Xenharmonic Wiki

Peppermint-24: Difference between revisions

← Older editNewer edit →

Wikispaces>xenjacob

VisualWikitext
Wikispaces>xenjacob
**Imported revision 5371407 - Original comment: formatted, hyperlinked.**
Wikispaces>xenjacob
**Imported revision 5371443 - Original comment: **
Line 1: Line 1:
<h2>IMPORTED REVISION FROM WIKISPACES</h2>
<h2>IMPORTED REVISION FROM WIKISPACES</h2>
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
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>
: This revision was by author [[User:xenjacob|xenjacob]] and made on <tt>2007-06-21 00:27:17 UTC</tt>.<br>
: The original revision id was <tt>5371407</tt>.<br>
: The original revision id was <tt>5371443</tt>.<br>
: The revision comment was: <tt>formatted, hyperlinked.</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>
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>
<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=  
<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.
[[http://launch.groups.yahoo.com/group/tuning/message/40057|Original article]] by Margo Schulter, on the Yahoo tuning forum, is quoted here.


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


Here I shall explore the mapping of approximate ratios, and especially
Here I shall explore the mapping of approximate ratios, and especially
of superparticular and other ratios within [[Harry Partch]]'s larger
of superparticular and other ratios within [[Harry Partch]]'s larger
17-limit set, in the tuning system and keyboard arrangement I call
17-limit set, in the tuning system and keyboard arrangement I call
Peppermint 24.
Peppermint 24.


Peppermint 24 takes as its basis a [[Regular Temperaments|regular temperament]] mentioned in
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]],
[[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
with a fifth of about 704.096 cents, and a precise ratio of
[[Phi]], the Golden Section (~1.618) between the larger chromatic semitone
[[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-C#) at about 128.669 cents and the smaller diatonic semitone
(e.g. C#-D) at about 79.522 cents.
(e.g. C#-D) at about 79.522 cents.


In Peppermint 24, two regular 12-note chains of this temperament are
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
placed at a distance of approximately 58.680 cents, so as to yield
some pure ratios of 6:7 (~266.871 cents).
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:


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
     187.349  346.393              683.253    891.445 1050.488
       C#*      Eb*                  F#*      G#*    Bb*
       C#*      Eb*                  F#*      G#*    Bb*
Line 46: Line 45:




In the following catalogue of some ratio equivalents and mappings, I
[[code]]
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;
In the following catalogue of some ratio equivalents and mappings, I
just ratios and tempered equivalents are given values in cents, shown
will focus on intervals no further from just than 8:9 or 9:16, which
in parentheses, with tempered variations in cents also shown.
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).


To describe the 58.68-cent interval between the two keyboards, whose
Octave numbers appear in a MIDI-style notation, with C4 as middle C;
addition or subtraction plays a role in obtaining or approximating
just ratios and tempered equivalents are given values in cents, shown
many ratios, I shall the term "quasi-diesis," or QD for short. This
in parentheses, with tempered variations in cents also shown.
"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).


As this partial catalogue might suggest, many ratios of 2-3-7-9-11-13
To describe the 58.68-cent interval between the two keyboards, whose
are represented quite accurately, with 14:17:21 and related ratios
addition or subtraction plays a role in obtaining or approximating
also closely approximated.
many ratios, I shall the term "quasi-diesis," or QD for short. This
"artificial" diesis-like interval is actually somewhat larger than the
natural diesis in the regular Wilson/Pepper temperament at about 49.15
cents (12 tempered fifths less 7 pure octaves).


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


2:3 (701.96) -- This is the usual fifth (e.g. F3-C4, 704.10, +2.14).
1:2 (1200) -- This is the usual octave (e.g. F3-F4), at a pure 1:2.


3:4 (498.04) -- Usual fourth (e.g. C4-F4, 495.90, -2.14).
2:3 (701.96) -- This is the usual fifth (e.g. F3-C4, 704.10, +2.14).


6:7 (266.87) -- Major second + QD (e.g. D4-E*4), at a pure 6:7.
3:4 (498.04) -- Usual fourth (e.g. C4-F4, 495.90, -2.14).


7:8 (231.17) -- Minor third - QD (e.g. C*4-Eb4, 229.03, -2.14)
6:7 (266.87) -- Major second + QD (e.g. D4-E*4), at a pure 6:7.


8:9 (203.91) -- Usual major second (e.g. C4-D4, 208.19, +4.28)
7:8 (231.17) -- Minor third - QD (e.g. C*4-Eb4, 229.03, -2.14)


11:12 (150.64) -- Major second - QD (e.g. C*4-D4, 149.51, -1.13)
8:9 (203.91) -- Usual major second (e.g. C4-D4, 208.19, +4.28)


12:13 (138.57) -- Minor second + QD (e.g. E4-F*4, 138.20, -0.37)
11:12 (150.64) -- Major second - QD (e.g. C*4-D4, 149.51, -1.13)


13:14 (128.30) -- Usual apotome (e.g. C4-C#4, 128.67, +0.37)
12:13 (138.57) -- Minor second + QD (e.g. E4-F*4, 138.20, -0.37)


17:18 (98.95) -- Diminished third - QD (e.g. G#*4-Bb4, 100.36, -1.41)
13:14 (128.30) -- Usual apotome (e.g. C4-C#4, 128.67, +0.37)


21:22 (80.54) -- Usual minor second (e.g. E4-F4, 79.52, -1.02)
17:18 (98.95) -- Diminished third - QD (e.g. G#*4-Bb4, 100.36, -1.41)


24:25 (70.67) -- Apotome - QD (e.g. E*4-Eb4, 69.99, -0.68)
21:22 (80.54) -- Usual minor second (e.g. E4-F4, 79.52, -1.02)


27:28 (62.96) -- QD (e.g. E4-E*4, 58.68, -4.28)
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)
**2. Other ratios -- many within 17-odd limit**


7:9 (435.08) -- Fourth - QD (e.g. G*4-C5, 437.22, +2.14)
4:7 (968.83) -- Major sixth + QD (e.g. G3-E*4, 970.97, +2.14)


7:12 (933.13) -- Minor seventh - QD (e.g. G*3-F4), at a pure 7:12.
7:9 (435.08) -- Fourth - QD (e.g. G*4-C5, 437.22, +2.14)


9:14 (764.92) -- Fifth + QD (e.g. G4-D*5, 762.78, -2.14)
7:12 (933.13) -- Minor seventh - QD (e.g. G*3-F4), at a pure 7:12.


9:16 (996.09) -- Usual minor seventh (e.g. G4-F4, 991.81, -4.28)
9:14 (764.92) -- Fifth + QD (e.g. G4-D*5, 762.78, -2.14)


6:11 (1049.36) -- Minor seventh + QD (e.g. G3-F*4, 1050.49, +1.13)
9:16 (996.09) -- Usual minor seventh (e.g. G4-F4, 991.81, -4.28)


7:11 (782.49) -- Usual minor sixth (e.g. A3-F4, 783.62, +1.13)
6:11 (1049.36) -- Minor seventh + QD (e.g. G3-F*4, 1050.49, +1.13)


8:11 (551.32) -- Fourth + QD (e.g. G3-C*4, 554.58, +3.27)
7:11 (782.49) -- Usual minor sixth (e.g. A3-F4, 783.62, +1.13)


9:11 (347.41) -- Minor third + QD (e.g. G3-Bb*3, 346.39, -1.02)
8:11 (551.32) -- Fourth + QD (e.g. G3-C*4, 554.58, +3.27)


8:13 (840.53) -- Minor sixth + QD (e.g. G3-Eb*3, 842.30, +1.77)
9:11 (347.41) -- Minor third + QD (e.g. G3-Bb*3, 346.39, -1.02)


9:13 (636.62) -- Diminished fifth + QD (e.g. A3-Eb*4, 634.11, -2.51)
8:13 (840.53) -- Minor sixth + QD (e.g. G3-Eb*3, 842.30, +1.77)


11:13 (289.21) -- Usual minor third (e.g. D3-F3, 287.71, -1.50)
9:13 (636.62) -- Diminished fifth + QD (e.g. A3-Eb*4, 634.11, -2.51)


11:14 (417.51) -- Usual major third (e.g. D3-F#3, 416.38, -1.13)
11:13 (289.21) -- Usual minor third (e.g. D3-F3, 287.71, -1.50)


11:16 (648.68) -- Fifth - QD (e.g. F*3-C4, 645.42, -3.27)
11:14 (417.51) -- Usual major third (e.g. D3-F#3, 416.38, -1.13)


11:18 (852.59) -- Major sixth - QD (e.g. G*4-E5, 853.61, +1.02)
11:16 (648.68) -- Fifth - QD (e.g. F*3-C4, 645.42, -3.27)


11:21 (1119.46) -- Usual major seventh (e.g. F3-E4, 1120.48, +1.02)
11:18 (852.59) -- Major sixth - QD (e.g. G*4-E5, 853.61, +1.02)


12:17 (603.00) -- Augmented third + QD (e.g. Eb4-G#*4, 603.73, +0.73)
11:21 (1119.46) -- Usual major seventh (e.g. F3-E4, 1120.48, +1.02)


13:16 (359.47) -- Major third - QD (e.g. C*4-E4, 357.70, -1.77)
12:17 (603.00) -- Augmented third + QD (e.g. Eb4-G#*4, 603.73, +0.73)


13:18 (563.38) -- Augmented fourth - QD (e.g. C*4-F#4, 565.89, +2.51)
13:16 (359.47) -- Major third - QD (e.g. C*4-E4, 357.70, -1.77)


13:21 (830.25) -- Usual augmented fifth (e.g. C4-G#4, 832.76, +2.51)
13:18 (563.38) -- Augmented fourth - QD (e.g. C*4-F#4, 565.89, +2.51)


13:22 (910.79) -- Usual major sixth (e.g. G3-E4, 912.29, +1.50)
13:21 (830.25) -- Usual augmented fifth (e.g. C4-G#4, 832.76, +2.51)


13:23 (987.75) -- Usual minor seventh (e.g. D4-C5, 991.81, +4.06)
13:22 (910.79) -- Usual major sixth (e.g. G3-E4, 912.29, +1.50)


13:24 (1061.43) -- Major seventh - QD (e.g. F*3-E4), 1061.80, +0.37)
13:23 (987.75) -- Usual minor seventh (e.g. D4-C5, 991.81, +4.06)


14:17 (336.13) -- Usual augmented second (e.g. F4-G#4, 336.86, +0.73)
13:24 (1061.43) -- Major seventh - QD (e.g. F*3-E4), 1061.80, +0.37)


14:27 (1137.04) -- Octave - QD (e.g. F*4-F5, 1141.32, +4.28)
14:17 (336.13) -- Usual augmented second (e.g. F4-G#4, 336.86, +0.73)


15:17 (216.69) -- Diminished third + QD (e.g. C#4-Eb*4, 217.72, +1.04)
14:27 (1137.04) -- Octave - QD (e.g. F*4-F5, 1141.32, +4.28)


16:21 (470.71) -- Major third + QD (e.g. C4-E*4, 475.06, +4.28)
15:17 (216.69) -- Diminished third + QD (e.g. C#4-Eb*4, 217.72, +1.04)


16:23 (628.27) -- Usual augmented fourth (e.g. C4-F#4, 624.57, -3.70)
16:21 (470.71) -- Major third + QD (e.g. C4-E*4, 475.06, +4.28)


18:23 (424.36) -- Diminished fourth + QD (e.g. B4-Eb*5, 425.91, +1.55)
16:23 (628.27) -- Usual augmented fourth (e.g. C4-F#4, 624.57, -3.70)


16:25 (772.63) -- Diminished fourth + QD (e.g. F#4-Bb*4, 774.09, +1.46)
18:23 (424.36) -- Diminished fourth + QD (e.g. B4-Eb*5, 425.91, +1.55)


17:20 (281.36) -- Augmented second - QD (e.g. F*4-G#4, 278.18, -3.18)
16:25 (772.63) -- Diminished fourth + QD (e.g. F#4-Bb*4, 774.09, +1.46)


17:21 (365.83) -- Usual diminished fourth (e.g. F#4-Bb4, 367.24, +1.41)
17:20 (281.36) -- Augmented second - QD (e.g. F*4-G#4, 278.18, -3.18)


17:28 (863.87) -- Usual diminished seventh (e.g. F#4-Eb4, 863.14, -0.73)
17:21 (365.83) -- Usual diminished fourth (e.g. F#4-Bb4, 367.24, +1.41)


21:34 (834.17) -- Usual augmented fifth (e.g. F3-C#4, 832.76, +1.41)
17:28 (863.87) -- Usual diminished seventh (e.g. F#4-Eb4, 863.14, -0.73)


28:51 (1038.08) -- Usual augmented sixth (e.g. Eb3-C#4, 1040.96, +2.87)
21:34 (834.17) -- Usual augmented fifth (e.g. F3-C#4, 832.76, +1.41)


21:23 (157.49) -- Usual diminished third (e.g. C#4-Eb4, 159.04, +1.55)
28:51 (1038.08) -- Usual augmented sixth (e.g. Eb3-C#4, 1040.96, +2.87)


21:26 (369.75) -- Usual diminished fourth (e.g. C#4-F4, 367.24, -1.51)
21:23 (157.49) -- Usual diminished third (e.g. C#4-Eb4, 159.04, +1.55)


23:27 (277.59) -- Augmented second - QD (e.g. Eb*4-F#4, 278.18, +0.59)
21:26 (369.75) -- Usual diminished fourth (e.g. C#4-F4, 367.24, -1.51)


26:33 (412.75) -- Usual major third (e.g. F4-A4, 416.38, +3.63)
23:27 (277.59) -- Augmented second - QD (e.g. Eb*4-F#4, 278.18, +0.59)


28:33 (284.45) -- Usual minor third (e.g. E4-G4, 287.71, +3.27)
26:33 (412.75) -- Usual major third (e.g. F4-A4, 416.38, +3.63)


33:56 (915.55) -- Usual major sixth (e.g. G4-E5, 912.29, -3.27)
28:33 (284.45) -- Usual minor third (e.g. E4-G4, 287.71, +3.27)


[[code]]</pre></div>
33:56 (915.55) -- Usual major sixth (e.g. G4-E5, 912.29, -3.27)</pre></div>
<h4>Original HTML content:</h4>
<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;
<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;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;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 here.&lt;br /&gt;
&lt;br /&gt;
&lt;br /&gt;
An interesting feature of tuning systems, as implemented on keyboards&lt;br /&gt;
(conventional or alternative), is the &lt;a class="wiki_link" href="/keyboard%20mappings"&gt;mapping&lt;/a&gt; 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 &lt;a class="wiki_link" href="/Harry%20Partch"&gt;Harry Partch&lt;/a&gt;'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 &lt;a class="wiki_link" href="/Regular%20Temperaments"&gt;regular temperament&lt;/a&gt; mentioned in&lt;br /&gt;
&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;,&lt;br /&gt;
with a fifth of about 704.096 cents, and a precise ratio of&lt;br /&gt;
&lt;a class="wiki_link" href="/Phi"&gt;Phi&lt;/a&gt;, 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;!-- ws:start:WikiTextCodeRule:0:
&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;
&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;
  --&gt;
  --&gt;
&lt;style type="text/css"&gt;&lt;!--
&lt;style type="text/css"&gt;&lt;!--
Line 203: Line 220:


--&gt;
--&gt;
&lt;/style&gt;&lt;pre class="text"&gt; An interesting feature of tuning systems, as implemented on keyboards
&lt;/style&gt;&lt;pre class="text"&gt;     187.349  346.393              683.253    891.445 1050.488
(conventional or alternative), is the [[keyboard mappings|mapping]] of pure or tempered
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
17-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;
    187.349  346.393              683.253    891.445 1050.488
       C#*      Eb*                  F#*      G#*    Bb*
       C#*      Eb*                  F#*      G#*    Bb*
   C*        D*          E*      F*        G*        A*      B*    C*
   C*        D*          E*      F*        G*        A*      B*    C*
Line 237: Line 231:
   0      208.191    416.382 495.904    704.096  912.287 1120.478 1200
   0      208.191    416.382 495.904    704.096  912.287 1120.478 1200
&amp;nbsp;
&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;
----
1. Multiplex (n:1) and [[superparticular]] (n+1:n) intervals
----
&amp;nbsp;
1:2 (1200) -- This is the usual octave (e.g. F3-F4), at a pure 1:2.
&amp;nbsp;
2:3 (701.96) -- This is the usual fifth (e.g. F3-C4, 704.10, +2.14).
&amp;nbsp;
3:4 (498.04) -- Usual fourth (e.g. C4-F4, 495.90, -2.14).
&amp;nbsp;
6:7 (266.87) -- Major second + QD (e.g. D4-E*4), at a pure 6:7.
&amp;nbsp;
7:8 (231.17) -- Minor third - QD (e.g. C*4-Eb4, 229.03, -2.14)
&amp;nbsp;
8:9 (203.91) -- Usual major second (e.g. C4-D4, 208.19, +4.28)
&amp;nbsp;
11:12 (150.64) -- Major second - QD (e.g. C*4-D4, 149.51, -1.13)
&amp;nbsp;
12:13 (138.57) -- Minor second + QD (e.g. E4-F*4, 138.20, -0.37)
&amp;nbsp;
13:14 (128.30) -- Usual apotome (e.g. C4-C#4, 128.67, +0.37)
&amp;nbsp;
17:18 (98.95) -- Diminished third - QD (e.g. G#*4-Bb4, 100.36, -1.41)
&amp;nbsp;
21:22 (80.54) -- Usual minor second (e.g. E4-F4, 79.52, -1.02)
&amp;nbsp;
24:25 (70.67) -- Apotome - QD (e.g. E*4-Eb4, 69.99, -0.68)
&amp;nbsp;
27:28 (62.96) -- QD (e.g. E4-E*4, 58.68, -4.28)
&amp;nbsp;
&amp;nbsp;
----
2. Other ratios -- many within 17-odd limit
----
&amp;nbsp;
4:7 (968.83) -- Major sixth + QD (e.g. G3-E*4, 970.97, +2.14)
&amp;nbsp;
7:9 (435.08) -- Fourth - QD (e.g. G*4-C5, 437.22, +2.14)
&amp;nbsp;
7:12 (933.13) -- Minor seventh - QD (e.g. G*3-F4), at a pure 7:12.
&amp;nbsp;
9:14 (764.92) -- Fifth + QD (e.g. G4-D*5, 762.78, -2.14)
&amp;nbsp;
9:16 (996.09) -- Usual minor seventh (e.g. G4-F4, 991.81, -4.28)
&amp;nbsp;
6:11 (1049.36) -- Minor seventh + QD (e.g. G3-F*4, 1050.49, +1.13)
&amp;nbsp;
7:11 (782.49) -- Usual minor sixth (e.g. A3-F4, 783.62, +1.13)
&amp;nbsp;
8:11 (551.32) -- Fourth + QD (e.g. G3-C*4, 554.58, +3.27)
&amp;nbsp;
9:11 (347.41) -- Minor third + QD (e.g. G3-Bb*3, 346.39, -1.02)
&amp;nbsp;
8:13 (840.53) -- Minor sixth + QD (e.g. G3-Eb*3, 842.30, +1.77)
&amp;nbsp;
9:13 (636.62) -- Diminished fifth + QD (e.g. A3-Eb*4, 634.11, -2.51)
&amp;nbsp;
11:13 (289.21) -- Usual minor third (e.g. D3-F3, 287.71, -1.50)
&amp;nbsp;
11:14 (417.51) -- Usual major third (e.g. D3-F#3, 416.38, -1.13)
&amp;nbsp;
11:16 (648.68) -- Fifth - QD (e.g. F*3-C4, 645.42, -3.27)
&amp;nbsp;
11:18 (852.59) -- Major sixth - QD (e.g. G*4-E5, 853.61, +1.02)
&amp;nbsp;
11:21 (1119.46) -- Usual major seventh (e.g. F3-E4, 1120.48, +1.02)
&amp;nbsp;
12:17 (603.00) -- Augmented third + QD (e.g. Eb4-G#*4, 603.73, +0.73)
&amp;nbsp;
13:16 (359.47) -- Major third - QD (e.g. C*4-E4, 357.70, -1.77)
&amp;nbsp;
13:18 (563.38) -- Augmented fourth - QD (e.g. C*4-F#4, 565.89, +2.51)
&amp;nbsp;
13:21 (830.25) -- Usual augmented fifth (e.g. C4-G#4, 832.76, +2.51)
&amp;nbsp;
13:22 (910.79) -- Usual major sixth (e.g. G3-E4, 912.29, +1.50)
&amp;nbsp;
13:23 (987.75) -- Usual minor seventh (e.g. D4-C5, 991.81, +4.06)
&amp;nbsp;
13:24 (1061.43) -- Major seventh - QD (e.g. F*3-E4), 1061.80, +0.37)
&amp;nbsp;
14:17 (336.13) -- Usual augmented second (e.g. F4-G#4, 336.86, +0.73)
&amp;nbsp;
14:27 (1137.04) -- Octave - QD (e.g. F*4-F5, 1141.32, +4.28)
&amp;nbsp;
15:17 (216.69) -- Diminished third + QD (e.g. C#4-Eb*4, 217.72, +1.04)
&amp;nbsp;
16:21 (470.71) -- Major third + QD (e.g. C4-E*4, 475.06, +4.28)
&amp;nbsp;
16:23 (628.27) -- Usual augmented fourth (e.g. C4-F#4, 624.57, -3.70)
&amp;nbsp;
18:23 (424.36) -- Diminished fourth + QD (e.g. B4-Eb*5, 425.91, +1.55)
&amp;nbsp;
16:25 (772.63) -- Diminished fourth + QD (e.g. F#4-Bb*4, 774.09, +1.46)
&amp;nbsp;
17:20 (281.36) -- Augmented second - QD (e.g. F*4-G#4, 278.18, -3.18)
&amp;nbsp;
17:21 (365.83) -- Usual diminished fourth (e.g. F#4-Bb4, 367.24, +1.41)
&amp;nbsp;
17:28 (863.87) -- Usual diminished seventh (e.g. F#4-Eb4, 863.14, -0.73)
&amp;nbsp;
21:34 (834.17) -- Usual augmented fifth (e.g. F3-C#4, 832.76, +1.41)
&amp;nbsp;
28:51 (1038.08) -- Usual augmented sixth (e.g. Eb3-C#4, 1040.96, +2.87)
&amp;nbsp;
21:23 (157.49) -- Usual diminished third (e.g. C#4-Eb4, 159.04, +1.55)
&amp;nbsp;
21:26 (369.75) -- Usual diminished fourth (e.g. C#4-F4, 367.24, -1.51)
&amp;nbsp;
23:27 (277.59) -- Augmented second - QD (e.g. Eb*4-F#4, 278.18, +0.59)
&amp;nbsp;
26:33 (412.75) -- Usual major third (e.g. F4-A4, 416.38, +3.63)
&amp;nbsp;
28:33 (284.45) -- Usual minor third (e.g. E4-G4, 287.71, +3.27)
&amp;nbsp;
33:56 (915.55) -- Usual major sixth (e.g. G4-E5, 912.29, -3.27)
&amp;nbsp;&lt;/pre&gt;
&amp;nbsp;&lt;/pre&gt;


&lt;!-- ws:end:WikiTextCodeRule:0 --&gt;&lt;/body&gt;&lt;/html&gt;</pre></div>
&lt;!-- ws:end:WikiTextCodeRule:0 --&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;
&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;
1:2 (1200) -- This is the usual octave (e.g. F3-F4), at a pure 1:2.&lt;br /&gt;
&lt;br /&gt;
2:3 (701.96) -- This is the usual fifth (e.g. F3-C4, 704.10, +2.14).&lt;br /&gt;
&lt;br /&gt;
3:4 (498.04) -- Usual fourth (e.g. C4-F4, 495.90, -2.14).&lt;br /&gt;
&lt;br /&gt;
6:7 (266.87) -- Major second + QD (e.g. D4-E*4), at a pure 6:7.&lt;br /&gt;
&lt;br /&gt;
7:8 (231.17) -- Minor third - QD (e.g. C*4-Eb4, 229.03, -2.14)&lt;br /&gt;
&lt;br /&gt;
8:9 (203.91) -- Usual major second (e.g. C4-D4, 208.19, +4.28)&lt;br /&gt;
&lt;br /&gt;
11:12 (150.64) -- Major second - QD (e.g. C*4-D4, 149.51, -1.13)&lt;br /&gt;
&lt;br /&gt;
12:13 (138.57) -- Minor second + QD (e.g. E4-F*4, 138.20, -0.37)&lt;br /&gt;
&lt;br /&gt;
13:14 (128.30) -- Usual apotome (e.g. C4-C#4, 128.67, +0.37)&lt;br /&gt;
&lt;br /&gt;
17:18 (98.95) -- Diminished third - QD (e.g. G#*4-Bb4, 100.36, -1.41)&lt;br /&gt;
&lt;br /&gt;
21:22 (80.54) -- Usual minor second (e.g. E4-F4, 79.52, -1.02)&lt;br /&gt;
&lt;br /&gt;
24:25 (70.67) -- Apotome - QD (e.g. E*4-Eb4, 69.99, -0.68)&lt;br /&gt;
&lt;br /&gt;
27:28 (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;
4:7 (968.83) -- Major sixth + QD (e.g. G3-E*4, 970.97, +2.14)&lt;br /&gt;
&lt;br /&gt;
7:9 (435.08) -- Fourth - QD (e.g. G*4-C5, 437.22, +2.14)&lt;br /&gt;
&lt;br /&gt;
7:12 (933.13) -- Minor seventh - QD (e.g. G*3-F4), at a pure 7:12.&lt;br /&gt;
&lt;br /&gt;
9:14 (764.92) -- Fifth + QD (e.g. G4-D*5, 762.78, -2.14)&lt;br /&gt;
&lt;br /&gt;
9:16 (996.09) -- Usual minor seventh (e.g. G4-F4, 991.81, -4.28)&lt;br /&gt;
&lt;br /&gt;
6:11 (1049.36) -- Minor seventh + QD (e.g. G3-F*4, 1050.49, +1.13)&lt;br /&gt;
&lt;br /&gt;
7:11 (782.49) -- Usual minor sixth (e.g. A3-F4, 783.62, +1.13)&lt;br /&gt;
&lt;br /&gt;
8:11 (551.32) -- Fourth + QD (e.g. G3-C*4, 554.58, +3.27)&lt;br /&gt;
&lt;br /&gt;
9:11 (347.41) -- Minor third + QD (e.g. G3-Bb*3, 346.39, -1.02)&lt;br /&gt;
&lt;br /&gt;
8:13 (840.53) -- Minor sixth + QD (e.g. G3-Eb*3, 842.30, +1.77)&lt;br /&gt;
&lt;br /&gt;
9:13 (636.62) -- Diminished fifth + QD (e.g. A3-Eb*4, 634.11, -2.51)&lt;br /&gt;
&lt;br /&gt;
11:13 (289.21) -- Usual minor third (e.g. D3-F3, 287.71, -1.50)&lt;br /&gt;
&lt;br /&gt;
11:14 (417.51) -- Usual major third (e.g. D3-F#3, 416.38, -1.13)&lt;br /&gt;
&lt;br /&gt;
11:16 (648.68) -- Fifth - QD (e.g. F*3-C4, 645.42, -3.27)&lt;br /&gt;
&lt;br /&gt;
11:18 (852.59) -- Major sixth - QD (e.g. G*4-E5, 853.61, +1.02)&lt;br /&gt;
&lt;br /&gt;
11:21 (1119.46) -- Usual major seventh (e.g. F3-E4, 1120.48, +1.02)&lt;br /&gt;
&lt;br /&gt;
12:17 (603.00) -- Augmented third + QD (e.g. Eb4-G#*4, 603.73, +0.73)&lt;br /&gt;
&lt;br /&gt;
13:16 (359.47) -- Major third - QD (e.g. C*4-E4, 357.70, -1.77)&lt;br /&gt;
&lt;br /&gt;
13:18 (563.38) -- Augmented fourth - QD (e.g. C*4-F#4, 565.89, +2.51)&lt;br /&gt;
&lt;br /&gt;
13:21 (830.25) -- Usual augmented fifth (e.g. C4-G#4, 832.76, +2.51)&lt;br /&gt;
&lt;br /&gt;
13:22 (910.79) -- Usual major sixth (e.g. G3-E4, 912.29, +1.50)&lt;br /&gt;
&lt;br /&gt;
13:23 (987.75) -- Usual minor seventh (e.g. D4-C5, 991.81, +4.06)&lt;br /&gt;
&lt;br /&gt;
13:24 (1061.43) -- Major seventh - QD (e.g. F*3-E4), 1061.80, +0.37)&lt;br /&gt;
&lt;br /&gt;
14:17 (336.13) -- Usual augmented second (e.g. F4-G#4, 336.86, +0.73)&lt;br /&gt;
&lt;br /&gt;
14:27 (1137.04) -- Octave - QD (e.g. F*4-F5, 1141.32, +4.28)&lt;br /&gt;
&lt;br /&gt;
15:17 (216.69) -- Diminished third + QD (e.g. C#4-Eb*4, 217.72, +1.04)&lt;br /&gt;
&lt;br /&gt;
16:21 (470.71) -- Major third + QD (e.g. C4-E*4, 475.06, +4.28)&lt;br /&gt;
&lt;br /&gt;
16:23 (628.27) -- Usual augmented fourth (e.g. C4-F#4, 624.57, -3.70)&lt;br /&gt;
&lt;br /&gt;
18:23 (424.36) -- Diminished fourth + QD (e.g. B4-Eb*5, 425.91, +1.55)&lt;br /&gt;
&lt;br /&gt;
16:25 (772.63) -- Diminished fourth + QD (e.g. F#4-Bb*4, 774.09, +1.46)&lt;br /&gt;
&lt;br /&gt;
17:20 (281.36) -- Augmented second - QD (e.g. F*4-G#4, 278.18, -3.18)&lt;br /&gt;
&lt;br /&gt;
17:21 (365.83) -- Usual diminished fourth (e.g. F#4-Bb4, 367.24, +1.41)&lt;br /&gt;
&lt;br /&gt;
17:28 (863.87) -- Usual diminished seventh (e.g. F#4-Eb4, 863.14, -0.73)&lt;br /&gt;
&lt;br /&gt;
21:34 (834.17) -- Usual augmented fifth (e.g. F3-C#4, 832.76, +1.41)&lt;br /&gt;
&lt;br /&gt;
28:51 (1038.08) -- Usual augmented sixth (e.g. Eb3-C#4, 1040.96, +2.87)&lt;br /&gt;
&lt;br /&gt;
21:23 (157.49) -- Usual diminished third (e.g. C#4-Eb4, 159.04, +1.55)&lt;br /&gt;
&lt;br /&gt;
21:26 (369.75) -- Usual diminished fourth (e.g. C#4-F4, 367.24, -1.51)&lt;br /&gt;
&lt;br /&gt;
23:27 (277.59) -- Augmented second - QD (e.g. Eb*4-F#4, 278.18, +0.59)&lt;br /&gt;
&lt;br /&gt;
26:33 (412.75) -- Usual major third (e.g. F4-A4, 416.38, +3.63)&lt;br /&gt;
&lt;br /&gt;
28:33 (284.45) -- Usual minor third (e.g. E4-G4, 287.71, +3.27)&lt;br /&gt;
&lt;br /&gt;
33:56 (915.55) -- Usual major sixth (e.g. G4-E5, 912.29, -3.27)&lt;/body&gt;&lt;/html&gt;</pre></div>
Retrieved from "https://en.xen.wiki/w/Peppermint-24"