Kai's House of Odds and Ends: 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:kai.lugheidh|kai.lugheidh]] and made on <tt>2018-01-17 04:29:47 UTC</tt>.

: This revision was by author [[User:kai.lugheidh|kai.lugheidh]] and made on <tt>2018-01-17 04:49:17 UTC</tt>.

: The original revision id was <tt>~~624971083~~</tt>.

: The original revision id was <tt>624971673</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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=La Monte Young's tuning from //The Well-Tuned Piano,// in Rational Comma Notation (and brand spankin' new KAIGI SIGNATURES!)=

With Concert Pitch (1/1) being C4@256Hz, Eb4[7] = 7/6 = (298+2/3)Hz serves nicely as the tonic. Originally, Young tuned his "A", 189/128, to 440Hz, giving Eb of (297+187/189)Hz. These two pitches are less than 0.7Hz apart, a difference that can be essentially ignored, unless you're playing along to a recording with the same reference pitch, or feel like giving an officially sanctioned performance of the WTP (yeah right.)

With Concert Pitch (1/1) being C4@256Hz, Eb4[7] = 7/6 = (298+2/3)Hz serves nicely as the tonic in [[xenharmonic/Rational Comma Notation (RCN)|Rational Comma Notation (RCN). ]]Originally, Young tuned his "A", 189/128, to 440Hz, giving Eb of (297+187/189)Hz. These two pitches are less than 0.7Hz apart, a difference that can be essentially ignored, unless you're playing along to a recording with the same reference pitch, or feel like giving an officially sanctioned performance of the WTP (yeah right.)

The octave above the Eb[7] reference pitch is as follows:

|| || "Eb" || "E" || "F" || "F#" || "G" || "G#" || "A" || "Bb" || "B" || "C" || "C#" || "D" ||

||= ||= **"Eb"** ||= **"E"** ||= **"F"** ||= **"F#"** ||= **"G"** ||= **"G#"** ||= **"A"** ||= **"Bb"** ||= **"B"** ||= **"C"** ||= **"C#"** ||= **"D"** ||= **"Eb"** ||

|| RATIO || 1/1 || 567/512 || 9/8 || 147/128 || 21/16 || 1323/1024 || 189/128 || 3/2 || 49/32 || 7/4 || 441/256 || 63/32 ||

||= **RATIO** ||= 1/1 ||= 567/512 ||= 9/8 ||= 147/128 ||= 21/16 ||= 1323/1024 ||= 189/128 ||= 3/2 ||= 49/32 ||= 7/4 ||= 441/256 ||= 63/32 ||= 2/1 ||

|| CENTS || 0.000000 || 176.645910 || 203.910002 || 239.606814 || 470.780907 || 443.516816 || 674.690909 || 701.955001 || 737.651813 || 968.825906 || 941.561815 || 1172.735908 ||

||= **CENTS** ||= 0.000000 ||= 176.645910 ||= 203.910002 ||= 239.606814 ||= 470.780907 ||= 443.516816 ||= 674.690909 ||= 701.955001 ||= 737.651813 ||= 968.825906 ||= 941.561815 ||= 1172.735908 ||= 1200.000000 ||

|| KAIGI ~~SIG.~~ || – || ~~◻4▲~~ || ◻2 || ~~◻▲2~~ || ◻▲ || ~~◻3▲2~~ || ~~◻3▲~~ || ◻ || ▲2 || ▲ || ~~◻2▲2~~ || ~~◻2▲~~ ||

||= **KAIGI SIGNATURE** ||= – ||= ◻4▲ ||= ◻2 ||= ◻▲2 ||= ◻▲ ||= ◻3▲2 ||= ◻3▲ ||= ◻ ||= ▲2 ||= ▲ ||= ◻2▲2 ||= ◻2▲ ||= – ||

|| ABS. PITCH (RCN) || Eb4[7] || F4[49] || || || || || || || || || || ||

||= **ABS. PITCH (RCN)** ||= Eb4[7] ||= F4[49] ||= F4[7] ||= Gb4[343] ||= Ab4[49] ||= Ab4[343] ||= Bb4[49] ||= Bb4[7] ||= Cb4[343] ||= Db4[49] ||= Db4[343] ||= Eb5[49] ||= Eb5[7] ||

|| FREQUENCY || 298+2/3 || || || || || || || || || || || ||

||= **FREQUENCY (Hz)** ||= 298+2/3 ||= 330+3/4 ||= 336 ||= 343 ||= 392 ||= 385+7/8 ||= 441 ||= 448 ||= 457+1/3 ||= 522+2/3 ||= 514+1/2 ||= 588 ||= 597+1/3 ||

The pitch classes of this tuning as a lattice in RCN:

||= Cb[343] ||= Gb[343] ||= Db[343] ||= Ab[343] ||= ||

</pre></div>

||= Db[49] ||= Ab[49] ||= Eb[49] ||= Bb[49] ||= F[49] ||

||= Eb[7] ||= Bb[7] ||= F[7] ||= ||= ||

The same lattice in tonic-agnostic Kaigi Signatures, which makes the logic of the system clear:

|| ▲2 || ◻▲2 || ◻2▲2 || ◻3▲2 || ||

|| ▲ || ◻▲ || ◻2▲ || ◻3▲ || ◻4▲ ||

|| – || ◻ || ◻2 || || ||</pre></div>

<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>Kai's House of Odds and Ends</title></head><body>Hai, I'm Kai! This is a place for musings, messings, and novelties of mine that I'm not sure deserve their own page, or that I'm having trouble fitting into an existing page, but that I think will be of interest to my fellow xen-heads anyways!

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<h1 id="toc0"><a name="La Monte Young's tuning from The Well-Tuned Piano, in Rational Comma Notation (and brand spankin' new KAIGI SIGNATURES!)"></a>La Monte Young's tuning from The Well-Tuned Piano, in Rational Comma Notation (and brand spankin' new KAIGI SIGNATURES!)</h1>

With Concert Pitch (1/1) being C4@256Hz, Eb4[7] = 7/6 = (298+2/3)Hz serves nicely as the tonic. Originally, Young tuned his &quot;A&quot;, 189/128, to 440Hz, giving Eb of (297+187/189)Hz. These two pitches are less than 0.7Hz apart, a difference that can be essentially ignored, unless you're playing along to a recording with the same reference pitch, or feel like giving an officially sanctioned performance of the WTP (yeah right.)

With Concert Pitch (1/1) being C4@256Hz, Eb4[7] = 7/6 = (298+2/3)Hz serves nicely as the tonic in <a class="wiki_link" href="http://xenharmonic.wikispaces.com/Rational%20Comma%20Notation%20%28RCN%29">Rational Comma Notation (RCN). </a>Originally, Young tuned his &quot;A&quot;, 189/128, to 440Hz, giving Eb of (297+187/189)Hz. These two pitches are less than 0.7Hz apart, a difference that can be essentially ignored, unless you're playing along to a recording with the same reference pitch, or feel like giving an officially sanctioned performance of the WTP (yeah right.)

The octave above the Eb[7] reference pitch is as follows:

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<tr>

<td>

</td>

<td style="text-align: center;">&quot;Eb&quot;

</td>

<td style="text-align: center;">&quot;E&quot;

</td>

<td style="text-align: center;">&quot;F&quot;

</td>

<td style="text-align: center;">&quot;F#&quot;

</td>

<td style="text-align: center;">&quot;G&quot;

</td>

<td style="text-align: center;">&quot;G#&quot;

</td>

<td style="text-align: center;">&quot;A&quot;

</td>

<td style="text-align: center;">&quot;Bb&quot;

</td>

<td style="text-align: center;">&quot;B&quot;

</td>

<td style="text-align: center;">&quot;C&quot;

</td>

<td style="text-align: center;">&quot;C#&quot;

</td>

<td style="text-align: center;">&quot;D&quot;

</td>

<td style="text-align: center;">&quot;Eb&quot;

</td>

</tr>

<tr>

<td style="text-align: center;">RATIO

</td>

<td style="text-align: center;">1/1

</td>

<td>~~&quot;Eb&quot;~~

<td style="text-align: center;">567/512

</td>

<td>~~&quot;E&quot;~~

<td style="text-align: center;">9/8

</td>

<td>~~&quot;F&quot;~~

<td style="text-align: center;">147/128

</td>

<td>~~&quot;F#&quot;~~

<td style="text-align: center;">21/16

</td>

<td>~~&quot;G&quot;~~

<td style="text-align: center;">1323/1024

</td>

<td>~~&quot;G#&quot;~~

<td style="text-align: center;">189/128

</td>

<td>~~&quot;A&quot;~~

<td style="text-align: center;">3/2

</td>

<td>~~&quot;Bb&quot;~~

<td style="text-align: center;">49/32

</td>

<td>~~&quot;B&quot;~~

<td style="text-align: center;">7/4

</td>

<td>~~&quot;C&quot;~~

<td style="text-align: center;">441/256

</td>

<td>~~&quot;C#&quot;~~

<td style="text-align: center;">63/32

</td>

<td>~~&quot;D&quot;~~

<td style="text-align: center;">2/1

</td>

</tr>

<tr>

<td>~~RATIO~~

<td style="text-align: center;">CENTS

</td>

<td style="text-align: center;">0.000000

</td>

<td>~~1/1~~

<td style="text-align: center;">176.645910

</td>

<td>~~567/512~~

<td style="text-align: center;">203.910002

</td>

<td>~~9/8~~

<td style="text-align: center;">239.606814

</td>

<td>~~147/128~~

<td style="text-align: center;">470.780907

</td>

<td>~~21/16~~

<td style="text-align: center;">443.516816

</td>

<td>~~1323/1024~~

<td style="text-align: center;">674.690909

</td>

<td>~~189/128~~

<td style="text-align: center;">701.955001

</td>

<td>~~3/2~~

<td style="text-align: center;">737.651813

</td>

<td>~~49/32~~

<td style="text-align: center;">968.825906

</td>

<td>~~7/4~~

<td style="text-align: center;">941.561815

</td>

<td>~~441/256~~

<td style="text-align: center;">1172.735908

</td>

<td>~~63/32~~

<td style="text-align: center;">1200.000000

</td>

</tr>

<tr>

<td>~~CENTS~~

<td style="text-align: center;">KAIGI SIGNATURE

</td>

<td style="text-align: center;">–

</td>

<td>~~0.000000~~

<td style="text-align: center;">◻4▲

</td>

<td>~~176.645910~~

<td style="text-align: center;">◻2

</td>

<td>~~203.910002~~

<td style="text-align: center;">◻▲2

</td>

<td>~~239.606814~~

<td style="text-align: center;">◻▲

</td>

<td>~~470.780907~~

<td style="text-align: center;">◻3▲2

</td>

<td>~~443.516816~~

<td style="text-align: center;">◻3▲

</td>

<td>~~674.690909~~

<td style="text-align: center;">◻

</td>

<td>~~701.955001~~

<td style="text-align: center;">▲2

</td>

<td>~~737.651813~~

<td style="text-align: center;">▲

</td>

<td>~~968.825906~~

<td style="text-align: center;">◻2▲2

</td>

<td>~~941.561815~~

<td style="text-align: center;">◻2▲

</td>

<td>~~1172.735908~~

<td style="text-align: center;">–

</td>

</tr>

<tr>

<td>~~KAIGI SIG~~.

<td style="text-align: center;">ABS. PITCH (RCN)

</td>

<td>–

<td style="text-align: center;">Eb4[7]

</td>

<td style="text-align: center;">F4[49]

</td>

<td>~~◻4▲~~

<td style="text-align: center;">F4[7]

</td>

<td>◻2

<td style="text-align: center;">Gb4[343]

</td>

<td>~~◻▲2~~

<td style="text-align: center;">Ab4[49]

</td>

<td>◻▲

<td style="text-align: center;">Ab4[343]

</td>

<td>~~◻3▲2~~

<td style="text-align: center;">Bb4[49]

</td>

<td>~~◻3▲~~

<td style="text-align: center;">Bb4[7]

</td>

<td>◻

<td style="text-align: center;">Cb4[343]

</td>

<td>▲2

<td style="text-align: center;">Db4[49]

</td>

<td>▲

<td style="text-align: center;">Db4[343]

</td>

<td>~~◻2▲2~~

<td style="text-align: center;">Eb5[49]

</td>

<td>~~◻2▲~~

<td style="text-align: center;">Eb5[7]

</td>

</tr>

<tr>

<td>~~ABS. PITCH~~ (~~RCN~~)

<td style="text-align: center;">FREQUENCY (Hz)

</td>

<td>~~Eb4[7]~~

<td style="text-align: center;">298+2/3

</td>

<td>~~F4[49]~~

<td style="text-align: center;">330+3/4

</td>

<td>

<td style="text-align: center;">336

</td>

<td style="text-align: center;">343

</td>

<td style="text-align: center;">392

</td>

<td style="text-align: center;">385+7/8

</td>

<td style="text-align: center;">441

</td>

<td style="text-align: center;">448

</td>

<td style="text-align: center;">457+1/3

</td>

<td style="text-align: center;">522+2/3

</td>

<td style="text-align: center;">514+1/2

</td>

<td style="text-align: center;">588

</td>

<td style="text-align: center;">597+1/3

</td>

</tr>

</table>

The pitch classes of this tuning as a lattice in RCN:

<tr>

<td style="text-align: center;">Cb[343]

</td>

<td>

<td style="text-align: center;">Gb[343]

</td>

<td>

<td style="text-align: center;">Db[343]

</td>

<td>

<td style="text-align: center;">Ab[343]

</td>

<td>

</td>

<td>

</tr>

<tr>

<td style="text-align: center;">Db[49]

</td>

<td>

<td style="text-align: center;">Ab[49]

</td>

<td>

<td style="text-align: center;">Eb[49]

</td>

<td>

<td style="text-align: center;">Bb[49]

</td>

<td>

<td style="text-align: center;">F[49]

</td>

</tr>

<tr>

<td>~~FREQUENCY~~

<td style="text-align: center;">Eb[7]

</td>

<td style="text-align: center;">Bb[7]

</td>

<td style="text-align: center;">F[7]

</td>

</td>

<td>~~298+~~2/3

</td>

</tr>

</table>

The same lattice in tonic-agnostic Kaigi Signatures, which makes the logic of the system clear:

<tr>

<td>▲2

</td>

<td>◻▲2

</td>

<td>◻2▲2

</td>

<td>◻3▲2

</td>

<td>

</td>

<td>

</tr>

<tr>

<td>▲

</td>

<td>

<td>◻▲

</td>

<td>

<td>◻2▲

</td>

<td>

<td>◻3▲

</td>

<td>

<td>◻4▲

</td>

<td>

</tr>

<tr>

<td>–

</td>

<td>

<td>◻

</td>

<td>

<td>◻2

</td>

<td>

@@ 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:kai.lugheidh|kai.lugheidh]] and made on <tt>2018-01-17 04:29:47 UTC</tt>.<br>
+: This revision was by author [[User:kai.lugheidh|kai.lugheidh]] and made on <tt>2018-01-17 04:49:17 UTC</tt>.<br>
-: The original revision id was <tt>624971083</tt>.<br>
+: The original revision id was <tt>624971673</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 10: / Line 10: @@
 =La Monte Young's tuning from //The Well-Tuned Piano,// in Rational Comma Notation (and brand spankin' new KAIGI SIGNATURES!)=
-With Concert Pitch (1/1) being C4@256Hz, Eb4[7] = 7/6 = (298+2/3)Hz serves nicely as the tonic. Originally, Young tuned his "A", 189/128, to 440Hz, giving Eb of (297+187/189)Hz. These two pitches are less than 0.7Hz apart, a difference that can be essentially ignored, unless you're playing along to a recording with the same reference pitch, or feel like giving an officially sanctioned performance of the WTP (yeah right.)
+With Concert Pitch (1/1) being C4@256Hz, Eb4[7] = 7/6 = (298+2/3)Hz serves nicely as the tonic in [[xenharmonic/Rational Comma Notation (RCN)|Rational Comma Notation (RCN). ]]Originally, Young tuned his "A", 189/128, to 440Hz, giving Eb of (297+187/189)Hz. These two pitches are less than 0.7Hz apart, a difference that can be essentially ignored, unless you're playing along to a recording with the same reference pitch, or feel like giving an officially sanctioned performance of the WTP (yeah right.)
 The octave above the Eb[7] reference pitch is as follows:
-||   || "Eb" || "E" || "F" || "F#" || "G" || "G#" || "A" || "Bb" || "B" || "C" || "C#" || "D" ||
+||=   ||= **&lt;span style="background-color: #000000; color: #ffffff;"&gt;"Eb"&lt;/span&gt;** ||= **"E"** ||= **"F"** ||= **&lt;span style="background-color: #000000; color: #ffffff;"&gt;"F#"&lt;/span&gt;** ||= **"G"** ||= **&lt;span style="background-color: #000000; color: #ffffff;"&gt;"G#"&lt;/span&gt;** ||= **"A"** ||= **&lt;span style="background-color: #000000; color: #ffffff;"&gt;"Bb"&lt;/span&gt;** ||= **"B"** ||= **"C"** ||= **&lt;span style="background-color: #000000; color: #ffffff;"&gt;"C#"&lt;/span&gt;** ||= **"D"** ||= **&lt;span style="background-color: #000000; color: #ffffff;"&gt;"Eb"&lt;/span&gt;** ||
-|| RATIO || 1/1 || 567/512 || 9/8 || 147/128 || 21/16 || 1323/1024 || 189/128 || 3/2 || 49/32 || 7/4 || 441/256 || 63/32 ||
+||= **RATIO** ||= 1/1 ||= 567/512 ||= 9/8 ||= 147/128 ||= 21/16 ||= 1323/1024 ||= 189/128 ||= 3/2 ||= 49/32 ||= 7/4 ||= 441/256 ||= 63/32 ||= 2/1 ||
-|| CENTS || 0.000000 || 176.645910 || 203.910002 || 239.606814 || 470.780907 || 443.516816 || 674.690909 || 701.955001 || 737.651813 || 968.825906 || 941.561815 || 1172.735908 ||
+||= **CENTS** ||= 0.000000 ||= 176.645910 ||= 203.910002 ||= 239.606814 ||= 470.780907 ||= 443.516816 ||= 674.690909 ||= 701.955001 ||= 737.651813 ||= 968.825906 ||= 941.561815 ||= 1172.735908 ||= 1200.000000 ||
-|| KAIGI SIG. || – || ◻4▲ || ◻2 || ◻▲2 || ◻▲ || ◻3▲2 ||  ◻3▲ || ◻ || ▲2 || ▲ || ◻2▲2 || ◻2▲ ||
+||= **KAIGI SIGNATURE** ||= – ||= ◻4&lt;span style="color: #00edff;"&gt;▲&lt;/span&gt; ||= ◻2 ||= ◻&lt;span style="color: #00edff;"&gt;▲&lt;/span&gt;2 ||= ◻&lt;span style="color: #00edff;"&gt;▲&lt;/span&gt; ||= ◻3&lt;span style="color: #00edff;"&gt;▲&lt;/span&gt;2 ||= ◻3&lt;span style="color: #00edff;"&gt;▲&lt;/span&gt; ||= ◻ ||= &lt;span style="color: #00edff;"&gt;▲&lt;/span&gt;2 ||= &lt;span style="color: #00edff;"&gt;▲&lt;/span&gt; ||= ◻2&lt;span style="color: #00edff;"&gt;▲&lt;/span&gt;2 ||= ◻2&lt;span style="color: #00edff;"&gt;▲&lt;/span&gt; ||= – ||
-|| ABS. PITCH (RCN) || Eb4[7] || F4[49] ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||
+||= **ABS. PITCH (RCN)** ||= Eb4[7] ||= F4[49] ||= F4[7] ||= Gb4[343] ||= Ab4[49] ||= Ab4[343] ||= Bb4[49] ||= Bb4[7] ||= Cb4[343] ||= Db4[49] ||= Db4[343] ||= Eb5[49] ||= Eb5[7] ||
-|| FREQUENCY || 298+2/3 ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||   ||
+||= **FREQUENCY (Hz)** ||= 298+2/3 ||= 330+3/4 ||= 336 ||= 343 ||= 392 ||= 385+7/8 ||= 441 ||= 448 ||= 457+1/3 ||= 522+2/3 ||= 514+1/2 ||= 588 ||= 597+1/3 ||
+The pitch classes of this tuning as a lattice in RCN:
+||= Cb[343] ||= Gb[343] ||= Db[343] ||= Ab[343] ||=   ||
-</pre></div>
+||= Db[49] ||= Ab[49] ||= Eb[49] ||= Bb[49] ||= F[49] ||
+||= Eb[7] ||= Bb[7] ||= F[7] ||=   ||=   ||
+The same lattice in tonic-agnostic Kaigi Signatures, which makes the logic of the system clear:
+|| &lt;span style="color: #00edff;"&gt;▲&lt;/span&gt;2 || ◻&lt;span style="color: #00edff;"&gt;▲&lt;/span&gt;2 || ◻2&lt;span style="color: #00edff;"&gt;▲&lt;/span&gt;2 || ◻3&lt;span style="color: #00edff;"&gt;▲&lt;/span&gt;2 ||   ||
+|| &lt;span style="color: #00edff;"&gt;▲&lt;/span&gt; || ◻&lt;span style="color: #00edff;"&gt;▲&lt;/span&gt; || ◻2&lt;span style="color: #00edff;"&gt;▲&lt;/span&gt; || ◻3&lt;span style="color: #00edff;"&gt;▲&lt;/span&gt; || ◻4&lt;span style="color: #00edff;"&gt;▲&lt;/span&gt; ||
+|| – || ◻ || ◻2 ||   ||   ||</pre></div>
 <h4>Original HTML content:</h4>
 <div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;Kai's House of Odds and Ends&lt;/title&gt;&lt;/head&gt;&lt;body&gt;Hai, I'm Kai! This is a place for musings, messings, and novelties of mine that I'm not sure deserve their own page, or that I'm having trouble fitting into an existing page, but that I think will be of interest to my fellow xen-heads anyways!&lt;br /&gt;
@@ Line 27: / Line 32: @@
 &lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="La Monte Young's tuning from The Well-Tuned Piano, in Rational Comma Notation (and brand spankin' new KAIGI SIGNATURES!)"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;La Monte Young's tuning from &lt;em&gt;The Well-Tuned Piano,&lt;/em&gt; in Rational Comma Notation (and brand spankin' new KAIGI SIGNATURES!)&lt;/h1&gt;
   &lt;br /&gt;
-With Concert Pitch (1/1) being C4@256Hz, Eb4[7] = 7/6 = (298+2/3)Hz serves nicely as the tonic. Originally, Young tuned his &amp;quot;A&amp;quot;, 189/128, to 440Hz, giving Eb of (297+187/189)Hz. These two pitches are less than 0.7Hz apart, a difference that can be essentially ignored, unless you're playing along to a recording with the same reference pitch, or feel like giving an officially sanctioned performance of the WTP (yeah right.)&lt;br /&gt;
+With Concert Pitch (1/1) being C4@256Hz, Eb4[7] = 7/6 = (298+2/3)Hz serves nicely as the tonic in &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Rational%20Comma%20Notation%20%28RCN%29"&gt;Rational Comma Notation (RCN). &lt;/a&gt;Originally, Young tuned his &amp;quot;A&amp;quot;, 189/128, to 440Hz, giving Eb of (297+187/189)Hz. These two pitches are less than 0.7Hz apart, a difference that can be essentially ignored, unless you're playing along to a recording with the same reference pitch, or feel like giving an officially sanctioned performance of the WTP (yeah right.)&lt;br /&gt;
 &lt;br /&gt;
 The octave above the Eb[7] reference pitch is as follows:&lt;br /&gt;
@@ Line 34: / Line 39: @@
 &lt;table class="wiki_table"&gt;
      &lt;tr&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;strong&gt;&lt;span style="background-color: #000000; color: #ffffff;"&gt;&amp;quot;Eb&amp;quot;&lt;/span&gt;&lt;/strong&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;strong&gt;&amp;quot;E&amp;quot;&lt;/strong&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;strong&gt;&amp;quot;F&amp;quot;&lt;/strong&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;strong&gt;&lt;span style="background-color: #000000; color: #ffffff;"&gt;&amp;quot;F#&amp;quot;&lt;/span&gt;&lt;/strong&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;strong&gt;&amp;quot;G&amp;quot;&lt;/strong&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;strong&gt;&lt;span style="background-color: #000000; color: #ffffff;"&gt;&amp;quot;G#&amp;quot;&lt;/span&gt;&lt;/strong&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;strong&gt;&amp;quot;A&amp;quot;&lt;/strong&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;strong&gt;&lt;span style="background-color: #000000; color: #ffffff;"&gt;&amp;quot;Bb&amp;quot;&lt;/span&gt;&lt;/strong&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;strong&gt;&amp;quot;B&amp;quot;&lt;/strong&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;strong&gt;&amp;quot;C&amp;quot;&lt;/strong&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;strong&gt;&lt;span style="background-color: #000000; color: #ffffff;"&gt;&amp;quot;C#&amp;quot;&lt;/span&gt;&lt;/strong&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;strong&gt;&amp;quot;D&amp;quot;&lt;/strong&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;strong&gt;&lt;span style="background-color: #000000; color: #ffffff;"&gt;&amp;quot;Eb&amp;quot;&lt;/span&gt;&lt;/strong&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;&lt;strong&gt;RATIO&lt;/strong&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;1/1&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&amp;quot;Eb&amp;quot;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;567/512&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&amp;quot;E&amp;quot;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;9/8&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&amp;quot;F&amp;quot;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;147/128&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&amp;quot;F#&amp;quot;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;21/16&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&amp;quot;G&amp;quot;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;1323/1024&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&amp;quot;G#&amp;quot;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;189/128&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&amp;quot;A&amp;quot;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;3/2&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&amp;quot;Bb&amp;quot;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;49/32&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&amp;quot;B&amp;quot;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;7/4&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&amp;quot;C&amp;quot;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;441/256&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&amp;quot;C#&amp;quot;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;63/32&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&amp;quot;D&amp;quot;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;2/1&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
      &lt;tr&gt;
-         &lt;td&gt;RATIO&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;&lt;strong&gt;CENTS&lt;/strong&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;0.000000&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;1/1&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;176.645910&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;567/512&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;203.910002&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;9/8&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;239.606814&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;147/128&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;470.780907&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;21/16&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;443.516816&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;1323/1024&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;674.690909&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;189/128&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;701.955001&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;3/2&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;737.651813&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;49/32&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;968.825906&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;7/4&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;941.561815&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;441/256&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;1172.735908&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;63/32&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;1200.000000&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
      &lt;tr&gt;
-         &lt;td&gt;CENTS&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;&lt;strong&gt;KAIGI SIGNATURE&lt;/strong&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;–&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;0.000000&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;◻4&lt;span style="color: #00edff;"&gt;▲&lt;/span&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;176.645910&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;◻2&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;203.910002&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;◻&lt;span style="color: #00edff;"&gt;▲&lt;/span&gt;2&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;239.606814&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;◻&lt;span style="color: #00edff;"&gt;▲&lt;/span&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;470.780907&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;◻3&lt;span style="color: #00edff;"&gt;▲&lt;/span&gt;2&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;443.516816&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;◻3&lt;span style="color: #00edff;"&gt;▲&lt;/span&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;674.690909&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;◻&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;701.955001&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;&lt;span style="color: #00edff;"&gt;▲&lt;/span&gt;2&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;737.651813&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;&lt;span style="color: #00edff;"&gt;▲&lt;/span&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;968.825906&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;◻2&lt;span style="color: #00edff;"&gt;▲&lt;/span&gt;2&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;941.561815&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;◻2&lt;span style="color: #00edff;"&gt;▲&lt;/span&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;1172.735908&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;–&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
      &lt;tr&gt;
-         &lt;td&gt;KAIGI SIG.&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;&lt;strong&gt;ABS. PITCH (RCN)&lt;/strong&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;–&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;Eb4[7]&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;F4[49]&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;◻4▲&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;F4[7]&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;◻2&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;Gb4[343]&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;◻▲2&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;Ab4[49]&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;◻▲&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;Ab4[343]&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;◻3▲2&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;Bb4[49]&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;◻3▲&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;Bb4[7]&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;◻&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;Cb4[343]&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;▲2&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;Db4[49]&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;▲&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;Db4[343]&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;◻2▲2&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;Eb5[49]&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;◻2▲&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;Eb5[7]&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
      &lt;tr&gt;
-         &lt;td&gt;ABS. PITCH (RCN)&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;&lt;strong&gt;FREQUENCY (Hz)&lt;/strong&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;Eb4[7]&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;298+2/3&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;F4[49]&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;330+3/4&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;336&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;343&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;392&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;385+7/8&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;441&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;448&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;457+1/3&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;522+2/3&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;514+1/2&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;588&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;597+1/3&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+&lt;/table&gt;
+The pitch classes of this tuning as a lattice in RCN:&lt;br /&gt;
+&lt;table class="wiki_table"&gt;
+    &lt;tr&gt;
+        &lt;td style="text-align: center;"&gt;Cb[343]&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;Gb[343]&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;Db[343]&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;Ab[343]&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+         &lt;td style="text-align: center;"&gt;Db[49]&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;Ab[49]&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;Eb[49]&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;Bb[49]&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;F[49]&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
      &lt;tr&gt;
-         &lt;td&gt;FREQUENCY&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;Eb[7]&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;Bb[7]&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;F[7]&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;298+2/3&lt;br /&gt;
+        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+&lt;/table&gt;
+The same lattice in tonic-agnostic Kaigi Signatures, which makes the logic of the system clear:&lt;br /&gt;
+&lt;table class="wiki_table"&gt;
+    &lt;tr&gt;
+         &lt;td&gt;&lt;span style="color: #00edff;"&gt;▲&lt;/span&gt;2&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;◻&lt;span style="color: #00edff;"&gt;▲&lt;/span&gt;2&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;◻2&lt;span style="color: #00edff;"&gt;▲&lt;/span&gt;2&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;◻3&lt;span style="color: #00edff;"&gt;▲&lt;/span&gt;2&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+         &lt;td&gt;&lt;span style="color: #00edff;"&gt;▲&lt;/span&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td&gt;◻&lt;span style="color: #00edff;"&gt;▲&lt;/span&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td&gt;◻2&lt;span style="color: #00edff;"&gt;▲&lt;/span&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td&gt;◻3&lt;span style="color: #00edff;"&gt;▲&lt;/span&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td&gt;◻4&lt;span style="color: #00edff;"&gt;▲&lt;/span&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+    &lt;/tr&gt;
+    &lt;tr&gt;
+         &lt;td&gt;–&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td&gt;◻&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td&gt;◻2&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;&lt;br /&gt;