;#45;- Diminished third + QD (e.g. C#4-Eb*4, 217.72, +1.04)&amp;lt;br/&amp;gt;&amp;lt;br/&amp;gt; 16:21 (470.71) &amp;amp;#45;- Major third + QD (e.g. C4-E*4, 475.06, +4.28)&amp;lt;br/&amp;gt;&amp;lt;br/&amp;gt; 16:23 (628.27) &amp;amp;#45;- Usual augmented fourth (e.g. C4-F#4, 624.57, -3.70)&amp;lt;br/&amp;gt;&amp;lt;br/&amp;gt; 18:23 (424.36) &amp;amp;#45;- Diminished fourth + QD (e.g. B4-Eb*5, 425.91, +1.55)&amp;lt;br/&amp;gt;&amp;lt;br/&amp;gt; 16:25 (772.63) &amp;amp;#45;- Diminished fourth + QD (e.g. F#4-Bb*4, 774.09, +1.46)&amp;lt;br/&amp;gt;&amp;lt;br/&amp;gt; 17:20 (281.36) &amp;amp;#45;- Augmented second - QD (e.g. F*4-G#4, 278.18, -3.18)&amp;lt;br/&amp;gt;&amp;lt;br/&amp;gt; 17:21 (365.83) &amp;amp;#45;- Usual diminished fourth (e.g. F#4-Bb4, 367.24, +1.41)&amp;lt;br/&amp;gt;&amp;lt;br/&amp;gt; 17:28 (863.87) &amp;amp;#45;- Usual diminished seventh (e.g. F#4-Eb4, 863.14, -0.73)&amp;lt;br/&amp;gt;&amp;lt;br/&amp;gt; 21:34 (834.17) &amp;amp;#45;- Usual augmented fifth (e.g. F3-C#4, 832.76, +1.41)&amp;lt;br/&amp;gt;&amp;lt;br/&amp;gt; 28:51 (1038.08) &amp;amp;#45;- Usual augmented sixth (e.g. Eb3-C#4, 1040.96, +2.87)&amp;lt;br/&amp;gt;&amp;lt;br/&amp;gt; 21:23 (157.49) &amp;amp;#45;- Usual diminished third (e.g. C#4-Eb4, 159.04, +1.55)&amp;lt;br/&amp;gt;&amp;lt;br/&amp;gt; 21:26 (369.75) &amp;amp;#45;- Usual diminished fourth (e.g. C#4-F4, 367.24, -1.51)&amp;lt;br/&amp;gt;&amp;lt;br/&amp;gt; 23:27 (277.59) &amp;amp;#45;- Augmented second - QD (e.g. Eb*4-F#4, 278.18, +0.59)&amp;lt;br/&amp;gt;&amp;lt;br/&amp;gt; 26:33 (412.75) &amp;amp;#45;- Usual major third (e.g. F4-A4, 416.38, +3.63)&amp;lt;br/&amp;gt;&amp;lt;br/&amp;gt; 28:33 (284.45) &amp;amp;#45;- Usual minor third (e.g. E4-G4, 287.71, +3.27)&amp;lt;br/&amp;gt;&amp;lt;br/&amp;gt; 33:56 (915.55) &amp;amp;#45;- Usual major sixth (e.g. G4-E5, 912.29, -3.27)&amp;lt;br/&amp;gt;&amp;lt;/pre&amp;gt;
- --&gt;
-&lt;style type="text/css"&gt;&lt;!--
-/**
- * GeSHi (C) 2004 - 2007 Nigel McNie, 2007 - 2008 Benny Baumann
- * (http://qbnz.com/highlighter/ and http://geshi.org/)
- */
-.text  {font-family:monospace;}
-.text .imp {font-weight: bold; color: red;}
-.text span.xtra { display:block; }
---&gt;
+; [[Claudi Meneghin]]
-&lt;/style&gt;&lt;pre class="text"&gt; An interesting feature of tuning systems, as implemented on keyboards
+* [https://www.youtube.com/watch?v=5vPvI6MXWFM ''ST LOUIS FUGUE (Fugue on St Louis Blues), for Baroque Ensemble - (Microtonal, PEPPERMINT)''] (2025)
- (conventional or alternative), is the [[keyboard mappings|mapping]] of pure or tempered
+* [https://www.youtube.com/watch?v=iZlvKLg4CoM ''PEPPERMINT FUGUE in 5 parts «Les Regrets»''] (2025)
- ratios to positions on the keyboard layout.
-&amp;nbsp;
- Here I shall explore the mapping of approximate ratios, and especially
- of superparticular and other ratios within [[Harry Partch]]'s larger
--limit set, in the tuning system and keyboard arrangement I call
- Peppermint 24.
-&amp;nbsp;
- 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]],
- 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.
-&amp;nbsp;
- 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).
-&amp;nbsp;
- Here is a 24-note keyboard arrangement, with an asterisk (*) showing a
- note on the upper keyboard:
-&amp;nbsp;
-.349  346.393              683.253    891.445 1050.488
-       C#*      Eb*                   F#*       G#*     Bb*
-  C*        D*          E*      F*        G*        A*       B*     C*
-.680   266.871    475.062  554.584   762.775  970.967  1179.158 1258.680
-/6
--------------------------------------------------------------------------
-.669  287.713              624.574    832.765 991.809
-        C#      Eb                    F#         G#     Bb
-  C         D           E       F          G         A        B     C
-       208.191    416.382 495.904    704.096   912.287 1120.478 1200
-&amp;nbsp;
-&amp;nbsp;
- 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).
-&amp;nbsp;
- 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.
-&amp;nbsp;
- 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 &amp;quot;quasi-diesis,&amp;quot; or QD for short. This
- &amp;quot;artificial&amp;quot; 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).
-&amp;nbsp;
- 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.
-&amp;nbsp;
-&amp;nbsp;
-----
-. Multiplex (n:1) and [[superparticular]] (n+1:n) intervals
-----
-&amp;nbsp;
-:2 (1200) -- This is the usual octave (e.g. F3-F4), at a pure 1:2.
-&amp;nbsp;
-:3 (701.96) -- This is the usual fifth (e.g. F3-C4, 704.10, +2.14).
-&amp;nbsp;
-:4 (498.04) -- Usual fourth (e.g. C4-F4, 495.90, -2.14).
-&amp;nbsp;
-:7 (266.87) -- Major second + QD (e.g. D4-E*4), at a pure 6:7.
-&amp;nbsp;
-:8 (231.17) -- Minor third - QD (e.g. C*4-Eb4, 229.03, -2.14)
-&amp;nbsp;
-:9 (203.91) -- Usual major second (e.g. C4-D4, 208.19, +4.28)
-&amp;nbsp;
-:12 (150.64) -- Major second - QD (e.g. C*4-D4, 149.51, -1.13)
-&amp;nbsp;
-:13 (138.57) -- Minor second + QD (e.g. E4-F*4, 138.20, -0.37)
-&amp;nbsp;
-:14 (128.30) -- Usual apotome (e.g. C4-C#4, 128.67, +0.37)
-&amp;nbsp;
-:18 (98.95) -- Diminished third - QD (e.g. G#*4-Bb4, 100.36, -1.41)
-&amp;nbsp;
-:22 (80.54) -- Usual minor second (e.g. E4-F4, 79.52, -1.02)
-&amp;nbsp;
-:25 (70.67) -- Apotome - QD (e.g. E*4-Eb4, 69.99, -0.68)
-&amp;nbsp;
-:28 (62.96) -- QD (e.g. E4-E*4, 58.68, -4.28)
-&amp;nbsp;
-&amp;nbsp;
-----
-. Other ratios -- many within 17-odd limit
-----
-&amp;nbsp;
-:7 (968.83) -- Major sixth + QD (e.g. G3-E*4, 970.97, +2.14)
-&amp;nbsp;
-:9 (435.08) -- Fourth - QD (e.g. G*4-C5, 437.22, +2.14)
-&amp;nbsp;
-:12 (933.13) -- Minor seventh - QD (e.g. G*3-F4), at a pure 7:12.
-&amp;nbsp;
-:14 (764.92) -- Fifth + QD (e.g. G4-D*5, 762.78, -2.14)
-&amp;nbsp;
-:16 (996.09) -- Usual minor seventh (e.g. G4-F4, 991.81, -4.28)
-&amp;nbsp;
-:11 (1049.36) -- Minor seventh + QD (e.g. G3-F*4, 1050.49, +1.13)
-&amp;nbsp;
-:11 (782.49) -- Usual minor sixth (e.g. A3-F4, 783.62, +1.13)
-&amp;nbsp;
-:11 (551.32) -- Fourth + QD (e.g. G3-C*4, 554.58, +3.27)
-&amp;nbsp;
-:11 (347.41) -- Minor third + QD (e.g. G3-Bb*3, 346.39, -1.02)
-&amp;nbsp;
-:13 (840.53) -- Minor sixth + QD (e.g. G3-Eb*3, 842.30, +1.77)
-&amp;nbsp;
-:13 (636.62) -- Diminished fifth + QD (e.g. A3-Eb*4, 634.11, -2.51)
-&amp;nbsp;
-:13 (289.21) -- Usual minor third (e.g. D3-F3, 287.71, -1.50)
-&amp;nbsp;
-:14 (417.51) -- Usual major third (e.g. D3-F#3, 416.38, -1.13)
-&amp;nbsp;
-:16 (648.68) -- Fifth - QD (e.g. F*3-C4, 645.42, -3.27)
-&amp;nbsp;
-:18 (852.59) -- Major sixth - QD (e.g. G*4-E5, 853.61, +1.02)
-&amp;nbsp;
-:21 (1119.46) -- Usual major seventh (e.g. F3-E4, 1120.48, +1.02)
-&amp;nbsp;
-:17 (603.00) -- Augmented third + QD (e.g. Eb4-G#*4, 603.73, +0.73)
-&amp;nbsp;
-:16 (359.47) -- Major third - QD (e.g. C*4-E4, 357.70, -1.77)
-&amp;nbsp;
-:18 (563.38) -- Augmented fourth - QD (e.g. C*4-F#4, 565.89, +2.51)
-&amp;nbsp;
-:21 (830.25) -- Usual augmented fifth (e.g. C4-G#4, 832.76, +2.51)
-&amp;nbsp;
-:22 (910.79) -- Usual major sixth (e.g. G3-E4, 912.29, +1.50)
-&amp;nbsp;
-:23 (987.75) -- Usual minor seventh (e.g. D4-C5, 991.81, +4.06)
-&amp;nbsp;
-:24 (1061.43) -- Major seventh - QD (e.g. F*3-E4), 1061.80, +0.37)
-&amp;nbsp;
-:17 (336.13) -- Usual augmented second (e.g. F4-G#4, 336.86, +0.73)
-&amp;nbsp;
-:27 (1137.04) -- Octave - QD (e.g. F*4-F5, 1141.32, +4.28)
-&amp;nbsp;
-:17 (216.69) -- Diminished third + QD (e.g. C#4-Eb*4, 217.72, +1.04)
-&amp;nbsp;
-:21 (470.71) -- Major third + QD (e.g. C4-E*4, 475.06, +4.28)
-&amp;nbsp;
-:23 (628.27) -- Usual augmented fourth (e.g. C4-F#4, 624.57, -3.70)
-&amp;nbsp;
-:23 (424.36) -- Diminished fourth + QD (e.g. B4-Eb*5, 425.91, +1.55)
-&amp;nbsp;
-:25 (772.63) -- Diminished fourth + QD (e.g. F#4-Bb*4, 774.09, +1.46)
-&amp;nbsp;
-:20 (281.36) -- Augmented second - QD (e.g. F*4-G#4, 278.18, -3.18)
-&amp;nbsp;
-:21 (365.83) -- Usual diminished fourth (e.g. F#4-Bb4, 367.24, +1.41)
-&amp;nbsp;
-:28 (863.87) -- Usual diminished seventh (e.g. F#4-Eb4, 863.14, -0.73)
-&amp;nbsp;
-:34 (834.17) -- Usual augmented fifth (e.g. F3-C#4, 832.76, +1.41)
-&amp;nbsp;
-:51 (1038.08) -- Usual augmented sixth (e.g. Eb3-C#4, 1040.96, +2.87)
-&amp;nbsp;
-:23 (157.49) -- Usual diminished third (e.g. C#4-Eb4, 159.04, +1.55)
-&amp;nbsp;
-:26 (369.75) -- Usual diminished fourth (e.g. C#4-F4, 367.24, -1.51)
-&amp;nbsp;
-:27 (277.59) -- Augmented second - QD (e.g. Eb*4-F#4, 278.18, +0.59)
-&amp;nbsp;
-:33 (412.75) -- Usual major third (e.g. F4-A4, 416.38, +3.63)
-&amp;nbsp;
-:33 (284.45) -- Usual minor third (e.g. E4-G4, 287.71, +3.27)
-&amp;nbsp;
-:56 (915.55) -- Usual major sixth (e.g. G4-E5, 912.29, -3.27)
-&amp;nbsp;&lt;/pre&gt;
-&lt;!-- ws:end:WikiTextCodeRule:0 --&gt;&lt;/body&gt;&lt;/html&gt;</pre></div>
+[[Category:24-tone scales]]
+[[Category:Tempered scales]]
+[[Category:Todo:clarify]]