2L 3s: 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:keenanpepper|keenanpepper]] and made on <tt>2011-11-~~25 20~~:32:02 UTC</tt>.

: This revision was by author [[User:keenanpepper|keenanpepper]] and made on <tt>2011-11-28 18:33:24 UTC</tt>.

: The original revision id was <tt>~~279101170~~</tt>.

: The original revision id was <tt>279892900</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.

Line 9:

The [[meantone]] pentatonic scale, in which the generator approximates 4/3 but other intervals in the scale approximate 6/5 and 5/4, has by far the lowest harmonic entropy of all 5-note MOS scales, which explains the worldwide popularity of these scales and their very long history of use. It is also strictly [[Rothenberg propriety|proper]].

||||||||||||~ Generator ||~ Cents ||~ Scale steps ||~ Comments ||

||||||||||||~ Generator ||~ Cents ||~ s ||~ L-s ||~ |L-2s| ||~ Scale steps ||~ Comments ||

|| 2\5 || || || || || || 480 || 1 1 1 1 1 ||= ||

|| 2\5 || || || || || || 480 || 240 || 0 || 240 || 1 1 1 1 1 ||= ||

|| || || || || || 11\27 || 488.89 || 6 5 5 6 5 ||= Slendro (insofar as it resembles a MOS)

|| || || || || || 11\27 || 488.89 || || || || 6 5 5 6 5 ||= Slendro (insofar as it resembles a MOS)

would be in this region ||

|| || || || || 9\22 || || 490.91 || 5 4 4 5 4 || ||

|| || || || || 9\22 || || 490.91 || 218.18 || 54.55 || 163.64 || 5 4 4 5 4 || ||

|| || || || || || 16\39 || 492.31 || 9 7 7 9 7 ||= No-5's superpyth/dominant is around here ||

|| || || || || || 16\39 || 492.31 || || || || 9 7 7 9 7 ||= No-5's superpyth/dominant is around here ||

|| || || || 7\17 || || || 494.12 || 4 3 3 4 3 || ||

|| || || || 7\17 || || || 494.12 || 211.76 || 70.59 || 141.18 || 4 3 3 4 3 || ||

|| || || || || 12\29 || || 496.55 || 7 5 5 7 5 || ||

|| || || || || 12\29 || || 496.55 || || || || 7 5 5 7 5 || ||

|| || || || || || 17\41 || 497.56 || 10 7 7 10 7 ||= Pythagorean pentatonic is around here ||

|| || || || || || 17\41 || 497.56 || || || || 10 7 7 10 7 ||= Pythagorean pentatonic is around here ||

|| || || 5\12 || || || || 500 || 3 2 2 3 2 ||= Familiar 12-equal pentatonic

|| || || 5\12 || || || || 500 || 200 || 100 || 100 || 3 2 2 3 2 ||= Familiar 12-equal pentatonic

(also optimum rank range: L/s=3/2) ||

|| || || || || 13\31 || || 503.23 || 8 5 5 8 5 ||= Optimal meantone pentatonic

|| || || || || 13\31 || || 503.23 || 193.55 || 116.13 || 77.42 || 8 5 5 8 5 ||= Optimal meantone pentatonic

is around here ||

|| || || || || || || 1200/(4-phi) || phi 1 1 phi 1 ||= Golden meantone ||

|| || || || || || || 1200/(4-phi) || 192.43 || 118.93 || 73.50 || phi 1 1 phi 1 ||= Golden meantone ||

|| || || || || || 21\50 || 504 || 13 8 8 13 8 ||= ||

|| || || || || || 21\50 || 504 || 192 || 120 || 72 || 13 8 8 13 8 ||= ||

|| || || || 8\19 || || || 505.26 || 5 3 3 5 3 || ||

|| || || || 8\19 || || || 505.26 || 189.47 || 126.32 || 63.16 || 5 3 3 5 3 || ||

|| || 3\7 || || || || || 514.29 || 2 1 1 2 1 ||= (Boundary of propriety: smaller

|| || 3\7 || || || || || 514.29 || 171.43 || 171.43 || 0 || 2 1 1 2 1 ||= (Boundary of propriety: smaller

generators than this are strictly proper) ||

|| || || || 7\16 || || || 525 || 5 2 2 5 2 ||= 5-note subset of pelog (insofar as it

|| || || || 7\16 || || || 525 || 150 || 225 || 75 || 5 2 2 5 2 ||= 5-note subset of pelog (insofar as it

resembles a MOS) would be in this region ||

|| || || 4\9 || || || || 533.33 || 3 1 1 3 1 || ||

|| || || 4\9 || || || || 533.33 || 133.33 || 266.67 || 133.33 || 3 1 1 3 1 || ||

|| || || || 5\11 || || || 545.45 || 4 1 1 4 1 || ||

|| || || || 5\11 || || || 545.45 || 109.09 || 327.27 || 218.18 || 4 1 1 4 1 || ||

|| 1\2 || || || || || || 600 || 1 0 0 1 0 || ||

|| 1\2 || || || || || || 600 || 0 || 600 || 600 || 1 0 0 1 0 || ||

From a [[3-limit]] perspective, just make a chain of four 4/3's and octave-reduce, and you end up with pentatonic.

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</th>

<th>Cents

</th>

<th>s

</th>

<th>L-s

</th>

<th>|L-2s|

</th>

<th>Scale steps

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</td>

<td>480

</td>

<td>240

</td>

<td>0

</td>

<td>240

</td>

<td>1 1 1 1 1

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</td>

<td>488.89

</td>

<td>

</td>

<td>

</td>

<td>

</td>

<td>6 5 5 6 5

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</td>

<td>490.91

</td>

<td>218.18

</td>

<td>54.55

</td>

<td>163.64

</td>

<td>5 4 4 5 4

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</td>

<td>492.31

</td>

<td>

</td>

<td>

</td>

<td>

</td>

<td>9 7 7 9 7

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</td>

<td>494.12

</td>

<td>211.76

</td>

<td>70.59

</td>

<td>141.18

</td>

<td>4 3 3 4 3

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</td>

<td>496.55

</td>

<td>

</td>

<td>

</td>

<td>

</td>

<td>7 5 5 7 5

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</td>

<td>497.56

</td>

<td>

</td>

<td>

</td>

<td>

</td>

<td>10 7 7 10 7

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</td>

<td>500

</td>

<td>200

</td>

<td>100

</td>

<td>100

</td>

<td>3 2 2 3 2

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</td>

<td>503.23

</td>

<td>193.55

</td>

<td>116.13

</td>

<td>77.42

</td>

<td>8 5 5 8 5

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</td>

<td>1200/(4-phi)

</td>

<td>192.43

</td>

<td>118.93

</td>

<td>73.50

</td>

<td>phi 1 1 phi 1

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</td>

<td>504

</td>

<td>192

</td>

<td>120

</td>

<td>72

</td>

<td>13 8 8 13 8

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</td>

<td>505.26

</td>

<td>189.47

</td>

<td>126.32

</td>

<td>63.16

</td>

<td>5 3 3 5 3

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</td>

<td>514.29

</td>

<td>171.43

</td>

<td>171.43

</td>

<td>0

</td>

<td>2 1 1 2 1

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</td>

<td>525

</td>

<td>150

</td>

<td>225

</td>

<td>75

</td>

<td>5 2 2 5 2

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</td>

<td>533.33

</td>

<td>133.33

</td>

<td>266.67

</td>

<td>133.33

</td>

<td>3 1 1 3 1

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</td>

<td>545.45

</td>

<td>109.09

</td>

<td>327.27

</td>

<td>218.18

</td>

<td>4 1 1 4 1

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</td>

<td>

</td>

<td>600

</td>

<td>0

</td>

<td>600

</td>

<td>600

@@ 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:keenanpepper|keenanpepper]] and made on <tt>2011-11-25 20:32:02 UTC</tt>.<br>
+: This revision was by author [[User:keenanpepper|keenanpepper]] and made on <tt>2011-11-28 18:33:24 UTC</tt>.<br>
-: The original revision id was <tt>279101170</tt>.<br>
+: The original revision id was <tt>279892900</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 9: / Line 9: @@
 The [[meantone]] pentatonic scale, in which the generator approximates 4/3 but other intervals in the scale approximate 6/5 and 5/4, has by far the lowest harmonic entropy of all 5-note MOS scales, which explains the worldwide popularity of these scales and their very long history of use. It is also strictly [[Rothenberg propriety|proper]].
-||||||||||||~ Generator ||~ Cents ||~ Scale steps ||~ Comments ||
+||||||||||||~ Generator ||~ Cents ||~ s ||~ L-s ||~ |L-2s| ||~ Scale steps ||~ Comments ||
-|| 2\5 ||   ||   ||   ||   ||   || 480 || 1 1 1 1 1 ||=   ||
+|| 2\5 ||   ||   ||   ||   ||   || 480 || 240 || 0 || 240 || 1 1 1 1 1 ||=   ||
-||   ||   ||   ||   ||   || 11\27 || 488.89 || 6 5 5 6 5 ||= Slendro (insofar as it resembles a MOS)
+||   ||   ||   ||   ||   || 11\27 || 488.89 ||   ||   ||   || 6 5 5 6 5 ||= Slendro (insofar as it resembles a MOS)
 would be in this region ||
-||   ||   ||   ||   || 9\22 ||   || 490.91 || 5 4 4 5 4 ||   ||
+||   ||   ||   ||   || 9\22 ||   || 490.91 || 218.18 || 54.55 || 163.64 || 5 4 4 5 4 ||   ||
-||   ||   ||   ||   ||   || 16\39 || 492.31 || 9 7 7 9 7 ||= No-5's superpyth/dominant is around here ||
+||   ||   ||   ||   ||   || 16\39 || 492.31 ||   ||   ||   || 9 7 7 9 7 ||= No-5's superpyth/dominant is around here ||
-||   ||   ||   || 7\17 ||   ||   || 494.12 || 4 3 3 4 3 ||   ||
+||   ||   ||   || 7\17 ||   ||   || 494.12 || 211.76 || 70.59 || 141.18 || 4 3 3 4 3 ||   ||
-||   ||   ||   ||   || 12\29 ||   || 496.55 || 7 5 5 7 5 ||   ||
+||   ||   ||   ||   || 12\29 ||   || 496.55 ||   ||   ||   || 7 5 5 7 5 ||   ||
-||   ||   ||   ||   ||   || 17\41 || 497.56 || 10 7 7 10 7 ||= Pythagorean pentatonic is around here ||
+||   ||   ||   ||   ||   || 17\41 || 497.56 ||   ||   ||   || 10 7 7 10 7 ||= Pythagorean pentatonic is around here ||
-||   ||   || 5\12 ||   ||   ||   || 500 || 3 2 2 3 2 ||= Familiar 12-equal pentatonic
+||   ||   || 5\12 ||   ||   ||   || 500 || 200 || 100 || 100 || 3 2 2 3 2 ||= Familiar 12-equal pentatonic
 (also optimum rank range: L/s=3/2) ||
-||   ||   ||   ||   || 13\31 ||   || 503.23 || 8 5 5 8 5 ||= Optimal meantone pentatonic
+||   ||   ||   ||   || 13\31 ||   || 503.23 || 193.55 || 116.13 || 77.42 || 8 5 5 8 5 ||= Optimal meantone pentatonic
 is around here ||
-||   ||   ||   ||   ||   ||   || 1200/(4-phi) || phi 1 1 phi 1 ||= Golden meantone ||
+||   ||   ||   ||   ||   ||   || 1200/(4-phi) || 192.43 || 118.93 || 73.50 || phi 1 1 phi 1 ||= Golden meantone ||
-||   ||   ||   ||   ||   || 21\50 || 504 || 13 8 8 13 8 ||=   ||
+||   ||   ||   ||   ||   || 21\50 || 504 || 192 || 120 || 72 || 13 8 8 13 8 ||=   ||
-||   ||   ||   || 8\19 ||   ||   || 505.26 || 5 3 3 5 3 ||   ||
+||   ||   ||   || 8\19 ||   ||   || 505.26 || 189.47 || 126.32 || 63.16 || 5 3 3 5 3 ||   ||
-||   || 3\7 ||   ||   ||   ||   || 514.29 || 2 1 1 2 1 ||= (Boundary of propriety: smaller
+||   || 3\7 ||   ||   ||   ||   || 514.29 || 171.43 || 171.43 || 0 || 2 1 1 2 1 ||= (Boundary of propriety: smaller
 generators than this are strictly proper) ||
-||   ||   ||   || 7\16 ||   ||   || 525 || 5 2 2 5 2 ||= 5-note subset of pelog (insofar as it
+||   ||   ||   || 7\16 ||   ||   || 525 || 150 || 225 || 75 || 5 2 2 5 2 ||= 5-note subset of pelog (insofar as it
 resembles a MOS) would be in this region ||
-||   ||   || 4\9 ||   ||   ||   || 533.33 || 3 1 1 3 1 ||   ||
+||   ||   || 4\9 ||   ||   ||   || 533.33 || 133.33 || 266.67 || 133.33 || 3 1 1 3 1 ||   ||
-||   ||   ||   || 5\11 ||   ||   || 545.45 || 4 1 1 4 1 ||   ||
+||   ||   ||   || 5\11 ||   ||   || 545.45 || 109.09 || 327.27 || 218.18 || 4 1 1 4 1 ||   ||
-|| 1\2 ||   ||   ||   ||   ||   || 600 || 1 0 0 1 0 ||   ||
+|| 1\2 ||   ||   ||   ||   ||   || 600 || 0 || 600 || 600 || 1 0 0 1 0 ||   ||
 From a [[3-limit]] perspective, just make a chain of four 4/3's and octave-reduce, and you end up with pentatonic.
@@ Line 49: / Line 49: @@
 &lt;/th&gt;
          &lt;th&gt;Cents&lt;br /&gt;
+&lt;/th&gt;
+        &lt;th&gt;s&lt;br /&gt;
+&lt;/th&gt;
+        &lt;th&gt;L-s&lt;br /&gt;
+&lt;/th&gt;
+        &lt;th&gt;|L-2s|&lt;br /&gt;
 &lt;/th&gt;
          &lt;th&gt;Scale steps&lt;br /&gt;
@@ Line 69: / Line 75: @@
 &lt;/td&gt;
          &lt;td&gt;480&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;240&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;0&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;240&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;1 1 1 1 1&lt;br /&gt;
@@ Line 89: / Line 101: @@
 &lt;/td&gt;
          &lt;td&gt;488.89&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;6 5 5 6 5&lt;br /&gt;
@@ Line 110: / Line 128: @@
 &lt;/td&gt;
          &lt;td&gt;490.91&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;218.18&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;54.55&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;163.64&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;5 4 4 5 4&lt;br /&gt;
@@ Line 130: / Line 154: @@
 &lt;/td&gt;
          &lt;td&gt;492.31&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;9 7 7 9 7&lt;br /&gt;
@@ Line 150: / Line 180: @@
 &lt;/td&gt;
          &lt;td&gt;494.12&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;211.76&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;70.59&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;141.18&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;4 3 3 4 3&lt;br /&gt;
@@ Line 170: / Line 206: @@
 &lt;/td&gt;
          &lt;td&gt;496.55&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;7 5 5 7 5&lt;br /&gt;
@@ Line 190: / Line 232: @@
 &lt;/td&gt;
          &lt;td&gt;497.56&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;10 7 7 10 7&lt;br /&gt;
@@ Line 210: / Line 258: @@
 &lt;/td&gt;
          &lt;td&gt;500&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;200&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;100&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;100&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;3 2 2 3 2&lt;br /&gt;
@@ Line 231: / Line 285: @@
 &lt;/td&gt;
          &lt;td&gt;503.23&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;193.55&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;116.13&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;77.42&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;8 5 5 8 5&lt;br /&gt;
@@ Line 252: / Line 312: @@
 &lt;/td&gt;
          &lt;td&gt;1200/(4-phi)&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;192.43&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;118.93&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;73.50&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;phi 1 1 phi 1&lt;br /&gt;
@@ Line 272: / Line 338: @@
 &lt;/td&gt;
          &lt;td&gt;504&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;192&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;120&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;72&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;13 8 8 13 8&lt;br /&gt;
@@ Line 292: / Line 364: @@
 &lt;/td&gt;
          &lt;td&gt;505.26&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;189.47&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;126.32&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;63.16&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;5 3 3 5 3&lt;br /&gt;
@@ Line 312: / Line 390: @@
 &lt;/td&gt;
          &lt;td&gt;514.29&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;171.43&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;171.43&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;0&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;2 1 1 2 1&lt;br /&gt;
@@ Line 333: / Line 417: @@
 &lt;/td&gt;
          &lt;td&gt;525&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;150&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;225&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;75&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;5 2 2 5 2&lt;br /&gt;
@@ Line 354: / Line 444: @@
 &lt;/td&gt;
          &lt;td&gt;533.33&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;133.33&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;266.67&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;133.33&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;3 1 1 3 1&lt;br /&gt;
@@ Line 374: / Line 470: @@
 &lt;/td&gt;
          &lt;td&gt;545.45&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;109.09&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;327.27&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;218.18&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;4 1 1 4 1&lt;br /&gt;
@@ Line 392: / Line 494: @@
 &lt;/td&gt;
          &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;600&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;0&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;600&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;600&lt;br /&gt;