Peppermint-24: Difference between revisions

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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, is quoted below. In addition to what it says, it may be noted that the Wilson/Pepper fifth it mentions, of size approximately 704.096 cents, has a precise value of (67 + sqrt(5))/118 octaves, which is (40200 + 600 sqrt(5))/59 cents.

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

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:
[[code]]
     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


[[code]]

In the following catalogue of some ratio equivalents and mappings, I will focus on intervals no further from just than 8:9 or 9:16, which vary from their pure sizes by about 4.282 cents (twice the tempering of the fifth, at about 2.141 cents wide of 2:3).

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)

Original HTML content:

<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>
 <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, is quoted below. In addition to what it says, it may be noted that the Wilson/Pepper fifth it mentions, of size approximately 704.096 cents, has a precise value of (67 + sqrt(5))/118 octaves, which is (40200 + 600 sqrt(5))/59 cents.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:3:&lt;h1&gt; --><h1 id="toc1"><a name="Margo Schulter's article"></a><!-- ws:end:WikiTextHeadingRule:3 -->Margo Schulter's article</h1>
An interesting feature of tuning systems, as implemented on keyboards (conventional or alternative), is the <a class="wiki_link" href="/keyboard%20mappings">mapping</a> of pure or tempered ratios to positions on the keyboard layout.<br />
<br />
Here I shall explore the mapping of approximate ratios, and especially of superparticular and other ratios within <a class="wiki_link" href="/Harry%20Partch">Harry Partch</a>'s larger 17-limit set, in the tuning system and keyboard arrangement I call Peppermint 24.<br />
<br />
Peppermint 24 takes as its basis a <a class="wiki_link" href="/Regular%20Temperaments">regular temperament</a> mentioned in <a class="wiki_link" href="/Erv%20Wilson">Ervin Wilson</a>'s Scale Tree and described on the Tuning List by <a class="wiki_link" href="/Keenan%20Pepper">Keenan Pepper</a>, with a fifth of about 704.096 cents, and a precise ratio of <a class="wiki_link" href="/Phi">Phi</a>, the Golden Section (~1.618) between the larger chromatic semitone (e.g. C-C#) at about 128.669 cents and the smaller diatonic semitone (e.g. C#-D) at about 79.522 cents.<br />
<br />
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).<br />
<br />
Here is a 24-note keyboard arrangement, with an asterisk (*) showing a note on the upper keyboard:<br />
<!-- ws:start:WikiTextCodeRule:0:
&lt;pre class=&quot;text&quot;&gt;     187.349  346.393              683.253    891.445 1050.488&lt;br/&gt;       C#*      Eb*                   F#*       G#*     Bb*&lt;br/&gt;  C*        D*          E*      F*        G*        A*       B*     C*&lt;br/&gt;58.680   266.871    475.062  554.584   762.775  970.967  1179.158 1258.680&lt;br/&gt;           7/6&lt;br/&gt;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;-&lt;br/&gt;     128.669  287.713              624.574    832.765 991.809&lt;br/&gt;        C#      Eb                    F#         G#     Bb&lt;br/&gt;  C         D           E       F          G         A        B     C&lt;br/&gt;  0       208.191    416.382 495.904    704.096   912.287 1120.478 1200&lt;br/&gt;&lt;br/&gt;&lt;/pre&gt;
 -->
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/**
 * GeSHi (C) 2004 - 2007 Nigel McNie, 2007 - 2008 Benny Baumann
 * (http://qbnz.com/highlighter/ and http://geshi.org/)
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-->
</style><pre class="text">     187.349  346.393              683.253    891.445 1050.488
       C#*      Eb*                   F#*       G#*     Bb*
  C*        D*          E*      F*        G*        A*       B*     C*
58.680   266.871    475.062  554.584   762.775  970.967  1179.158 1258.680
           7/6
-------------------------------------------------------------------------
     128.669  287.713              624.574    832.765 991.809
        C#      Eb                    F#         G#     Bb
  C         D           E       F          G         A        B     C
  0       208.191    416.382 495.904    704.096   912.287 1120.478 1200
&nbsp;
&nbsp;</pre>

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

Peppermint-24: Difference between revisions

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