18edo: 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:~~xenwolf~~|~~xenwolf~~]] and made on <tt>2016-11-~~01 18~~:36:31 UTC</tt>.

: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2016-11-21 04:30:30 UTC</tt>.

: The original revision id was <tt>~~597693000~~</tt>.

: The original revision id was <tt>599948016</tt>.

: The revision comment was: <tt>~~removed tel links~~</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.

<h4>Original Wikitext content:</h4>

Line 10:

[[18平均律|日本語]]

**18 Equal Divisions of the Octave**

**AKA The Third-Tone System**

Line 23:

==Representations of Just Intervals==

|| Degree || Cents coarse/fine

|| Degree || Cents

DMS ||= 5L3s Notation || Nearest Ratio || Error (cents coarse/fine)

coarse/fine

DMS ||= 5L3s Notation ||||= [[Ups and Downs Notation|up/down ]]

[[Ups and Downs Notation|notation]] ** || Nearest Ratio || Error (cents

coarse/fine)

(DMS) ||< 17-Limit Ratios* ||

|| 0 || 0 ||= **C** || 1/1 || 0 ||< **1/1** ||

|| 0 || 0 ||= **C** ||= C ||= P1 || 1/1 || 0 ||< **1/1** ||

|| 1 || 66.667

80

20° ||= Db || 27/26 || +1.329

20° ||= Db ||= C^,

Dv ||= ^P1,

vM2 || 27/26 || +1.329

+1.595

+23'56" ||< 78/75, 75/72 ||

|| 2 || 133.333

160

40° ||= C# || 27/25 || +0.096

40° ||= C# ||= D ||= M2 || 27/25 || +0.096

+0.115

+1'43" ||< 51/55, 42/39 ||

|| 3 || 200

240

60° ||= **D** || 9/8 || -3.910

60° ||= **D** ||= D^,

Ev ||= ~2 || 9/8 || -3.910

-4.692

-1°10'23" ||< **9/8** ||

|| 4 || 266.667

320

80° ||= Eb || 7/6 || -0.204

80° ||= Eb ||= E ||= m2,

M3 || 7/6 || -0.204

-0.245

-1"50" ||< **75/64** ||

|| 5 || 333.333

400

100° ||= D# || 17/14 or 40/33 || -2.796 +0.293

100° ||= D# ||= E^ ||= ~3 || 17/14 or 40/33 || -2.796 +0.293

-3.355 +0.351

-50'20" +5'16" ||< **39/32** ||

|| 6 || 400

480

120° ||= **E** || 5/4 or 44/35 || +13.686 +3.822

120° ||= **E** ||= Eb,

F# ||= m3,

A4 || 5/4 or 44/35 || +13.686 +3.822

+16.4235 +4.586

+4°6'21" +1°8'47" ||< 64/55 ||

|| 7 || 466.667

560

140° ||= **F** || 21/16 || -4.114

140° ||= **F** ||= Fv ||= ^m3,

vP4 || 21/16 || -4.114

-4.937

-1°14'3" ||< **21/16** ||

|| 8 || 533.333

640

160° ||= Gb || 15/11 || -3.617

160° ||= Gb ||= F ||= P4 || 15/11 || -3.617

-4.341

-1°5'7" ||< 102/75 ||

|| 9 || 600

720

180° ||= F# || 17/12 or 24/17 || -3.000 +3.000

180° ||= F# ||= F^,

Gv ||= ^P4,

vP5 || 17/12 or 24/17 || -3.000 +3.000

-3.6005 +3.6005

-54' +54' ||< 17/12 ||

|| 10 || 666.667

800

200° ||= **G** || 22/15 || +3.617

200° ||= **G** ||= G ||= P5 || 22/15 || +3.617

+4.341

+1°5'7" ||< 75/51 ||

|| 11 || 733.333

880

220° ||= Hb || 32/21 || +4.114

220° ||= Hb ||= G^ ||= ^P5,

vM6 || 32/21 || +4.114

+4.937

+1°14'3" ||< 32/21 ||

|| 12 || 800

960

240° ||= G# || 8/5 or 35/22 || -13.686 -3.822

240° ||= G# ||= A ||= d5,

M6 || 8/5 or 35/22 || -13.686 -3.822

-16.4235 -4.586

-4°6'21" -1°8'47" ||< **51/32** ||

|| 13 || 866.667

1040

260° ||= **H** || 28/17 or 33/20 || +2.796 -0.293

260° ||= **H** ||= A^,

Bv ||= ~6 || 28/17 or 33/20 || +2.796 -0.293

+3.355 -0.351

+50'20" -5'16" ||< 64/39 ||

|| 14 || 933.333

1120

280° ||= **A** || 12/7 || +0.204

280° ||= **A** ||= B ||= m6,

M7 || 12/7 || +0.204

+0.245

+1"50" ||< **55/32** ||

|| 15 || 1000

1200

300° ||= Bb || 16/9 || +3.910

300° ||= Bb ||= B^ ||= ~7 || 16/9 || +3.910

+4.692

+1°10'23" ||< 16/9 ||

|| 16 || 1066.667

1280

320° ||= A# || 50/27 || -0.096

320° ||= A# ||= Bb,

C# ||= m7 || 50/27 || -0.096

-0.115

-1'43" ||< 39/21 ||

|| 17 || 1133.333

1360

340° ||= **B** || 52/27 || -1.329

340° ||= **B** ||= Cv ||= ^m7,

vP8 || 52/27 || -1.329

-1.595

-23'56" ||< 75/39 ||

|| 18 || 1200

1440

360° ||= **C** || 2/1 || 0 ||< **2/1** ||

360° ||= **C** ||= C ||= P8 || 2/1 || 0 ||< **2/1** ||

*based on the above description of 18-EDO as a 2.9.75.21.55.39.51 subgroup temperament

** based on 18b-edo with sharp lower than flat and major narrower than minor

For alternative notations, see [[xenharmonic/Ups and Downs Notation#Summary%20of%20EDO%20notation-%22Superflat%22%20EDOs|Ups and Downs Notation -"Superflat" EDOs]] and [[xenharmonic/Ups and Downs Notation#Summary%20of%20EDO%20notation-%22Supersharp%22%20EDOs|Ups and Downs Notation -"Supersharp" EDOs]] and [[xenharmonic/Ups and Downs Notation#Natural%20Generators|Ups and Downs Notation - Natural Generators]].

==Useful Moment-of-Symmetry Scales==

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18 Equal Divisions of the Octave

18 Equal Divisions of the Octave

AKA The Third-Tone System

Line 186:

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<td>Degree

</td>

<td>Cents coarse/fine

<td>Cents

coarse/fine

DMS

</td>

<td style="text-align: center;">5L3s Notation

</td>

<a class="wiki_link" href="/Ups%20and%20Downs%20Notation">notation</a>

</td>

<td>Nearest Ratio

</td>

<td>Error (cents coarse/fine)

<td>Error (cents

coarse/fine)

(DMS)

</td>

Line 204:

Line 231:

<td>0

</td>

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

</td>

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

</td>

<td style="text-align: center;">P1

</td>

<td>1/1

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<td>0

</td>

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

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

</td>

</tr>

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</td>

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

</td>

<td style="text-align: center;">C^,

Dv

</td>

<td style="text-align: center;">^P1,

vM2

</td>

<td>27/26

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</td>

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

</td>

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

</td>

<td style="text-align: center;">M2

</td>

<td>27/25

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60°

</td>

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

</td>

<td style="text-align: center;">D^,

Ev

</td>

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

</td>

<td>9/8

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-1°10'23&quot;

</td>

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

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

</td>

</tr>

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</td>

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

</td>

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

</td>

<td style="text-align: center;">m2,

M3

</td>

<td>7/6

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-1&quot;50&quot;

</td>

<td style="text-align: left;">75/64

<td style="text-align: left;">75/64

</td>

</tr>

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</td>

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

</td>

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

</td>

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

</td>

<td>17/14 or 40/33

Line 300:

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-50'20&quot; +5'16&quot;

</td>

<td style="text-align: left;">39/32

<td style="text-align: left;">39/32

</td>

</tr>

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120°

</td>

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

</td>

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

F#

</td>

<td style="text-align: center;">m3,

A4

</td>

<td>5/4 or 44/35

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Line 389:

140°

</td>

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

</td>

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

</td>

<td style="text-align: center;">^m3,

vP4

</td>

<td>21/16

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Line 402:

-1°14'3&quot;

</td>

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

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

</td>

</tr>

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</td>

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

</td>

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

</td>

<td style="text-align: center;">P4

</td>

<td>15/11

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Line 435:

</td>

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

</td>

<td style="text-align: center;">F^,

Gv

</td>

<td style="text-align: center;">^P4,

vP5

</td>

<td>17/12 or 24/17

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200°

</td>

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

</td>

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

</td>

<td style="text-align: center;">P5

</td>

<td>22/15

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</td>

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

</td>

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

</td>

<td style="text-align: center;">^P5,

vM6

</td>

<td>32/21

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Line 504:

</td>

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

</td>

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

</td>

<td style="text-align: center;">d5,

M6

</td>

<td>8/5 or 35/22

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-4°6'21&quot; -1°8'47&quot;

</td>

<td style="text-align: left;">51/32

<td style="text-align: left;">51/32

</td>

</tr>

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260°

</td>

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

</td>

<td style="text-align: center;">A^,

Bv

</td>

<td style="text-align: center;">~6

</td>

<td>28/17 or 33/20

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Line 549:

280°

</td>

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

</td>

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

</td>

<td style="text-align: center;">m6,

M7

</td>

<td>12/7

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+1&quot;50&quot;

</td>

<td style="text-align: left;">55/32

<td style="text-align: left;">55/32

</td>

</tr>

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Line 573:

</td>

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

</td>

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

</td>

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

</td>

<td>16/9

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</td>

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

</td>

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

C#

</td>

<td style="text-align: center;">m7

</td>

<td>50/27

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340°

</td>

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

</td>

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

</td>

<td style="text-align: center;">^m7,

vP8

</td>

<td>52/27

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360°

</td>

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

</td>

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

</td>

<td style="text-align: center;">P8

</td>

<td>2/1

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<td>0

</td>

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

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

</td>

</tr>

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*based on the above description of 18-EDO as a 2.9.75.21.55.39.51 subgroup temperament

<ul><ul><li>based on 18b-edo with sharp lower than flat and major narrower than minor</li></ul></ul>

For alternative notations, see <a class="wiki_link" href="http://xenharmonic.wikispaces.com/Ups%20and%20Downs%20Notation#Summary%20of%20EDO%20notation-%22Superflat%22%20EDOs">Ups and Downs Notation -&quot;Superflat&quot; EDOs</a> and <a class="wiki_link" href="http://xenharmonic.wikispaces.com/Ups%20and%20Downs%20Notation#Summary%20of%20EDO%20notation-%22Supersharp%22%20EDOs">Ups and Downs Notation -&quot;Supersharp&quot; EDOs</a> and <a class="wiki_link" href="http://xenharmonic.wikispaces.com/Ups%20and%20Downs%20Notation#Natural%20Generators">Ups and Downs Notation - Natural Generators</a>.

<h2 id="toc2"><a name="Basic Properties-Useful Moment-of-Symmetry Scales"></a>Useful Moment-of-Symmetry Scales</h2>

@@ 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:xenwolf|xenwolf]] and made on <tt>2016-11-01 18:36:31 UTC</tt>.<br>
+: This revision was by author [[User:TallKite|TallKite]] and made on <tt>2016-11-21 04:30:30 UTC</tt>.<br>
-: The original revision id was <tt>597693000</tt>.<br>
+: The original revision id was <tt>599948016</tt>.<br>
-: The revision comment was: <tt>removed tel links</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>
 <h4>Original Wikitext content:</h4>
@@ Line 10: / Line 10: @@
 &lt;span style="display: block; text-align: right;"&gt;[[18平均律|日本語]]
 &lt;/span&gt;
 **18 Equal Divisions of the Octave**
 **AKA The Third-Tone System**
@@ Line 23: / Line 23: @@
 ==Representations of Just Intervals==
-|| Degree || Cents coarse/fine
+|| Degree || Cents
-DMS ||= 5L3s Notation || Nearest Ratio || Error (cents coarse/fine)
+coarse/fine
+DMS ||= 5L3s Notation ||||= [[Ups and Downs Notation|up/down ]]
+[[Ups and Downs Notation|notation]] ** || Nearest Ratio || Error (cents
+coarse/fine)
 (DMS) ||&lt; 17-Limit Ratios* ||
-|| 0 || 0 ||= **C** || 1/1 || 0 ||&lt; **1/1** ||
+|| 0 || 0 ||= **C** ||= C ||= P1 || 1/1 || 0 ||&lt; **1/1** ||
 || 1 || 66.667
-° ||= Db || 27/26 || +1.329
+° ||= Db ||= C^,
+Dv ||= ^P1,
+vM2 || 27/26 || +1.329
 +1.595
 +23'56" ||&lt; 78/75, 75/72 ||
 || 2 || 133.333
-° ||= C# || 27/25 || +0.096
+° ||= C# ||= D ||= M2 || 27/25 || +0.096
 +0.115
 +1'43" ||&lt; 51/55, 42/39 ||
 || 3 || 200
-° ||= **D** || 9/8 || -3.910
+° ||= **D** ||= D^,
+Ev ||= ~2 || 9/8 || -3.910
 -4.692
 -1°10'23" ||&lt; **9/8** ||
 || 4 || 266.667
-° ||= Eb || 7/6 || -0.204
+° ||= Eb ||= E ||= m2,
+M3 || 7/6 || -0.204
 -0.245
 -1"50" ||&lt; **75/64** ||
 || 5 || 333.333
-° ||= D# || 17/14 or 40/33 || -2.796 +0.293
+° ||= D# ||= E^ ||= ~3 || 17/14 or 40/33 || -2.796 +0.293
 -3.355 +0.351
 -50'20" +5'16" ||&lt; **39/32** ||
 || 6 || 400
-° ||= **E** || 5/4 or 44/35 || +13.686 +3.822
+° ||= **E** ||= Eb,
+F# ||= m3,
+A4 || 5/4 or 44/35 || +13.686 +3.822
 +16.4235 +4.586
 +4°6'21" +1°8'47" ||&lt; 64/55 ||
 || 7 || 466.667
-° ||= **F** || 21/16 || -4.114
+° ||= **F** ||= Fv ||= ^m3,
+vP4 || 21/16 || -4.114
 -4.937
 -1°14'3" ||&lt; **21/16** ||
 || 8 || 533.333
-° ||= Gb || 15/11 || -3.617
+° ||= Gb ||= F ||= P4 || 15/11 || -3.617
 -4.341
 -1°5'7" ||&lt; 102/75 ||
 || 9 || 600
-° ||= F# || 17/12 or 24/17 || -3.000 +3.000
+° ||= F# ||= F^,
+Gv ||= ^P4,
+vP5 || 17/12 or 24/17 || -3.000 +3.000
 -3.6005 +3.6005
 -54' +54' ||&lt; 17/12 ||
 || 10 || 666.667
-° ||= **G** || 22/15 || +3.617
+° ||= **G** ||= G ||= P5 || 22/15 || +3.617
 +4.341
 +1°5'7" ||&lt; 75/51 ||
 || 11 || 733.333
-° ||= Hb || 32/21 || +4.114
+° ||= Hb ||= G^ ||= ^P5,
+vM6 || 32/21 || +4.114
 +4.937
 +1°14'3" ||&lt; 32/21 ||
 || 12 || 800
-° ||= G# || 8/5 or 35/22 || -13.686 -3.822
+° ||= G# ||= A ||= d5,
+M6 || 8/5 or 35/22 || -13.686 -3.822
 -16.4235 -4.586
 -4°6'21" -1°8'47" ||&lt; **51/32** ||
 || 13 || 866.667
-° ||= **H** || 28/17 or 33/20 || +2.796 -0.293
+° ||= **H** ||= A^,
+Bv ||= ~6 || 28/17 or 33/20 || +2.796 -0.293
 +3.355 -0.351
 &lt;span style="line-height: 15.6000003814697px;"&gt;+&lt;/span&gt;50'20" &lt;span style="line-height: 15.6000003814697px;"&gt;-&lt;/span&gt;&lt;span style="line-height: 1.5;"&gt;5'16"&lt;/span&gt; ||&lt; 64/39 ||
 || 14 || 933.333
-° ||= **A** || 12/7 || +0.204
+° ||= **A** ||= B ||= m6,
+M7 || 12/7 || +0.204
 +0.245
 +1"50" ||&lt; **55/32** ||
 || 15 || 1000
-° ||= Bb || 16/9 || +3.910
+° ||= Bb ||= B^ ||= ~7 || 16/9 || +3.910
 +4.692
 +1°10'23" ||&lt; 16/9 ||
 || 16 || 1066.667
-° ||= A# || 50/27 || -0.096
+° ||= A# ||= Bb,
+C# ||= m7 || 50/27 || -0.096
 -0.115
 -1'43" ||&lt; 39/21 ||
 || 17 || 1133.333
-° ||= **B** || 52/27 || -1.329
+° ||= **B** ||= Cv ||= ^m7,
+vP8 || 52/27 || -1.329
 -1.595
 -23'56" ||&lt; 75/39 ||
 || 18 || 1200
-° ||= **C** || 2/1 || 0 ||&lt; **2/1** ||
+° ||= **C** ||= C ||= P8 || 2/1 || 0 ||&lt; **2/1** ||
 *based on the above description of 18-EDO as a 2.9.75.21.55.39.51 subgroup temperament
+** based on 18b-edo with sharp lower than flat and major narrower than minor
+For alternative notations, see [[xenharmonic/Ups and Downs Notation#Summary%20of%20EDO%20notation-%22Superflat%22%20EDOs|Ups and Downs Notation -"Superflat" EDOs]] and [[xenharmonic/Ups and Downs Notation#Summary%20of%20EDO%20notation-%22Supersharp%22%20EDOs|Ups and Downs Notation -"Supersharp" EDOs]] and [[xenharmonic/Ups and Downs Notation#Natural%20Generators|Ups and Downs Notation - Natural Generators]].
 ==&lt;span style="font-size: 1.3em;"&gt;Useful Moment-of-Symmetry Scales&lt;/span&gt;==
@@ Line 167: / Line 189: @@
 &lt;span style="display: block; text-align: right;"&gt;&lt;a class="wiki_link" href="/18%E5%B9%B3%E5%9D%87%E5%BE%8B"&gt;日本語&lt;/a&gt;&lt;br /&gt;
 &lt;/span&gt;&lt;br /&gt;
-&lt;strong&gt;18 Equal Divisions of the Octave&lt;/strong&gt; &lt;br /&gt;
+&lt;strong&gt;18 Equal Divisions of the Octave&lt;/strong&gt;&lt;br /&gt;
 &lt;strong&gt;AKA The Third-Tone System&lt;/strong&gt;&lt;br /&gt;
 &lt;br /&gt;
@@ Line 186: / Line 208: @@
          &lt;td&gt;Degree&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;Cents coarse/fine&lt;br /&gt;
+         &lt;td&gt;Cents &lt;br /&gt;
+coarse/fine&lt;br /&gt;
 DMS&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;5L3s Notation&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td colspan="2" style="text-align: center;"&gt;&lt;a class="wiki_link" href="/Ups%20and%20Downs%20Notation"&gt;up/down &lt;/a&gt;&lt;br /&gt;
+&lt;a class="wiki_link" href="/Ups%20and%20Downs%20Notation"&gt;notation&lt;/a&gt; &lt;strong&gt;&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;Nearest Ratio&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;Error (cents coarse/fine)&lt;br /&gt;
+         &lt;td&gt;Error (cents &lt;br /&gt;
+coarse/fine)&lt;br /&gt;
 (DMS)&lt;br /&gt;
 &lt;/td&gt;
@@ Line 204: / Line 231: @@
          &lt;td&gt;0&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;&lt;strong&gt;C&lt;/strong&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;&lt;/strong&gt;C&lt;strong&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;C&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;P1&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;1/1&lt;br /&gt;
@@ Line 210: / Line 241: @@
          &lt;td&gt;0&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: left;"&gt;&lt;strong&gt;1/1&lt;/strong&gt;&lt;br /&gt;
+         &lt;td style="text-align: left;"&gt;&lt;/strong&gt;1/1&lt;strong&gt;&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 221: / Line 252: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;Db&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;C^,&lt;br /&gt;
+Dv&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;^P1,&lt;br /&gt;
+vM2&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;27/26&lt;br /&gt;
@@ Line 239: / Line 276: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;C#&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;D&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;M2&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;27/25&lt;br /&gt;
@@ Line 256: / Line 297: @@
 °&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;&lt;strong&gt;D&lt;/strong&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;&lt;/strong&gt;D&lt;strong&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;D^,&lt;br /&gt;
+Ev&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;~2&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;9/8&lt;br /&gt;
@@ Line 264: / Line 310: @@
 -1°10'23&amp;quot;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: left;"&gt;&lt;strong&gt;9/8&lt;/strong&gt;&lt;br /&gt;
+         &lt;td style="text-align: left;"&gt;&lt;/strong&gt;9/8&lt;strong&gt;&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 275: / Line 321: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;Eb&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;E&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;m2,&lt;br /&gt;
+M3&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;7/6&lt;br /&gt;
@@ Line 282: / Line 333: @@
 -1&amp;quot;50&amp;quot;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: left;"&gt;&lt;strong&gt;75/64&lt;/strong&gt;&lt;br /&gt;
+         &lt;td style="text-align: left;"&gt;&lt;/strong&gt;75/64&lt;strong&gt;&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 293: / Line 344: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;D#&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;E^&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;~3&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;17/14 or 40/33&lt;br /&gt;
@@ Line 300: / Line 355: @@
 -50'20&amp;quot; +5'16&amp;quot;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: left;"&gt;&lt;strong&gt;39/32&lt;/strong&gt;&lt;br /&gt;
+         &lt;td style="text-align: left;"&gt;&lt;/strong&gt;39/32&lt;strong&gt;&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 310: / Line 365: @@
 °&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;&lt;strong&gt;E&lt;/strong&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;&lt;/strong&gt;E&lt;strong&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;Eb,&lt;br /&gt;
+F#&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;m3,&lt;br /&gt;
+A4&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;5/4 or 44/35&lt;br /&gt;
@@ Line 328: / Line 389: @@
 °&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;&lt;strong&gt;F&lt;/strong&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;&lt;/strong&gt;F&lt;strong&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;Fv&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;^m3,&lt;br /&gt;
+vP4&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;21/16&lt;br /&gt;
@@ Line 336: / Line 402: @@
 -1°14'3&amp;quot;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: left;"&gt;&lt;strong&gt;21/16&lt;/strong&gt;&lt;br /&gt;
+         &lt;td style="text-align: left;"&gt;&lt;/strong&gt;21/16&lt;strong&gt;&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 347: / Line 413: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;Gb&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;F&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;P4&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;15/11&lt;br /&gt;
@@ Line 365: / Line 435: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;F#&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;F^,&lt;br /&gt;
+Gv&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;^P4,&lt;br /&gt;
+vP5&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;17/12 or 24/17&lt;br /&gt;
@@ Line 382: / Line 458: @@
 °&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;&lt;strong&gt;G&lt;/strong&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;&lt;/strong&gt;G&lt;strong&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;G&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;P5&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;22/15&lt;br /&gt;
@@ Line 401: / Line 481: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;Hb&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;G^&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;^P5,&lt;br /&gt;
+vM6&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;32/21&lt;br /&gt;
@@ Line 419: / Line 504: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;G#&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;A&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;d5,&lt;br /&gt;
+M6&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;8/5 or 35/22&lt;br /&gt;
@@ Line 426: / Line 516: @@
 -4°6'21&amp;quot; -1°8'47&amp;quot;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: left;"&gt;&lt;strong&gt;51/32&lt;/strong&gt;&lt;br /&gt;
+         &lt;td style="text-align: left;"&gt;&lt;/strong&gt;51/32&lt;strong&gt;&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 436: / Line 526: @@
 °&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;&lt;strong&gt;H&lt;/strong&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;&lt;/strong&gt;H&lt;strong&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;A^,&lt;br /&gt;
+Bv&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;~6&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;28/17 or 33/20&lt;br /&gt;
@@ Line 454: / Line 549: @@
 °&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;&lt;strong&gt;A&lt;/strong&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;&lt;/strong&gt;A&lt;strong&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;B&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;m6,&lt;br /&gt;
+M7&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;12/7&lt;br /&gt;
@@ Line 462: / Line 562: @@
 +1&amp;quot;50&amp;quot;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: left;"&gt;&lt;strong&gt;55/32&lt;/strong&gt;&lt;br /&gt;
+         &lt;td style="text-align: left;"&gt;&lt;/strong&gt;55/32&lt;strong&gt;&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 473: / Line 573: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;Bb&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;B^&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;~7&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;16/9&lt;br /&gt;
@@ Line 491: / Line 595: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;A#&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;Bb,&lt;br /&gt;
+C#&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;m7&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;50/27&lt;br /&gt;
@@ Line 508: / Line 617: @@
 °&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;&lt;strong&gt;B&lt;/strong&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;&lt;/strong&gt;B&lt;strong&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;Cv&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;^m7,&lt;br /&gt;
+vP8&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;52/27&lt;br /&gt;
@@ Line 526: / Line 640: @@
 °&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;&lt;strong&gt;C&lt;/strong&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;&lt;/strong&gt;C&lt;strong&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;C&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;P8&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;2/1&lt;br /&gt;
@@ Line 532: / Line 650: @@
          &lt;td&gt;0&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: left;"&gt;&lt;strong&gt;2/1&lt;/strong&gt;&lt;br /&gt;
+         &lt;td style="text-align: left;"&gt;&lt;/strong&gt;2/1**&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 538: / Line 656: @@
 *based on the above description of 18-EDO as a 2.9.75.21.55.39.51 subgroup temperament&lt;br /&gt;
+&lt;br /&gt;
+&lt;ul&gt;&lt;ul&gt;&lt;li&gt;based on 18b-edo with sharp lower than flat and major narrower than minor&lt;/li&gt;&lt;/ul&gt;&lt;/ul&gt;&lt;br /&gt;
+For alternative notations, see &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Ups%20and%20Downs%20Notation#Summary%20of%20EDO%20notation-%22Superflat%22%20EDOs"&gt;Ups and Downs Notation -&amp;quot;Superflat&amp;quot; EDOs&lt;/a&gt; and &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Ups%20and%20Downs%20Notation#Summary%20of%20EDO%20notation-%22Supersharp%22%20EDOs"&gt;Ups and Downs Notation -&amp;quot;Supersharp&amp;quot; EDOs&lt;/a&gt; and &lt;a class="wiki_link" href="http://xenharmonic.wikispaces.com/Ups%20and%20Downs%20Notation#Natural%20Generators"&gt;Ups and Downs Notation - Natural Generators&lt;/a&gt;.&lt;br /&gt;
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc2"&gt;&lt;a name="Basic Properties-Useful Moment-of-Symmetry Scales"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;&lt;span style="font-size: 1.3em;"&gt;Useful Moment-of-Symmetry Scales&lt;/span&gt;&lt;/h2&gt;