Peppermint-24: Difference between revisions

Line 1:

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

=Peppermint 24=

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

~~: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-11-22 21:13:42 UTC</tt>. ~~

~~: The original revision id was <tt>278347106</tt>. ~~

~~: The revision comment was: <tt></tt> ~~

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

~~<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, 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.

[http://launch.groups.yahoo.com/group/tuning/message/40057 Original article] by Margo Schulter, on the Yahoo tuning forum, is quoted below. In addition to what it says, it may be noted that the Wilson/Pepper fifth it mentions, of size approximately 704.096 cents, has a precise value of (67 + √5)/118 octaves, which is (40200 + 600 √5)/59 cents.

=Margo Schulter's article=

An interesting feature of tuning systems, as implemented on keyboards (conventional or alternative), is the [[~~keyboard mappings~~|mapping]] of pure or tempered ratios to positions on the keyboard layout.

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.

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]], 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.

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).

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

~~[[code]]~~

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* D* E* F* G* A* B* C*

Line 31:

Line 24:

C D E F G A B C

0 208.191 416.382 495.904 704.096 912.287 1120.478 1200

</pre>

~~[[code]]~~

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).

Line 43:

Line 35:

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|superparticular]] (n+1:n) intervals'''

**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.

Line 72:

Line 63:

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

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

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

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

Line 155:

Line 145:

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)~~</pre></div>~~

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

~~<h4>Original HTML 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;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>

~~ ~~

<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, 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.

~~ ~~

<h1 id="toc1"><a name="Margo Schulter's article"></a>Margo Schulter's article</h1>

An interesting feature of tuning systems, as implemented on keyboards (conventional or alternative), is the <a class="wiki_link" href="/keyboard%20mappings">mapping</a> 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 <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Harry_Partch" rel="nofollow">Harry Partch's</a> larger 17-limit set, in the tuning system and keyboard arrangement I call Peppermint 24.

~~ ~~

Peppermint 24 takes as its basis a <a class="wiki_link" href="/Regular%20Temperaments">regular temperament</a> mentioned in <a class="wiki_link" href="/Erv%20Wilson">Ervin Wilson</a>'s Scale Tree and described on the Tuning List by <a class="wiki_link" href="/Keenan%20Pepper">Keenan Pepper</a>, with a fifth of about 704.096 cents, and a precise ratio of <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Golden_ratio" rel="nofollow">Phi</a>, 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: ~~

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

&lt;pre class=&quot;text&quot;&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;/pre&gt;

~~-->~~

~~<style type="text/css"><!--~~

/**

* (http://qbnz.com/highlighter/ and http://geshi.org/)

*/

~~.text {font-family:monospace;}~~

~~.text .imp {font-weight: bold; color: red;}~~

~~.text span.xtra { display:block; }~~

~~-->~~

~~</style><pre class="text"> 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;~~

~~&nbsp;</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).

~~ ~~

Octave numbers appear in a MIDI-style notation, with C4 as middle C; just ratios and tempered equivalents are given values in cents, shown in parentheses, with tempered variations in cents also shown.

~~ ~~

To describe the 58.68-cent interval between the two keyboards, whose addition or subtraction plays a role in obtaining or approximating many ratios, I shall the term &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).

~~ ~~

~~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 <a class="wiki_link" href="/superparticular">superparticular</a> (n+1:n) intervals ~~

~~ ~~

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

~~ ~~

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

~~ ~~

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

~~ ~~

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

~~ ~~

~~7:8 (231.17) -- Minor third - QD (e.g. C*4-Eb4, 229.03, -2.14) ~~

~~ ~~

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

~~ ~~

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

~~ ~~

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

~~ ~~

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

~~ ~~

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

~~ ~~

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

~~ ~~

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

~~ ~~

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

~~ ~~

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

~~ ~~

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

~~ ~~

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

~~ ~~

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

~~ ~~

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

~~ ~~

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

~~ ~~

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

~~ ~~

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

~~ ~~

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

~~ ~~

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

~~ ~~

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

~~ ~~

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

~~ ~~

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

~~ ~~

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

~~ ~~

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

~~ ~~

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

~~ ~~

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

~~ ~~

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

~~ ~~

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

~~ ~~

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

~~ ~~

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

~~ ~~

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

~~ ~~

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

~~ ~~

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

~~ ~~

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

~~ ~~

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

~~ ~~

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

~~ ~~

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

~~ ~~

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

~~ ~~

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

~~ ~~

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

~~ ~~

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

~~ ~~

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

~~ ~~

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

~~ ~~

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

~~ ~~

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

~~ ~~

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

~~ ~~

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

~~ ~~

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

~~ ~~

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

~~ ~~

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

~~ ~~

~~33:56 (915.55) -- Usual major sixth (e.g. G4-E5, 912.29, -3.27)</body></html></pre></div>~~

@@ Line 1: / Line 1: @@
-<h2>IMPORTED REVISION FROM WIKISPACES</h2>
+=Peppermint 24=
-This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
-: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-11-22 21:13:42 UTC</tt>.<br>
-: The original revision id was <tt>278347106</tt>.<br>
-: The revision comment was: <tt></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, 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.
+[http://launch.groups.yahoo.com/group/tuning/message/40057 Original article] by Margo Schulter, on the Yahoo tuning forum, is quoted below. In addition to what it says, it may be noted that the Wilson/Pepper fifth it mentions, of size approximately 704.096 cents, has a precise value of (67 + √5)/118 octaves, which is (40200 + 600 √5)/59 cents.
 =Margo Schulter's article=
-An interesting feature of tuning systems, as implemented on keyboards (conventional or alternative), is the [[keyboard mappings|mapping]] of pure or tempered ratios to positions on the keyboard layout.
+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.
+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]], 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.
+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).
 Here is a 24-note keyboard arrangement, with an asterisk (*) showing a note on the upper keyboard:
-[[code]]
-.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*        D*          E*      F*        G*        A*       B*     C*
@@ Line 31: / Line 24: @@
    C         D           E       F          G         A        B     C
        208.191    416.382 495.904    704.096   912.287 1120.478 1200
+ </pre>
-[[code]]
 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).
@@ Line 43: / Line 35: @@
 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|superparticular]] (n+1:n) intervals'''
-**1. 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.
@@ Line 72: / Line 63: @@
 :28 (62.96) -- QD (e.g. E4-E*4, 58.68, -4.28)
+'''2. Other ratios -- many within 17-odd limit'''
-**2. Other ratios -- many within 17-odd limit**
 :7 (968.83) -- Major sixth + QD (e.g. G3-E*4, 970.97, +2.14)
@@ Line 155: / Line 145: @@
 :33 (284.45) -- Usual minor third (e.g. E4-G4, 287.71, +3.27)
-:56 (915.55) -- Usual major sixth (e.g. G4-E5, 912.29, -3.27)</pre></div>
+:56 (915.55) -- Usual major sixth (e.g. G4-E5, 912.29, -3.27)
-<h4>Original HTML 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;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;
- &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, 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.&lt;br /&gt;
-&lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:3:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc1"&gt;&lt;a name="Margo Schulter's article"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:3 --&gt;Margo Schulter's article&lt;/h1&gt;
-An interesting feature of tuning systems, as implemented on keyboards (conventional or alternative), is the &lt;a class="wiki_link" href="/keyboard%20mappings"&gt;mapping&lt;/a&gt; of pure or tempered ratios to positions on the keyboard layout.&lt;br /&gt;
-&lt;br /&gt;
-Here I shall explore the mapping of approximate ratios, and especially of superparticular and other ratios within &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Harry_Partch" rel="nofollow"&gt;Harry Partch's&lt;/a&gt; larger 17-limit set, in the tuning system and keyboard arrangement I call Peppermint 24.&lt;br /&gt;
-&lt;br /&gt;
-Peppermint 24 takes as its basis a &lt;a class="wiki_link" href="/Regular%20Temperaments"&gt;regular temperament&lt;/a&gt; mentioned in &lt;a class="wiki_link" href="/Erv%20Wilson"&gt;Ervin Wilson&lt;/a&gt;'s Scale Tree and described on the Tuning List by &lt;a class="wiki_link" href="/Keenan%20Pepper"&gt;Keenan Pepper&lt;/a&gt;, with a fifth of about 704.096 cents, and a precise ratio of &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Golden_ratio" rel="nofollow"&gt;Phi&lt;/a&gt;, 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.&lt;br /&gt;
-&lt;br /&gt;
-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).&lt;br /&gt;
-&lt;br /&gt;
-Here is a 24-note keyboard arrangement, with an asterisk (*) showing a note on the upper keyboard:&lt;br /&gt;
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-&amp;lt;pre class=&amp;quot;text&amp;quot;&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;/pre&amp;gt;
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-&lt;/style&gt;&lt;pre class="text"&gt;     187.349  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;&lt;/pre&gt;
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-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).&lt;br /&gt;
-&lt;br /&gt;
-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.&lt;br /&gt;
-&lt;br /&gt;
-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).&lt;br /&gt;
-&lt;br /&gt;
-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.&lt;br /&gt;
-&lt;br /&gt;
-&lt;br /&gt;
-&lt;strong&gt;1. Multiplex (n:1) and &lt;a class="wiki_link" href="/superparticular"&gt;superparticular&lt;/a&gt; (n+1:n) intervals&lt;/strong&gt;&lt;br /&gt;
-&lt;br /&gt;
-:2 (1200) -- This is the usual octave (e.g. F3-F4), at a pure 1:2.&lt;br /&gt;
-&lt;br /&gt;
-:3 (701.96) -- This is the usual fifth (e.g. F3-C4, 704.10, +2.14).&lt;br /&gt;
-&lt;br /&gt;
-:4 (498.04) -- Usual fourth (e.g. C4-F4, 495.90, -2.14).&lt;br /&gt;
-&lt;br /&gt;
-:7 (266.87) -- Major second + QD (e.g. D4-E*4), at a pure 6:7.&lt;br /&gt;
-&lt;br /&gt;
-:8 (231.17) -- Minor third - QD (e.g. C*4-Eb4, 229.03, -2.14)&lt;br /&gt;
-&lt;br /&gt;
-:9 (203.91) -- Usual major second (e.g. C4-D4, 208.19, +4.28)&lt;br /&gt;
-&lt;br /&gt;
-:12 (150.64) -- Major second - QD (e.g. C*4-D4, 149.51, -1.13)&lt;br /&gt;
-&lt;br /&gt;
-:13 (138.57) -- Minor second + QD (e.g. E4-F*4, 138.20, -0.37)&lt;br /&gt;
-&lt;br /&gt;
-:14 (128.30) -- Usual apotome (e.g. C4-C#4, 128.67, +0.37)&lt;br /&gt;
-&lt;br /&gt;
-:18 (98.95) -- Diminished third - QD (e.g. G#*4-Bb4, 100.36, -1.41)&lt;br /&gt;
-&lt;br /&gt;
-:22 (80.54) -- Usual minor second (e.g. E4-F4, 79.52, -1.02)&lt;br /&gt;
-&lt;br /&gt;
-:25 (70.67) -- Apotome - QD (e.g. E*4-Eb4, 69.99, -0.68)&lt;br /&gt;
-&lt;br /&gt;
-:28 (62.96) -- QD (e.g. E4-E*4, 58.68, -4.28)&lt;br /&gt;
-&lt;br /&gt;
-&lt;br /&gt;
-&lt;strong&gt;2. Other ratios -- many within 17-odd limit&lt;/strong&gt;&lt;br /&gt;
-&lt;br /&gt;
-:7 (968.83) -- Major sixth + QD (e.g. G3-E*4, 970.97, +2.14)&lt;br /&gt;
-&lt;br /&gt;
-:9 (435.08) -- Fourth - QD (e.g. G*4-C5, 437.22, +2.14)&lt;br /&gt;
-&lt;br /&gt;
-:12 (933.13) -- Minor seventh - QD (e.g. G*3-F4), at a pure 7:12.&lt;br /&gt;
-&lt;br /&gt;
-:14 (764.92) -- Fifth + QD (e.g. G4-D*5, 762.78, -2.14)&lt;br /&gt;
-&lt;br /&gt;
-:16 (996.09) -- Usual minor seventh (e.g. G4-F4, 991.81, -4.28)&lt;br /&gt;
-&lt;br /&gt;
-:11 (1049.36) -- Minor seventh + QD (e.g. G3-F*4, 1050.49, +1.13)&lt;br /&gt;
-&lt;br /&gt;
-:11 (782.49) -- Usual minor sixth (e.g. A3-F4, 783.62, +1.13)&lt;br /&gt;
-&lt;br /&gt;
-:11 (551.32) -- Fourth + QD (e.g. G3-C*4, 554.58, +3.27)&lt;br /&gt;
-&lt;br /&gt;
-:11 (347.41) -- Minor third + QD (e.g. G3-Bb*3, 346.39, -1.02)&lt;br /&gt;
-&lt;br /&gt;
-:13 (840.53) -- Minor sixth + QD (e.g. G3-Eb*3, 842.30, +1.77)&lt;br /&gt;
-&lt;br /&gt;
-:13 (636.62) -- Diminished fifth + QD (e.g. A3-Eb*4, 634.11, -2.51)&lt;br /&gt;
-&lt;br /&gt;
-:13 (289.21) -- Usual minor third (e.g. D3-F3, 287.71, -1.50)&lt;br /&gt;
-&lt;br /&gt;
-:14 (417.51) -- Usual major third (e.g. D3-F#3, 416.38, -1.13)&lt;br /&gt;
-&lt;br /&gt;
-:16 (648.68) -- Fifth - QD (e.g. F*3-C4, 645.42, -3.27)&lt;br /&gt;
-&lt;br /&gt;
-:18 (852.59) -- Major sixth - QD (e.g. G*4-E5, 853.61, +1.02)&lt;br /&gt;
-&lt;br /&gt;
-:21 (1119.46) -- Usual major seventh (e.g. F3-E4, 1120.48, +1.02)&lt;br /&gt;
-&lt;br /&gt;
-:17 (603.00) -- Augmented third + QD (e.g. Eb4-G#*4, 603.73, +0.73)&lt;br /&gt;
-&lt;br /&gt;
-:16 (359.47) -- Major third - QD (e.g. C*4-E4, 357.70, -1.77)&lt;br /&gt;
-&lt;br /&gt;
-:18 (563.38) -- Augmented fourth - QD (e.g. C*4-F#4, 565.89, +2.51)&lt;br /&gt;
-&lt;br /&gt;
-:21 (830.25) -- Usual augmented fifth (e.g. C4-G#4, 832.76, +2.51)&lt;br /&gt;
-&lt;br /&gt;
-:22 (910.79) -- Usual major sixth (e.g. G3-E4, 912.29, +1.50)&lt;br /&gt;
-&lt;br /&gt;
-:23 (987.75) -- Usual minor seventh (e.g. D4-C5, 991.81, +4.06)&lt;br /&gt;
-&lt;br /&gt;
-:24 (1061.43) -- Major seventh - QD (e.g. F*3-E4), 1061.80, +0.37)&lt;br /&gt;
-&lt;br /&gt;
-:17 (336.13) -- Usual augmented second (e.g. F4-G#4, 336.86, +0.73)&lt;br /&gt;
-&lt;br /&gt;
-:27 (1137.04) -- Octave - QD (e.g. F*4-F5, 1141.32, +4.28)&lt;br /&gt;
-&lt;br /&gt;
-:17 (216.69) -- Diminished third + QD (e.g. C#4-Eb*4, 217.72, +1.04)&lt;br /&gt;
-&lt;br /&gt;
-:21 (470.71) -- Major third + QD (e.g. C4-E*4, 475.06, +4.28)&lt;br /&gt;
-&lt;br /&gt;
-:23 (628.27) -- Usual augmented fourth (e.g. C4-F#4, 624.57, -3.70)&lt;br /&gt;
-&lt;br /&gt;
-:23 (424.36) -- Diminished fourth + QD (e.g. B4-Eb*5, 425.91, +1.55)&lt;br /&gt;
-&lt;br /&gt;
-:25 (772.63) -- Diminished fourth + QD (e.g. F#4-Bb*4, 774.09, +1.46)&lt;br /&gt;
-&lt;br /&gt;
-:20 (281.36) -- Augmented second - QD (e.g. F*4-G#4, 278.18, -3.18)&lt;br /&gt;
-&lt;br /&gt;
-:21 (365.83) -- Usual diminished fourth (e.g. F#4-Bb4, 367.24, +1.41)&lt;br /&gt;
-&lt;br /&gt;
-:28 (863.87) -- Usual diminished seventh (e.g. F#4-Eb4, 863.14, -0.73)&lt;br /&gt;
-&lt;br /&gt;
-:34 (834.17) -- Usual augmented fifth (e.g. F3-C#4, 832.76, +1.41)&lt;br /&gt;
-&lt;br /&gt;
-:51 (1038.08) -- Usual augmented sixth (e.g. Eb3-C#4, 1040.96, +2.87)&lt;br /&gt;
-&lt;br /&gt;
-:23 (157.49) -- Usual diminished third (e.g. C#4-Eb4, 159.04, +1.55)&lt;br /&gt;
-&lt;br /&gt;
-:26 (369.75) -- Usual diminished fourth (e.g. C#4-F4, 367.24, -1.51)&lt;br /&gt;
-&lt;br /&gt;
-:27 (277.59) -- Augmented second - QD (e.g. Eb*4-F#4, 278.18, +0.59)&lt;br /&gt;
-&lt;br /&gt;
-:33 (412.75) -- Usual major third (e.g. F4-A4, 416.38, +3.63)&lt;br /&gt;
-&lt;br /&gt;
-:33 (284.45) -- Usual minor third (e.g. E4-G4, 287.71, +3.27)&lt;br /&gt;
-&lt;br /&gt;
-:56 (915.55) -- Usual major sixth (e.g. G4-E5, 912.29, -3.27)&lt;/body&gt;&lt;/html&gt;</pre></div>