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Wikispaces>xenjacob **Imported revision 5371407 - Original comment: formatted, hyperlinked.** |
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<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: | : 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> | : The original revision id was <tt>5371407</tt>.<br> | ||
: The revision comment was: <tt></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> | 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. | ||
An interesting feature of tuning systems, as implemented on keyboards | [[code]] | ||
(conventional or alternative), is the mapping of pure or tempered | An interesting feature of tuning systems, as implemented on keyboards | ||
ratios to positions on the keyboard layout. | (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 | 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 temperament mentioned in | Peppermint 24 takes as its basis a [[Regular Temperaments|regular temperament]] mentioned in | ||
Ervin Wilson's Scale Tree and described on the Tuning List by Keenan | [[Erv Wilson|Ervin Wilson]]'s Scale Tree and described on the Tuning List by [[Keenan Pepper]], | ||
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 | Here is a 24-note keyboard arrangement, with an asterisk (*) showing a | ||
note on the upper keyboard: | note on the upper keyboard: | ||
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* | ||
C* D* E* F* G* A* B* C* | C* D* E* F* G* A* B* C* | ||
58.680 266.871 475.062 554.584 762.775 970.967 1179.158 1258.680 | 58.680 266.871 475.062 554.584 762.775 970.967 1179.158 1258.680 | ||
7/6 | 7/6 | ||
------------------------------------------------------------------------- | ------------------------------------------------------------------------- | ||
128.669 287.713 624.574 832.765 991.809 | 128.669 287.713 624.574 832.765 991.809 | ||
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 | ||
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 | ||
In the following catalogue of some ratio equivalents and mappings, I | |||
just | 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. | |||
cents | |||
To describe the 58.68-cent interval between the two keyboards, whose | |||
addition or subtraction plays a role in obtaining or approximating | |||
many ratios, I shall the term "quasi-diesis," or QD for short. This | |||
"artificial" diesis-like interval is actually somewhat larger than the | |||
natural diesis in the regular Wilson/Pepper temperament at about 49.15 | |||
cents (12 tempered fifths less 7 pure octaves). | |||
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: | ---- | ||
1. Multiplex (n:1) and [[superparticular]] (n+1:n) intervals | |||
---- | |||
2 | 1:2 (1200) -- This is the usual octave (e.g. F3-F4), at a pure 1:2. | ||
3 | 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). | |||
7 | 6:7 (266.87) -- Major second + QD (e.g. D4-E*4), at a pure 6:7. | ||
8 | 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) | |||
12 | 11:12 (150.64) -- Major second - QD (e.g. C*4-D4, 149.51, -1.13) | ||
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 | |||
---- | |||
7 | 4:7 (968.83) -- Major sixth + QD (e.g. G3-E*4, 970.97, +2.14) | ||
7: | 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: | 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: | 11:13 (289.21) -- Usual minor third (e.g. D3-F3, 287.71, -1.50) | ||
11: | 11:14 (417.51) -- Usual major third (e.g. D3-F#3, 416.38, -1.13) | ||
11: | 11:16 (648.68) -- Fifth - QD (e.g. F*3-C4, 645.42, -3.27) | ||
11: | 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: | 13:16 (359.47) -- Major third - QD (e.g. C*4-E4, 357.70, -1.77) | ||
13: | 13:18 (563.38) -- Augmented fourth - QD (e.g. C*4-F#4, 565.89, +2.51) | ||
13: | 13:21 (830.25) -- Usual augmented fifth (e.g. C4-G#4, 832.76, +2.51) | ||
13: | 13:22 (910.79) -- Usual major sixth (e.g. G3-E4, 912.29, +1.50) | ||
13: | 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: | 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: | 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: | 17:20 (281.36) -- Augmented second - QD (e.g. F*4-G#4, 278.18, -3.18) | ||
17: | 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: | 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) | |||
33:56 (915.55) -- Usual major sixth (e.g. G4-E5, 912.29, -3.27)</pre></div> | 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) | |||
[[code]]</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"><html><head><title>Peppermint-24</title></head><body><!-- ws:start:WikiTextHeadingRule: | <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><!-- ws:start:WikiTextHeadingRule:1:&lt;h1&gt; --><h1 id="toc0"><a name="Peppermint 24"></a><!-- ws:end:WikiTextHeadingRule:1 -->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.<br /> | <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 /> | |||
<br /> | <br /> | ||
An interesting feature of tuning systems, as implemented on keyboards<br /> | <!-- ws:start:WikiTextCodeRule:0: | ||
(conventional or alternative), is the mapping of pure or tempered<br /> | &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; | ||
ratios to positions on the keyboard layout.<br /> | --> | ||
<br /> | <style type="text/css"><!-- | ||
Here I shall explore the mapping of approximate ratios, and especially<br /> | /** | ||
of superparticular and other ratios within Harry Partch's larger<br /> | * GeSHi (C) 2004 - 2007 Nigel McNie, 2007 - 2008 Benny Baumann | ||
17-limit set, in the tuning system and keyboard arrangement I call<br /> | * (http://qbnz.com/highlighter/ and http://geshi.org/) | ||
Peppermint 24.<br /> | */ | ||
<br /> | .text {font-family:monospace;} | ||
Peppermint 24 takes as its basis a regular temperament mentioned in<br /> | .text .imp {font-weight: bold; color: red;} | ||
Ervin Wilson's Scale Tree and described on the Tuning List by Keenan<br /> | .text span.xtra { display:block; } | ||
Phi, the Golden Section (~1.618) between the larger chromatic semitone<br /> | --> | ||
(e.g. C-C#) at about 128.669 cents and the smaller diatonic semitone<br /> | </style><pre class="text"> An interesting feature of tuning systems, as implemented on keyboards | ||
(e.g. C#-D) at about 79.522 cents.<br /> | (conventional or alternative), is the [[keyboard mappings|mapping]] of pure or tempered | ||
<br /> | ratios to positions on the keyboard layout. | ||
In Peppermint 24, two regular 12-note chains of this temperament are<br /> | &nbsp; | ||
placed at a distance of approximately 58.680 cents, so as to yield<br /> | Here I shall explore the mapping of approximate ratios, and especially | ||
some pure ratios of 6:7 (~266.871 cents).<br /> | of superparticular and other ratios within [[Harry Partch]]'s larger | ||
<br /> | 17-limit set, in the tuning system and keyboard arrangement I call | ||
Here is a 24-note keyboard arrangement, with an asterisk (*) showing a<br /> | Peppermint 24. | ||
note on the upper keyboard:<br /> | &nbsp; | ||
<br /> | Peppermint 24 takes as its basis a [[Regular Temperaments|regular temperament]] mentioned in | ||
187.349 346.393 683.253 891.445 1050.488<br /> | [[Erv Wilson|Ervin Wilson]]'s Scale Tree and described on the Tuning List by [[Keenan Pepper]], | ||
C#* Eb* F#* G#* Bb*<br /> | with a fifth of about 704.096 cents, and a precise ratio of | ||
C* D* E* F* G* A* B* C*<br /> | [[Phi]], the Golden Section (~1.618) between the larger chromatic semitone | ||
58.680 266.871 475.062 554.584 762.775 970.967 1179.158 1258.680<br /> | (e.g. C-C#) at about 128.669 cents and the smaller diatonic semitone | ||
7/6<br /> | (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; | |||
&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; | |||
& | &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; | ||
<br /> | 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; | |||
<br /> | 17:18 (98.95) -- Diminished third - QD (e.g. G#*4-Bb4, 100.36, -1.41) | ||
&nbsp; | |||
<br /> | 21:22 (80.54) -- Usual minor second (e.g. E4-F4, 79.52, -1.02) | ||
&nbsp; | |||
<br /> | 24:25 (70.67) -- Apotome - QD (e.g. E*4-Eb4, 69.99, -0.68) | ||
&nbsp; | |||
<br /> | 27:28 (62.96) -- QD (e.g. E4-E*4, 58.68, -4.28) | ||
&nbsp; | |||
<br /> | &nbsp; | ||
---- | |||
<br /> | 2. Other ratios -- many within 17-odd limit | ||
---- | |||
<br /> | &nbsp; | ||
4:7 (968.83) -- Major sixth + QD (e.g. G3-E*4, 970.97, +2.14) | |||
<br /> | &nbsp; | ||
7:9 (435.08) -- Fourth - QD (e.g. G*4-C5, 437.22, +2.14) | |||
<br /> | &nbsp; | ||
7:12 (933.13) -- Minor seventh - QD (e.g. G*3-F4), at a pure 7:12. | |||
<br /> | &nbsp; | ||
9:14 (764.92) -- Fifth + QD (e.g. G4-D*5, 762.78, -2.14) | |||
<br /> | &nbsp; | ||
9:16 (996.09) -- Usual minor seventh (e.g. G4-F4, 991.81, -4.28) | |||
<br /> | &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) | |||
<br /> | &nbsp; | ||
9:11 (347.41) -- Minor third + QD (e.g. G3-Bb*3, 346.39, -1.02) | |||
<br /> | &nbsp; | ||
8:13 (840.53) -- Minor sixth + QD (e.g. G3-Eb*3, 842.30, +1.77) | |||
<br /> | &nbsp; | ||
9:13 (636.62) -- Diminished fifth + QD (e.g. A3-Eb*4, 634.11, -2.51) | |||
<br /> | &nbsp; | ||
11:13 (289.21) -- Usual minor third (e.g. D3-F3, 287.71, -1.50) | |||
<br /> | &nbsp; | ||
11:14 (417.51) -- Usual major third (e.g. D3-F#3, 416.38, -1.13) | |||
<br /> | &nbsp; | ||
11:16 (648.68) -- Fifth - QD (e.g. F*3-C4, 645.42, -3.27) | |||
<br /> | &nbsp; | ||
11:18 (852.59) -- Major sixth - QD (e.g. G*4-E5, 853.61, +1.02) | |||
<br /> | &nbsp; | ||
13 | 11:21 (1119.46) -- Usual major seventh (e.g. F3-E4, 1120.48, +1.02) | ||
<br /> | &nbsp; | ||
14 | 12:17 (603.00) -- Augmented third + QD (e.g. Eb4-G#*4, 603.73, +0.73) | ||
<br /> | &nbsp; | ||
13:16 (359.47) -- Major third - QD (e.g. C*4-E4, 357.70, -1.77) | |||
<br /> | &nbsp; | ||
13:18 (563.38) -- Augmented fourth - QD (e.g. C*4-F#4, 565.89, +2.51) | |||
<br /> | &nbsp; | ||
13:21 (830.25) -- Usual augmented fifth (e.g. C4-G#4, 832.76, +2.51) | |||
<br /> | &nbsp; | ||
13:22 (910.79) -- Usual major sixth (e.g. G3-E4, 912.29, +1.50) | |||
<br /> | &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) | |||
<br /> | &nbsp; | ||
14:27 (1137.04) -- Octave - QD (e.g. F*4-F5, 1141.32, +4.28) | |||
<br /> | &nbsp; | ||
15:17 (216.69) -- Diminished third + QD (e.g. C#4-Eb*4, 217.72, +1.04) | |||
<br /> | &nbsp; | ||
16:21 (470.71) -- Major third + QD (e.g. C4-E*4, 475.06, +4.28) | |||
<br /> | &nbsp; | ||
16:23 (628.27) -- Usual augmented fourth (e.g. C4-F#4, 624.57, -3.70) | |||
<br /> | &nbsp; | ||
18:23 (424.36) -- Diminished fourth + QD (e.g. B4-Eb*5, 425.91, +1.55) | |||
<br /> | &nbsp; | ||
16:25 (772.63) -- Diminished fourth + QD (e.g. F#4-Bb*4, 774.09, +1.46) | |||
<br /> | &nbsp; | ||
17:20 (281.36) -- Augmented second - QD (e.g. F*4-G#4, 278.18, -3.18) | |||
<br /> | &nbsp; | ||
17:21 (365.83) -- Usual diminished fourth (e.g. F#4-Bb4, 367.24, +1.41) | |||
<br /> | &nbsp; | ||
17:28 (863.87) -- Usual diminished seventh (e.g. F#4-Eb4, 863.14, -0.73) | |||
<br /> | &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> | |||
Revision as of 00:24, 21 June 2007
IMPORTED REVISION FROM WIKISPACES
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=Peppermint 24=
[[http://launch.groups.yahoo.com/group/tuning/message/40057|Original article]] by Margo Schulter, on the Yahoo tuning forum.
[[code]]
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 [[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
[[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:
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
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 "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.
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)
[[code]]Original HTML content:
<html><head><title>Peppermint-24</title></head><body><!-- ws:start:WikiTextHeadingRule:1:<h1> --><h1 id="toc0"><a name="Peppermint 24"></a><!-- ws:end:WikiTextHeadingRule:1 -->Peppermint 24</h1>
<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 />
<br />
<!-- ws:start:WikiTextCodeRule:0:
<pre class="text"> An interesting feature of tuning systems, as implemented on keyboards<br/> (conventional or alternative), is the [[keyboard mappings|mapping]] of pure or tempered<br/> ratios to positions on the keyboard layout.<br/><br/> Here I shall explore the mapping of approximate ratios, and especially<br/> of superparticular and other ratios within [[Harry Partch]]'s larger<br/> 17-limit set, in the tuning system and keyboard arrangement I call<br/> Peppermint 24.<br/><br/> Peppermint 24 takes as its basis a [[Regular Temperaments|regular temperament]] mentioned in<br/> [[Erv Wilson|Ervin Wilson]]'s Scale Tree and described on the Tuning List by [[Keenan Pepper]],<br/> with a fifth of about 704.096 cents, and a precise ratio of<br/> [[Phi]], the Golden Section (~1.618) between the larger chromatic semitone<br/> (e.g. C-C#) at about 128.669 cents and the smaller diatonic semitone<br/> (e.g. C#-D) at about 79.522 cents.<br/><br/> In Peppermint 24, two regular 12-note chains of this temperament are<br/> placed at a distance of approximately 58.680 cents, so as to yield<br/> some pure ratios of 6:7 (~266.871 cents).<br/><br/> Here is a 24-note keyboard arrangement, with an asterisk (*) showing a<br/> note on the upper keyboard:<br/><br/> 187.349 346.393 683.253 891.445 1050.488<br/> C#* Eb* F#* G#* Bb*<br/> C* D* E* F* G* A* B* C*<br/>58.680 266.871 475.062 554.584 762.775 970.967 1179.158 1258.680<br/> 7/6<br/>&#45;&#45;&#45;&#45;&#45;&#45;&#45;&#45;&#45;&#45;&#45;&#45;&#45;&#45;&#45;&#45;&#45;&#45;&#45;&#45;&#45;&#45;&#45;&#45;&#45;&#45;&#45;&#45;&#45;&#45;&#45;&#45;&#45;&#45;&#45;&#45;&#45;&#45;&#45;&#45;&#45;&#45;&#45;&#45;&#45;&#45;&#45;&#45;&#45;&#45;&#45;&#45;&#45;&#45;&#45;&#45;&#45;&#45;&#45;&#45;&#45;&#45;&#45;&#45;&#45;&#45;&#45;&#45;&#45;&#45;&#45;&#45;-<br/> 128.669 287.713 624.574 832.765 991.809<br/> C# Eb F# G# Bb<br/> C D E F G A B C<br/> 0 208.191 416.382 495.904 704.096 912.287 1120.478 1200<br/><br/><br/> In the following catalogue of some ratio equivalents and mappings, I<br/> will focus on intervals no further from just than 8:9 or 9:16, which<br/> vary from their pure sizes by about 4.282 cents (twice the tempering<br/> of the fifth, at about 2.141 cents wide of 2:3).<br/><br/> Octave numbers appear in a MIDI-style notation, with C4 as middle C;<br/> just ratios and tempered equivalents are given values in cents, shown<br/> in parentheses, with tempered variations in cents also shown.<br/><br/> To describe the 58.68-cent interval between the two keyboards, whose<br/> addition or subtraction plays a role in obtaining or approximating<br/> many ratios, I shall the term &quot;quasi-diesis,&quot; or QD for short. This<br/> &quot;artificial&quot; diesis-like interval is actually somewhat larger than the<br/> natural diesis in the regular Wilson/Pepper temperament at about 49.15<br/> cents (12 tempered fifths less 7 pure octaves).<br/><br/> As this partial catalogue might suggest, many ratios of 2-3-7-9-11-13<br/> are represented quite accurately, with 14:17:21 and related ratios<br/> also closely approximated.<br/><br/><br/>&#45;&#45;&#45;-<br/> 1. Multiplex (n:1) and [[superparticular]] (n+1:n) intervals<br/>&#45;&#45;&#45;-<br/><br/> 1:2 (1200) &#45;- This is the usual octave (e.g. F3-F4), at a pure 1:2.<br/><br/> 2:3 (701.96) &#45;- This is the usual fifth (e.g. F3-C4, 704.10, +2.14).<br/><br/> 3:4 (498.04) &#45;- Usual fourth (e.g. C4-F4, 495.90, -2.14).<br/><br/> 6:7 (266.87) &#45;- Major second + QD (e.g. D4-E*4), at a pure 6:7.<br/><br/> 7:8 (231.17) &#45;- Minor third - QD (e.g. C*4-Eb4, 229.03, -2.14)<br/><br/> 8:9 (203.91) &#45;- Usual major second (e.g. C4-D4, 208.19, +4.28)<br/><br/> 11:12 (150.64) &#45;- Major second - QD (e.g. C*4-D4, 149.51, -1.13)<br/><br/> 12:13 (138.57) &#45;- Minor second + QD (e.g. E4-F*4, 138.20, -0.37)<br/><br/> 13:14 (128.30) &#45;- Usual apotome (e.g. C4-C#4, 128.67, +0.37)<br/><br/> 17:18 (98.95) &#45;- Diminished third - QD (e.g. G#*4-Bb4, 100.36, -1.41)<br/><br/> 21:22 (80.54) &#45;- Usual minor second (e.g. E4-F4, 79.52, -1.02)<br/><br/> 24:25 (70.67) &#45;- Apotome - QD (e.g. E*4-Eb4, 69.99, -0.68)<br/><br/> 27:28 (62.96) &#45;- QD (e.g. E4-E*4, 58.68, -4.28)<br/><br/><br/>&#45;&#45;&#45;-<br/> 2. Other ratios &#45;- many within 17-odd limit<br/>&#45;&#45;&#45;-<br/><br/> 4:7 (968.83) &#45;- Major sixth + QD (e.g. G3-E*4, 970.97, +2.14)<br/><br/> 7:9 (435.08) &#45;- Fourth - QD (e.g. G*4-C5, 437.22, +2.14)<br/><br/> 7:12 (933.13) &#45;- Minor seventh - QD (e.g. G*3-F4), at a pure 7:12.<br/><br/> 9:14 (764.92) &#45;- Fifth + QD (e.g. G4-D*5, 762.78, -2.14)<br/><br/> 9:16 (996.09) &#45;- Usual minor seventh (e.g. G4-F4, 991.81, -4.28)<br/><br/> 6:11 (1049.36) &#45;- Minor seventh + QD (e.g. G3-F*4, 1050.49, +1.13)<br/><br/> 7:11 (782.49) &#45;- Usual minor sixth (e.g. A3-F4, 783.62, +1.13)<br/><br/> 8:11 (551.32) &#45;- Fourth + QD (e.g. G3-C*4, 554.58, +3.27)<br/><br/> 9:11 (347.41) &#45;- Minor third + QD (e.g. G3-Bb*3, 346.39, -1.02)<br/><br/> 8:13 (840.53) &#45;- Minor sixth + QD (e.g. G3-Eb*3, 842.30, +1.77)<br/><br/> 9:13 (636.62) &#45;- Diminished fifth + QD (e.g. A3-Eb*4, 634.11, -2.51)<br/><br/> 11:13 (289.21) &#45;- Usual minor third (e.g. D3-F3, 287.71, -1.50)<br/><br/> 11:14 (417.51) &#45;- Usual major third (e.g. D3-F#3, 416.38, -1.13)<br/><br/> 11:16 (648.68) &#45;- Fifth - QD (e.g. F*3-C4, 645.42, -3.27)<br/><br/> 11:18 (852.59) &#45;- Major sixth - QD (e.g. G*4-E5, 853.61, +1.02)<br/><br/> 11:21 (1119.46) &#45;- Usual major seventh (e.g. F3-E4, 1120.48, +1.02)<br/><br/> 12:17 (603.00) &#45;- Augmented third + QD (e.g. Eb4-G#*4, 603.73, +0.73)<br/><br/> 13:16 (359.47) &#45;- Major third - QD (e.g. C*4-E4, 357.70, -1.77)<br/><br/> 13:18 (563.38) &#45;- Augmented fourth - QD (e.g. C*4-F#4, 565.89, +2.51)<br/><br/> 13:21 (830.25) &#45;- Usual augmented fifth (e.g. C4-G#4, 832.76, +2.51)<br/><br/> 13:22 (910.79) &#45;- Usual major sixth (e.g. G3-E4, 912.29, +1.50)<br/><br/> 13:23 (987.75) &#45;- Usual minor seventh (e.g. D4-C5, 991.81, +4.06)<br/><br/> 13:24 (1061.43) &#45;- Major seventh - QD (e.g. F*3-E4), 1061.80, +0.37)<br/><br/> 14:17 (336.13) &#45;- Usual augmented second (e.g. F4-G#4, 336.86, +0.73)<br/><br/> 14:27 (1137.04) &#45;- Octave - QD (e.g. F*4-F5, 1141.32, +4.28)<br/><br/> 15:17 (216.69) &#45;- Diminished third + QD (e.g. C#4-Eb*4, 217.72, +1.04)<br/><br/> 16:21 (470.71) &#45;- Major third + QD (e.g. C4-E*4, 475.06, +4.28)<br/><br/> 16:23 (628.27) &#45;- Usual augmented fourth (e.g. C4-F#4, 624.57, -3.70)<br/><br/> 18:23 (424.36) &#45;- Diminished fourth + QD (e.g. B4-Eb*5, 425.91, +1.55)<br/><br/> 16:25 (772.63) &#45;- Diminished fourth + QD (e.g. F#4-Bb*4, 774.09, +1.46)<br/><br/> 17:20 (281.36) &#45;- Augmented second - QD (e.g. F*4-G#4, 278.18, -3.18)<br/><br/> 17:21 (365.83) &#45;- Usual diminished fourth (e.g. F#4-Bb4, 367.24, +1.41)<br/><br/> 17:28 (863.87) &#45;- Usual diminished seventh (e.g. F#4-Eb4, 863.14, -0.73)<br/><br/> 21:34 (834.17) &#45;- Usual augmented fifth (e.g. F3-C#4, 832.76, +1.41)<br/><br/> 28:51 (1038.08) &#45;- Usual augmented sixth (e.g. Eb3-C#4, 1040.96, +2.87)<br/><br/> 21:23 (157.49) &#45;- Usual diminished third (e.g. C#4-Eb4, 159.04, +1.55)<br/><br/> 21:26 (369.75) &#45;- Usual diminished fourth (e.g. C#4-F4, 367.24, -1.51)<br/><br/> 23:27 (277.59) &#45;- Augmented second - QD (e.g. Eb*4-F#4, 278.18, +0.59)<br/><br/> 26:33 (412.75) &#45;- Usual major third (e.g. F4-A4, 416.38, +3.63)<br/><br/> 28:33 (284.45) &#45;- Usual minor third (e.g. E4-G4, 287.71, +3.27)<br/><br/> 33:56 (915.55) &#45;- Usual major sixth (e.g. G4-E5, 912.29, -3.27)<br/></pre>
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</style><pre class="text"> 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 [[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
[[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:
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
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 "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.
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)
</pre>
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