Peppermint-24: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>

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

: This revision was by author [[User:~~xenjacob~~|~~xenjacob~~]] and made on <tt>~~2007~~-06-~~21 00~~:27:17 UTC</tt>.

: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-03-18 12:10:17 UTC</tt>.

: The original revision id was <tt>~~5371443~~</tt>.

: The original revision id was <tt>211791132</tt>.

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

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

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

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

[[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.

An interesting feature of tuning systems, as implemented on keyboards

=Margo Schulter's article=

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

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.

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 17-limit set, in the tuning system and keyboard arrangement I call Peppermint 24.

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

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.

[[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

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).

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

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

note on the upper keyboard:

[[code]]

187.349 346.393 683.253 891.445 1050.488

Line 47:

Line 35:

[[code]]

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

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).

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;

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.

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

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).

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

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.

are represented quite accurately, with 14:17:21 and related ratios

also closely approximated.

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

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

<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.

An interesting feature of tuning systems, as implemented on keyboards~~ ~~

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

(conventional or alternative), is the <a class="wiki_link" href="/keyboard%20mappings">mapping</a> of pure or tempered~~ ~~

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.

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 <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.

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.

Peppermint 24 takes as its basis a <a class="wiki_link" href="/Regular%20Temperaments">regular temperament</a> mentioned in~~ ~~

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.

<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.

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 some pure ratios of 6:7 (~266.871 cents).

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~~ ~~

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

note on the upper keyboard:

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

&lt;pre class=&quot;text&quot;&gt; 187.349 346.393 683.253 891.445 1050.488&lt;br/&gt; C#* Eb* F#* G#* Bb*&lt;br/&gt; C* D* E* F* G* A* B* C*&lt;br/&gt;58.680 266.871 475.062 554.584 762.775 970.967 1179.158 1258.680&lt;br/&gt; 7/6&lt;br/&gt;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;&amp;#45;-&lt;br/&gt; 128.669 287.713 624.574 832.765 991.809&lt;br/&gt; C# Eb F# G# Bb&lt;br/&gt; C D E F G A B C&lt;br/&gt; 0 208.191 416.382 495.904 704.096 912.287 1120.478 1200&lt;br/&gt;&lt;br/&gt;&lt;/pre&gt;

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

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).

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;~~ ~~

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.

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~~ ~~

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).

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

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

&quot;artificial&quot; diesis-like interval is actually somewhat larger than the~~ ~~

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

cents (12 tempered fifths less 7 pure octaves).

As this partial catalogue might suggest, many ratios of 2-3-7-9-11-13~~ ~~

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.

are represented quite accurately, with 14:17:21 and related ratios~~ ~~

also closely approximated.

@@ Line 1: / Line 1: @@
 <h2>IMPORTED REVISION FROM WIKISPACES</h2>
 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 00:27:17 UTC</tt>.<br>
+: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-03-18 12:10:17 UTC</tt>.<br>
-: The original revision id was <tt>5371443</tt>.<br>
+: The original revision id was <tt>211791132</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>
@@ Line 8: / Line 8: @@
 <div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">=Peppermint 24=
-[[http://launch.groups.yahoo.com/group/tuning/message/40057|Original article]] by Margo Schulter, on the Yahoo tuning forum, is quoted here.
+[[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.
-An interesting feature of tuning systems, as implemented on keyboards
+=Margo Schulter's article=
-(conventional or alternative), is the [[keyboard mappings|mapping]] of pure or tempered
+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.
-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 17-limit set, in the tuning system and keyboard arrangement I call Peppermint 24.
-of superparticular and other ratios within [[Harry Partch]]'s larger
--limit set, in the tuning system and keyboard arrangement I call
-Peppermint 24.
-Peppermint 24 takes as its basis a [[Regular Temperaments|regular temperament]] mentioned in
+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.
-[[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
+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).
-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
+Here is a 24-note keyboard arrangement, with an asterisk (*) showing a note on the upper keyboard:
-note on the upper keyboard:
 [[code]]
 .349  346.393              683.253    891.445 1050.488
@@ Line 47: / Line 35: @@
 [[code]]
-In the following catalogue of some ratio equivalents and mappings, I
+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).
-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;
+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.
-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
+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).
-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
+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.
-are represented quite accurately, with 14:17:21 and related ratios
-also closely approximated.
@@ Line 183: / Line 159: @@
 <div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;Peppermint-24&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextHeadingRule:1:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Peppermint 24"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:1 --&gt;Peppermint 24&lt;/h1&gt;
   &lt;br /&gt;
-&lt;a class="wiki_link_ext" href="http://launch.groups.yahoo.com/group/tuning/message/40057" rel="nofollow"&gt;Original article&lt;/a&gt; by Margo Schulter, on the Yahoo tuning forum, is quoted here.&lt;br /&gt;
+&lt;a class="wiki_link_ext" href="http://launch.groups.yahoo.com/group/tuning/message/40057" rel="nofollow"&gt;Original article&lt;/a&gt; by Margo Schulter, on the Yahoo tuning forum, is quoted below. In addition to what it says, it may be noted that the Wilson/Pepper fifth it mentions, of size approximately 704.096 cents, has a precise value of (67 + sqrt(5))/118 octaves, which is (40200 + 600 sqrt(5))/59 cents.&lt;br /&gt;
 &lt;br /&gt;
-An interesting feature of tuning systems, as implemented on keyboards&lt;br /&gt;
+&lt;!-- ws:start:WikiTextHeadingRule:3:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc1"&gt;&lt;a name="Margo Schulter's article"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:3 --&gt;Margo Schulter's article&lt;/h1&gt;
-(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;
+An interesting feature of tuning systems, as implemented on keyboards (conventional or alternative), is the &lt;a class="wiki_link" href="/keyboard%20mappings"&gt;mapping&lt;/a&gt; of pure or tempered ratios to positions on the keyboard layout.&lt;br /&gt;
-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;
+Here I shall explore the mapping of approximate ratios, and especially of superparticular and other ratios within &lt;a class="wiki_link" href="/Harry%20Partch"&gt;Harry Partch&lt;/a&gt;'s larger 17-limit set, in the tuning system and keyboard arrangement I call Peppermint 24.&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;
--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;
+Peppermint 24 takes as its basis a &lt;a class="wiki_link" href="/Regular%20Temperaments"&gt;regular temperament&lt;/a&gt; mentioned in &lt;a class="wiki_link" href="/Erv%20Wilson"&gt;Ervin Wilson&lt;/a&gt;'s Scale Tree and described on the Tuning List by &lt;a class="wiki_link" href="/Keenan%20Pepper"&gt;Keenan Pepper&lt;/a&gt;, with a fifth of about 704.096 cents, and a precise ratio of &lt;a class="wiki_link" href="/Phi"&gt;Phi&lt;/a&gt;, the Golden Section (~1.618) between the larger chromatic semitone (e.g. C-C#) at about 128.669 cents and the smaller diatonic semitone (e.g. C#-D) at about 79.522 cents.&lt;br /&gt;
-&lt;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;
+In Peppermint 24, two regular 12-note chains of this temperament are placed at a distance of approximately 58.680 cents, so as to yield some pure ratios of 6:7 (~266.871 cents).&lt;br /&gt;
-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;
+Here is a 24-note keyboard arrangement, with an asterisk (*) showing a note on the upper keyboard:&lt;br /&gt;
-note on the upper keyboard:&lt;br /&gt;
 &lt;!-- ws:start:WikiTextCodeRule:0:
 &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;
@@ Line 234: / Line 198: @@
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-In the following catalogue of some ratio equivalents and mappings, I&lt;br /&gt;
+In the following catalogue of some ratio equivalents and mappings, I will focus on intervals no further from just than 8:9 or 9:16, which vary from their pure sizes by about 4.282 cents (twice the tempering of the fifth, at about 2.141 cents wide of 2:3).&lt;br /&gt;
-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;
+Octave numbers appear in a MIDI-style notation, with C4 as middle C; just ratios and tempered equivalents are given values in cents, shown in parentheses, with tempered variations in cents also shown.&lt;br /&gt;
-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;
+To describe the 58.68-cent interval between the two keyboards, whose addition or subtraction plays a role in obtaining or approximating many ratios, I shall the term &amp;quot;quasi-diesis,&amp;quot; or QD for short. This &amp;quot;artificial&amp;quot; diesis-like interval is actually somewhat larger than the natural diesis in the regular Wilson/Pepper temperament at about 49.15 cents (12 tempered fifths less 7 pure octaves).&lt;br /&gt;
-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;
+As this partial catalogue might suggest, many ratios of 2-3-7-9-11-13 are represented quite accurately, with 14:17:21 and related ratios also closely approximated.&lt;br /&gt;
-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;