2L 3s: Difference between revisions

Line 1:

<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:~~JosephRuhf~~|~~JosephRuhf~~]] and made on <tt>2016-11-~~06 17~~:01:46 UTC</tt>.

: This revision was by author [[User:xenwolf|xenwolf]] and made on <tt>2016-11-07 03:11:27 UTC</tt>.

: The original revision id was <tt>~~598619904~~</tt>.

: The original revision id was <tt>598650408</tt>.

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

: The revision comment was: <tt>Reverted to Nov 6, 2016 9:01 pm: The problem is that you hide information by changing from "Scale steps" to "Trichord": you don't see the trichord distance any more. So better add another column for the Trichord view.</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 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 ||~ s ||~ L-s ||~ |L-2s| ||~ ~~Trichord~~ ||~ Comments ||

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

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

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

|| || || || || || 11\27 || 488.89 || 222.22 || 44.44 || 177.78 || 6 5 ||= Slendro ~~pentatonic~~ (insofar as it resembles a MOS) ||

|| || || || || || 11\27 || 488.89 || 222.22 || 44.44 || 177.78 || 6 5 5 6 5 ||= Slendro (insofar as it resembles a MOS)

|| || || || || 9\22 || || 490.91 || 218.18 || 54.545 || 163.64 || 5 4 ||= ||

would be in this region ||

|| || || || || || 16\39 || 492.31 || 215.38 || 61.54 || 153.85 || 9 7 ||= No-5's superpyth/dominant is around here ||

|| || || || || 9\22 || || 490.91 || 218.18 || 54.545 || 163.64 || 5 4 4 5 4 ||= ||

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

|| || || || || || 16\39 || 492.31 || 215.38 || 61.54 || 153.85 || 9 7 7 9 7 ||= No-5's superpyth/dominant is around here ||

|| || || || || || 19\46 || 495.65 || 208.7 || 78.26 || 130.435 || 11 8 || ||

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

|| || || || || 12\29 || || 496.55 || 206.9 || 82.76 || 124.14 || 7 5 ||= ||

|| || || || || || 19\46 || 495.65 || 208.7 || 78.26 || 130.435 || 11 8 8 11 8 || ||

|| || || || || || 17\41 || 497.56 || 204.88 || 87.8 || 117.07 || 10 7 ||= Pythagorean pentatonic is around here ||

|| || || || || 12\29 || || 496.55 || 206.9 || 82.76 || 124.14 || 7 5 5 7 5 ||= ||

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

|| || || || || || 17\41 || 497.56 || 204.88 || 87.8 || 117.07 || 10 7 7 10 7 ||= Pythagorean pentatonic is around here ||

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

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

|| || || || || || || 502.305 || 195.39 || 111.53 || 83.86 || pi 2 || ||

|| || || || || || || 502.305 || 195.39 || 111.53 || 83.86 || pi 2 pi 2 2 || ||

|| || || || || || 18\43 || 502.33 || 195.35 || 111.63 || 83.72 || 11 7 || ||

|| || || || || || 18\43 || 502.33 || 195.35 || 111.63 || 83.72 || 11 7 7 11 7 || ||

|| || || || || 13\31 || || 503.23 || 193.55 || 116.13 || 77.42 || 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) || 192.43 || 118.93 || 73.50 || phi 1 ||= Golden meantone ||

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

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

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

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

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

|| || || || || || 19\45 || 506.67 || 186.67 || 133.33 || 53.33 || 12 7 || ||

|| || || || || || 19\45 || 506.67 || 186.67 || 133.33 || 53.33 || 12 7 7 12 7 || ||

|| || || || || || || 507.18 || 185.64 || 135.9 || 49.74 || √3 1 || ||

|| || || || || || || 507.18 || 185.64 || 135.9 || 49.74 || √3 1 √3 1 1 || ||

|| || || || || 11\26 || || 507.69 || 184.615 || 138.46 || 46.15 || 7 4 || ||

|| || || || || 11\26 || || 507.69 || 184.615 || 138.46 || 46.15 || 7 4 4 7 4 || ||

|| || || || || || 14\33 || 509.09 || 181.82 || 145.455 || 36.36 || 9 5 || ||

|| || || || || || 14\33 || 509.09 || 181.82 || 145.455 || 36.36 || 9 5 5 9 5 || ||

|| || 3\7 || || || || || 514.29 || 171.43 || 171.43 || 0 || 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) ||

|| || || || || || 13\30 || 520 || 160 || 200 || 40 || 9 4 || ||

|| || || || || || 13\30 || 520 || 160 || 200 || 40 || 9 4 4 9 4 || ||

||< ||< ||< ||< ||< 10\23 ||< ||< 521.74 ||< 156.52 ||< 208.7 ||< 52.17 ||< 7 3 ||< ||

||< ||< ||< ||< ||< 10\23 ||< ||< 521.74 ||< 156.52 ||< 208.7 ||< 52.17 ||< 7 3 3 7 3 ||< ||

|| || || || || || 17\39 || 523.08 || 153.84 || 215.385 || 61.54 || 12 5 || ||

|| || || || || || 17\39 || 523.08 || 153.84 || 215.385 || 61.54 || 12 5 5 12 5 || ||

|| || || || 7\16 || || || 525 || 150 || 225 || 75 || 5 2 ||= ~~Pelog pentatonic~~ (insofar as it resembles a MOS)~~~~ ||

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

|| || || || || || 18\41 || 526.83 || 146.34 || 234.15 || 87.8 || 13 5 || ||

resembles a MOS) would be in this region ||

|| || || || || || || 600(25+√5)/31 || 145.7 || 235.75 || 90.05 || phi+1 1 || ||

|| || || || || || 18\41 || 526.83 || 146.34 || 234.15 || 87.8 || 13 5 5 13 5 || ||

|| || || || || 11\25 || || 528 || 144 || 240 || 96 || 8 3 || ||

|| || || || || || || 600(25+√5)/31 || 145.7 || 235.75 || 90.05 || phi+1 1 1 phi+1 1 || ||

|| || || || || || || 528.88 || 142.24 || 244.405 || 102.17 || e 1 ||= L/s = e ||

|| || || || || 11\25 || || 528 || 144 || 240 || 96 || 8 3 3 8 3 || ||

|| || || || || || 15\34 || 529.41 || 141.18 || 247.06 || 105.88 || 11 4 || ||

|| || || || || || || 528.88 || 142.24 || 244.405 || 102.17 || e 1 e 1 1 ||= L/s = e ||

|| || || 4\9 || || || || 533.33 || 133.33 || 266.67 || 133.33 || 3 1 ||= L/s = 3 ||

|| || || || || || 15\34 || 529.41 || 141.18 || 247.06 || 105.88 || 11 4 4 11 4 || ||

|| || || || || || || 535.36 || 129.26 || 276.835 || 147.57 || pi 1 ||= L/s = pi ||

|| || || 4\9 || || || || 533.33 || 133.33 || 266.67 || 133.33 || 3 1 1 3 1 ||= L/s = 3 ||

|| || || || || || 13\29 || 537.93 || 124.14 || 289.655 || 165.52 || 10 3 || ||

|| || || || || || || 535.36 || 129.26 || 276.835 || 147.57 || pi 1 pi 1 1 ||= L/s = pi ||

|| || || || || 9\20 || || 540 || 120 || 240 || 180 || 7 2 || ||

|| || || || || || 13\29 || 537.93 || 124.14 || 289.655 || 165.52 || 10 3 3 10 3 || ||

|| || || || || || 14\31 || 541.935 || 116.13 || 309.68 || 193.55 || 11 3 || ||

|| || || || || 9\20 || || 540 || 120 || 240 || 180 || 7 2 2 7 2 || ||

|| || || || 5\11 || || || 545.45 || 109.09 || 327.27 || 218.18 || 4 1 ||= L/s = 4 ||

|| || || || || || 14\31 || 541.935 || 116.13 || 309.68 || 193.55 || 11 3 3 11 3 || ||

|| || || || || || 11\24 || 550 || 100 || 350 || 250 || 9 2 || ||

|| || || || 5\11 || || || 545.45 || 109.09 || 327.27 || 218.18 || 4 1 1 4 1 ||= L/s = 4 ||

|| || || || || 6\13 || || 553.85 || 92.31 || 369.23 || 276.92 || 5 1 || ||

|| || || || || || 11\24 || 550 || 100 || 350 || 250 || 9 2 2 9 2 || ||

|| || || || || || 7\15 || 560 || 80 || 480 || 400 || 6 1 || ||

|| || || || || 6\13 || || 553.85 || 92.31 || 369.23 || 276.92 || 5 1 1 5 1 || ||

|| || || || || || 7\15 || 560 || 80 || 480 || 400 || 6 1 1 6 1 || ||

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

Line 76:

Line 78:

<th>|L-2s|

</th>

<th>~~Trichord~~

<th>Scale steps

</th>

<th>Comments

Line 102:

Line 104:

<td>240

</td>

<td>1 1

<td>1 1 1 1 1

</td>

Line 128:

Line 130:

<td>177.78

</td>

<td>6 5

<td>6 5 5 6 5

</td>

<td style="text-align: center;">Slendro ~~pentatonic~~ (insofar as it resembles a MOS)

<td style="text-align: center;">Slendro (insofar as it resembles a MOS)

would be in this region

</td>

</tr>

Line 154:

Line 157:

<td>163.64

</td>

<td>5 4

<td>5 4 4 5 4

</td>

Line 180:

Line 183:

<td>153.85

</td>

<td>9 7

<td>9 7 7 9 7

</td>

<td style="text-align: center;">No-5's superpyth/dominant is around here

Line 206:

Line 209:

<td>141.18

</td>

<td>4 3

<td>4 3 3 4 3

</td>

Line 232:

Line 235:

<td>130.435

</td>

<td>11 8

<td>11 8 8 11 8

</td>

<td>

Line 258:

Line 261:

<td>124.14

</td>

<td>7 5

<td>7 5 5 7 5

</td>

Line 284:

Line 287:

<td>117.07

</td>

<td>10 7

<td>10 7 7 10 7

</td>

<td style="text-align: center;">Pythagorean pentatonic is around here

Line 310:

Line 313:

<td>100

</td>

<td>3 2

<td>3 2 2 3 2

</td>

<td style="text-align: center;">Familiar 12-equal pentatonic

Line 337:

Line 340:

<td>83.86

</td>

<td>pi 2

<td>pi 2 pi 2 2

</td>

<td>

Line 363:

Line 366:

<td>83.72

</td>

<td>11 7

<td>11 7 7 11 7

</td>

<td>

Line 389:

Line 392:

<td>77.42

</td>

<td>8 5

<td>8 5 5 8 5

</td>

<td style="text-align: center;">Optimal meantone pentatonic

Line 416:

Line 419:

<td>73.50

</td>

<td>phi 1

<td>phi 1 1 phi 1

</td>

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

Line 442:

Line 445:

<td>72

</td>

<td>13 8

<td>13 8 8 13 8

</td>

Line 468:

Line 471:

<td>63.16

</td>

<td>5 3

<td>5 3 3 5 3

</td>

Line 494:

Line 497:

<td>53.33

</td>

<td>12 7

<td>12 7 7 12 7

</td>

<td>

Line 520:

Line 523:

<td>49.74

</td>

<td>√3 1

<td>√3 1 √3 1 1

</td>

<td>

Line 546:

Line 549:

<td>46.15

</td>

<td>7 4

<td>7 4 4 7 4

</td>

<td>

Line 572:

Line 575:

<td>36.36

</td>

<td>9 5

<td>9 5 5 9 5

</td>

<td>

Line 598:

Line 601:

<td>0

</td>

<td>2 1

<td>2 1 1 2 1

</td>

<td style="text-align: center;">(Boundary of propriety: smaller

Line 625:

Line 628:

<td>40

</td>

<td>9 4

<td>9 4 4 9 4

</td>

<td>

Line 651:

Line 654:

<td style="text-align: left;">52.17

</td>

<td style="text-align: left;">7 3

<td style="text-align: left;">7 3 3 7 3

</td>

Line 677:

Line 680:

<td>61.54

</td>

<td>12 5

<td>12 5 5 12 5

</td>

<td>

Line 703:

Line 706:

<td>75

</td>

<td>5 2

<td>5 2 2 5 2

</td>

<td style="text-align: center;">~~Pelog pentatonic~~ (insofar as it ~~resembles a MOS)~~</~~span~~>

<td style="text-align: center;">5-note subset of pelog (insofar as it

resembles a MOS) would be in this region

</td>

</tr>

Line 729:

Line 733:

<td>87.8

</td>

<td>13 5

<td>13 5 5 13 5

</td>

<td>

Line 755:

Line 759:

<td>90.05

</td>

<td>phi+1 1

<td>phi+1 1 1 phi+1 1

</td>

<td>

Line 781:

Line 785:

<td>96

</td>

<td>8 3

<td>8 3 3 8 3

</td>

<td>

Line 807:

Line 811:

<td>102.17

</td>

<td>e 1

<td>e 1 e 1 1

</td>

<td style="text-align: center;">L/s = e

Line 833:

Line 837:

<td>105.88

</td>

<td>11 4

<td>11 4 4 11 4

</td>

<td>

Line 859:

Line 863:

<td>133.33

</td>

<td>3 1

<td>3 1 1 3 1

</td>

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

Line 885:

Line 889:

<td>147.57

</td>

<td>pi 1

<td>pi 1 pi 1 1

</td>

<td style="text-align: center;">L/s = pi

Line 911:

Line 915:

<td>165.52

</td>

<td>10 3

<td>10 3 3 10 3

</td>

<td>

Line 937:

Line 941:

<td>180

</td>

<td>7 2

<td>7 2 2 7 2

</td>

<td>

Line 963:

Line 967:

<td>193.55

</td>

<td>11 3

<td>11 3 3 11 3

</td>

<td>

Line 989:

Line 993:

<td>218.18

</td>

<td>4 1

<td>4 1 1 4 1

</td>

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

Line 1,015:

Line 1,019:

<td>250

</td>

<td>9 2

<td>9 2 2 9 2

</td>

<td>

Line 1,041:

Line 1,045:

<td>276.92

</td>

<td>5 1

<td>5 1 1 5 1

</td>

<td>

Line 1,067:

Line 1,071:

<td>400

</td>

<td>6 1

<td>6 1 1 6 1

</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:JosephRuhf|JosephRuhf]] and made on <tt>2016-11-06 17:01:46 UTC</tt>.<br>
+: This revision was by author [[User:xenwolf|xenwolf]] and made on <tt>2016-11-07 03:11:27 UTC</tt>.<br>
-: The original revision id was <tt>598619904</tt>.<br>
+: The original revision id was <tt>598650408</tt>.<br>
-: The revision comment was: <tt></tt><br>
+: The revision comment was: <tt>Reverted to Nov 6, 2016 9:01 pm: The problem is that you hide information by changing from "Scale steps" to "Trichord": you don't see the trichord distance any more. So better add another column for the Trichord view.</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 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 ||~ s ||~ L-s ||~ |L-2s| ||~ Trichord ||~ Comments ||
+||||||||||||~ Generator ||~ Cents ||~ s ||~ L-s ||~ |L-2s| ||~ Scale steps ||~ Comments ||
-|| 2\5 ||   ||   ||   ||   ||   || 480 || 240 || 0 || 240 || 1 1 ||=   ||
+|| 2\5 ||   ||   ||   ||   ||   || 480 || 240 || 0 || 240 || 1 1 1 1 1 ||=   ||
-||   ||   ||   ||   ||   || 11\27 || 488.89 || 222.22 || 44.44 || 177.78 || 6 5 ||= Slendro pentatonic (insofar as it resembles a MOS) ||
+||   ||   ||   ||   ||   || 11\27 || 488.89 || 222.22 || 44.44 || 177.78 || 6 5 5 6 5 ||= Slendro (insofar as it resembles a MOS)
-||   ||   ||   ||   || 9\22 ||   || 490.91 || 218.18 || 54.545 || 163.64 || 5 4 ||=   ||
+would be in this region ||
-||   ||   ||   ||   ||   || 16\39 || 492.31 || 215.38 || 61.54 || 153.85 || 9 7 ||= No-5's superpyth/dominant is around here ||
+||   ||   ||   ||   || 9\22 ||   || 490.91 || 218.18 || 54.545 || 163.64 || 5 4 4 5 4 ||=   ||
-||   ||   ||   || 7\17 ||   ||   || 494.12 || 211.76 || 70.59 || 141.18 || 4 3 ||=   ||
+||   ||   ||   ||   ||   || 16\39 || 492.31 || 215.38 || 61.54 || 153.85 || 9 7 7 9 7 ||= No-5's superpyth/dominant is around here ||
-||   ||   ||   ||   ||   || 19\46 || 495.65 || 208.7 || 78.26 || 130.435 || 11 8 ||   ||
+||   ||   ||   || 7\17 ||   ||   || 494.12 || 211.76 || 70.59 || 141.18 || 4 3 3 4 3 ||=   ||
-||   ||   ||   ||   || 12\29 ||   || 496.55 || 206.9 || 82.76 || 124.14 || 7 5 ||=   ||
+||   ||   ||   ||   ||   || 19\46 || 495.65 || 208.7 || 78.26 || 130.435 || 11 8 8 11 8 ||   ||
-||   ||   ||   ||   ||   || 17\41 || 497.56 || 204.88 || 87.8 || 117.07 || 10 7 ||= Pythagorean pentatonic is around here ||
+||   ||   ||   ||   || 12\29 ||   || 496.55 || 206.9 || 82.76 || 124.14 || 7 5 5 7 5 ||=   ||
-||   ||   || 5\12 ||   ||   ||   || 500 || 200 || 100 || 100 || 3 2 ||= Familiar 12-equal pentatonic
+||   ||   ||   ||   ||   || 17\41 || 497.56 || 204.88 || 87.8 || 117.07 || 10 7 7 10 7 ||= Pythagorean pentatonic is around here ||
+||   ||   || 5\12 ||   ||   ||   || 500 || 200 || 100 || 100 || 3 2 2 3 2 ||= Familiar 12-equal pentatonic
 (also optimum rank range: L/s=3/2) ||
-||   ||   ||   ||   ||   ||   || 502.305 || 195.39 || 111.53 || 83.86 || pi 2 ||   ||
+||   ||   ||   ||   ||   ||   || 502.305 || 195.39 || 111.53 || 83.86 || pi 2 pi 2 2 ||   ||
-||   ||   ||   ||   ||   || 18\43 || 502.33 || 195.35 || 111.63 || 83.72 || 11 7 ||   ||
+||   ||   ||   ||   ||   || 18\43 || 502.33 || 195.35 || 111.63 || 83.72 || 11 7 7 11 7 ||   ||
-||   ||   ||   ||   || 13\31 ||   || 503.23 || 193.55 || 116.13 || 77.42 || 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) || 192.43 || 118.93 || 73.50 || phi 1 ||= Golden meantone ||
+||   ||   ||   ||   ||   ||   || 1200/(4-phi) || 192.43 || 118.93 || 73.50 || phi 1 1 phi 1 ||= Golden meantone ||
-||   ||   ||   ||   ||   || 21\50 || 504 || 192 || 120 || 72 || 13 8 ||=   ||
+||   ||   ||   ||   ||   || 21\50 || 504 || 192 || 120 || 72 || 13 8 8 13 8 ||=   ||
-||   ||   ||   || 8\19 ||   ||   || 505.26 || 189.47 || 126.32 || 63.16 || 5 3 ||=   ||
+||   ||   ||   || 8\19 ||   ||   || 505.26 || 189.47 || 126.32 || 63.16 || 5 3 3 5 3 ||=   ||
-||   ||   ||   ||   ||   || 19\45 || 506.67 || 186.67 || 133.33 || 53.33 || 12 7 ||   ||
+||   ||   ||   ||   ||   || 19\45 || 506.67 || 186.67 || 133.33 || 53.33 || 12 7 7 12 7 ||   ||
-||   ||   ||   ||   ||   ||   || 507.18 || 185.64 || 135.9 || 49.74 || √3 1 ||   ||
+||   ||   ||   ||   ||   ||   || 507.18 || 185.64 || 135.9 || 49.74 || √3 1 √3 1 1 ||   ||
-||   ||   ||   ||   || 11\26 ||   || 507.69 || 184.615 || 138.46 || 46.15 || 7 4 ||   ||
+||   ||   ||   ||   || 11\26 ||   || 507.69 || 184.615 || 138.46 || 46.15 || 7 4 4 7 4 ||   ||
-||   ||   ||   ||   ||   || 14\33 || 509.09 || 181.82 || 145.455 || 36.36 || 9 5 ||   ||
+||   ||   ||   ||   ||   || 14\33 || 509.09 || 181.82 || 145.455 || 36.36 || 9 5 5 9 5 ||   ||
-||   || 3\7 ||   ||   ||   ||   || 514.29 || 171.43 || 171.43 || 0 || 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) ||
-||   ||   ||   ||   ||   || 13\30 || 520 || 160 || 200 || 40 || 9 4 ||   ||
+||   ||   ||   ||   ||   || 13\30 || 520 || 160 || 200 || 40 || 9 4 4 9 4 ||   ||
-||&lt;   ||&lt;   ||&lt;   ||&lt;   ||&lt; 10\23 ||&lt;   ||&lt; 521.74 ||&lt; 156.52 ||&lt; 208.7 ||&lt; 52.17 ||&lt; 7 3 ||&lt;   ||
+||&lt;   ||&lt;   ||&lt;   ||&lt;   ||&lt; 10\23 ||&lt;   ||&lt; 521.74 ||&lt; 156.52 ||&lt; 208.7 ||&lt; 52.17 ||&lt; 7 3 3 7 3 ||&lt;   ||
-||   ||   ||   ||   ||   || 17\39 || 523.08 || 153.84 || 215.385 || 61.54 || 12 5 ||   ||
+||   ||   ||   ||   ||   || 17\39 || 523.08 || 153.84 || 215.385 || 61.54 || 12 5 5 12 5 ||   ||
-||   ||   ||   || 7\16 ||   ||   || 525 || 150 || 225 || 75 || 5 2 ||= &lt;span style="display: block; text-align: center;"&gt;Pelog pentatonic (insofar as it resembles a MOS)&lt;/span&gt; ||
+||   ||   ||   || 7\16 ||   ||   || 525 || 150 || 225 || 75 || 5 2 2 5 2 ||= 5-note subset of pelog (insofar as it
-||   ||   ||   ||   ||   || 18\41 || 526.83 || 146.34 || 234.15 || 87.8 || 13 5 ||   ||
+resembles a MOS) would be in this region ||
-||   ||   ||   ||   ||   ||   || 600(25+√5)/31 || 145.7 || 235.75 || 90.05 || phi+1 1 ||   ||
+||   ||   ||   ||   ||   || 18\41 || 526.83 || 146.34 || 234.15 || 87.8 || 13 5 5 13 5 ||   ||
-||   ||   ||   ||   || 11\25 ||   || 528 || 144 || 240 || 96 || 8 3 ||   ||
+||   ||   ||   ||   ||   ||   || 600(25+√5)/31 || 145.7 || 235.75 || 90.05 || phi+1 1 1 phi+1 1 ||   ||
-||   ||   ||   ||   ||   ||   || 528.88 || 142.24 || 244.405 || 102.17 || e 1 ||= L/s = e ||
+||   ||   ||   ||   || 11\25 ||   || 528 || 144 || 240 || 96 || 8 3 3 8 3 ||   ||
-||   ||   ||   ||   ||   || 15\34 || 529.41 || 141.18 || 247.06 || 105.88 || 11 4 ||   ||
+||   ||   ||   ||   ||   ||   || 528.88 || 142.24 || 244.405 || 102.17 || e 1 e 1 1 ||= L/s = e ||
-||   ||   || 4\9 ||   ||   ||   || 533.33 || 133.33 || 266.67 || 133.33 || 3 1 ||= L/s = 3 ||
+||   ||   ||   ||   ||   || 15\34 || 529.41 || 141.18 || 247.06 || 105.88 || 11 4 4 11 4 ||   ||
-||   ||   ||   ||   ||   ||   || 535.36 || 129.26 || 276.835 || 147.57 || pi 1 ||= &lt;span style="display: block; text-align: center;"&gt;L/s = pi&lt;/span&gt; ||
+||   ||   || 4\9 ||   ||   ||   || 533.33 || 133.33 || 266.67 || 133.33 || 3 1 1 3 1 ||= L/s = 3 ||
-||   ||   ||   ||   ||   || 13\29 || 537.93 || 124.14 || 289.655 || 165.52 || 10 3 ||   ||
+||   ||   ||   ||   ||   ||   || 535.36 || 129.26 || 276.835 || 147.57 || pi 1 pi 1 1 ||= &lt;span style="display: block; text-align: center;"&gt;L/s = pi&lt;/span&gt; ||
-||   ||   ||   ||   || 9\20 ||   || 540 || 120 || 240 || 180 || 7 2 ||   ||
+||   ||   ||   ||   ||   || 13\29 || 537.93 || 124.14 || 289.655 || 165.52 || 10 3 3 10 3 ||   ||
-||   ||   ||   ||   ||   || 14\31 || 541.935 || 116.13 || 309.68 || 193.55 || 11 3 ||   ||
+||   ||   ||   ||   || 9\20 ||   || 540 || 120 || 240 || 180 || 7 2 2 7 2 ||   ||
-||   ||   ||   || 5\11 ||   ||   || 545.45 || 109.09 || 327.27 || 218.18 || 4 1 ||= L/s = 4 ||
+||   ||   ||   ||   ||   || 14\31 || 541.935 || 116.13 || 309.68 || 193.55 || 11 3 3 11 3 ||   ||
-||   ||   ||   ||   ||   || 11\24 || 550 || 100 || 350 || 250 || 9 2 ||   ||
+||   ||   ||   || 5\11 ||   ||   || 545.45 || 109.09 || 327.27 || 218.18 || 4 1 1 4 1 ||= L/s = 4 ||
-||   ||   ||   ||   || 6\13 ||   || 553.85 || 92.31 || 369.23 || 276.92 || 5 1 ||   ||
+||   ||   ||   ||   ||   || 11\24 || 550 || 100 || 350 || 250 || 9 2 2 9 2 ||   ||
-||   ||   ||   ||   ||   || 7\15 || 560 || 80 || 480 || 400 || 6 1 ||   ||
+||   ||   ||   ||   || 6\13 ||   || 553.85 || 92.31 || 369.23 || 276.92 || 5 1 1 5 1 ||   ||
+||   ||   ||   ||   ||   || 7\15 || 560 || 80 || 480 || 400 || 6 1 1 6 1 ||   ||
 || 1\2 ||   ||   ||   ||   ||   || 600 || 0 || 600 || 600 || 1 0 0 1 0 ||=   ||
@@ Line 76: / Line 78: @@
          &lt;th&gt;|L-2s|&lt;br /&gt;
 &lt;/th&gt;
-         &lt;th&gt;Trichord&lt;br /&gt;
+         &lt;th&gt;Scale steps&lt;br /&gt;
 &lt;/th&gt;
          &lt;th&gt;Comments&lt;br /&gt;
@@ Line 102: / Line 104: @@
          &lt;td&gt;240&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;1 1&lt;br /&gt;
+         &lt;td&gt;1 1 1 1 1&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;&lt;br /&gt;
@@ Line 128: / Line 130: @@
          &lt;td&gt;177.78&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;6 5&lt;br /&gt;
+         &lt;td&gt;6 5 5 6 5&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;Slendro pentatonic (insofar as it resembles a MOS)&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;Slendro (insofar as it resembles a MOS)&lt;br /&gt;
+would be in this region&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 154: / Line 157: @@
          &lt;td&gt;163.64&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;5 4&lt;br /&gt;
+         &lt;td&gt;5 4 4 5 4&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;&lt;br /&gt;
@@ Line 180: / Line 183: @@
          &lt;td&gt;153.85&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;9 7&lt;br /&gt;
+         &lt;td&gt;9 7 7 9 7&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;No-5's superpyth/dominant is around here&lt;br /&gt;
@@ Line 206: / Line 209: @@
          &lt;td&gt;141.18&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;4 3&lt;br /&gt;
+         &lt;td&gt;4 3 3 4 3&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;&lt;br /&gt;
@@ Line 232: / Line 235: @@
          &lt;td&gt;130.435&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;11 8&lt;br /&gt;
+         &lt;td&gt;11 8 8 11 8&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;&lt;br /&gt;
@@ Line 258: / Line 261: @@
          &lt;td&gt;124.14&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;7 5&lt;br /&gt;
+         &lt;td&gt;7 5 5 7 5&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;&lt;br /&gt;
@@ Line 284: / Line 287: @@
          &lt;td&gt;117.07&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;10 7&lt;br /&gt;
+         &lt;td&gt;10 7 7 10 7&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;Pythagorean pentatonic is around here&lt;br /&gt;
@@ Line 310: / Line 313: @@
          &lt;td&gt;100&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;3 2&lt;br /&gt;
+         &lt;td&gt;3 2 2 3 2&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;Familiar 12-equal pentatonic&lt;br /&gt;
@@ Line 337: / Line 340: @@
          &lt;td&gt;83.86&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;pi 2&lt;br /&gt;
+         &lt;td&gt;pi 2 pi 2 2&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;&lt;br /&gt;
@@ Line 363: / Line 366: @@
          &lt;td&gt;83.72&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;11 7&lt;br /&gt;
+         &lt;td&gt;11 7 7 11 7&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;&lt;br /&gt;
@@ Line 389: / Line 392: @@
          &lt;td&gt;77.42&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;8 5&lt;br /&gt;
+         &lt;td&gt;8 5 5 8 5&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;Optimal meantone pentatonic&lt;br /&gt;
@@ Line 416: / Line 419: @@
          &lt;td&gt;73.50&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;phi 1&lt;br /&gt;
+         &lt;td&gt;phi 1 1 phi 1&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;Golden meantone&lt;br /&gt;
@@ Line 442: / Line 445: @@
          &lt;td&gt;72&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;13 8&lt;br /&gt;
+         &lt;td&gt;13 8 8 13 8&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;&lt;br /&gt;
@@ Line 468: / Line 471: @@
          &lt;td&gt;63.16&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;5 3&lt;br /&gt;
+         &lt;td&gt;5 3 3 5 3&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;&lt;br /&gt;
@@ Line 494: / Line 497: @@
          &lt;td&gt;53.33&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;12 7&lt;br /&gt;
+         &lt;td&gt;12 7 7 12 7&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;&lt;br /&gt;
@@ Line 520: / Line 523: @@
          &lt;td&gt;49.74&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;√3 1&lt;br /&gt;
+         &lt;td&gt;√3 1 √3 1 1&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;&lt;br /&gt;
@@ Line 546: / Line 549: @@
          &lt;td&gt;46.15&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;7 4&lt;br /&gt;
+         &lt;td&gt;7 4 4 7 4&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;&lt;br /&gt;
@@ Line 572: / Line 575: @@
          &lt;td&gt;36.36&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;9 5&lt;br /&gt;
+         &lt;td&gt;9 5 5 9 5&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;&lt;br /&gt;
@@ Line 598: / Line 601: @@
          &lt;td&gt;0&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;2 1&lt;br /&gt;
+         &lt;td&gt;2 1 1 2 1&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;(Boundary of propriety: smaller&lt;br /&gt;
@@ Line 625: / Line 628: @@
          &lt;td&gt;40&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;9 4&lt;br /&gt;
+         &lt;td&gt;9 4 4 9 4&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;&lt;br /&gt;
@@ Line 651: / Line 654: @@
          &lt;td style="text-align: left;"&gt;52.17&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: left;"&gt;7 3&lt;br /&gt;
+         &lt;td style="text-align: left;"&gt;7 3 3 7 3&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: left;"&gt;&lt;br /&gt;
@@ Line 677: / Line 680: @@
          &lt;td&gt;61.54&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;12 5&lt;br /&gt;
+         &lt;td&gt;12 5 5 12 5&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;&lt;br /&gt;
@@ Line 703: / Line 706: @@
          &lt;td&gt;75&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;5 2&lt;br /&gt;
+         &lt;td&gt;5 2 2 5 2&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;&lt;span style="display: block; text-align: center;"&gt;Pelog pentatonic (insofar as it resembles a MOS)&lt;/span&gt;&lt;br /&gt;
+         &lt;td style="text-align: center;"&gt;5-note subset of pelog (insofar as it&lt;br /&gt;
+resembles a MOS) would be in this region&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 729: / Line 733: @@
          &lt;td&gt;87.8&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;13 5&lt;br /&gt;
+         &lt;td&gt;13 5 5 13 5&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;&lt;br /&gt;
@@ Line 755: / Line 759: @@
          &lt;td&gt;90.05&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;phi+1 1&lt;br /&gt;
+         &lt;td&gt;phi+1 1 1 phi+1 1&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;&lt;br /&gt;
@@ Line 781: / Line 785: @@
          &lt;td&gt;96&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;8 3&lt;br /&gt;
+         &lt;td&gt;8 3 3 8 3&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;&lt;br /&gt;
@@ Line 807: / Line 811: @@
          &lt;td&gt;102.17&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;e 1&lt;br /&gt;
+         &lt;td&gt;e 1 e 1 1&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;L/s = e&lt;br /&gt;
@@ Line 833: / Line 837: @@
          &lt;td&gt;105.88&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;11 4&lt;br /&gt;
+         &lt;td&gt;11 4 4 11 4&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;&lt;br /&gt;
@@ Line 859: / Line 863: @@
          &lt;td&gt;133.33&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;3 1&lt;br /&gt;
+         &lt;td&gt;3 1 1 3 1&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;L/s = 3&lt;br /&gt;
@@ Line 885: / Line 889: @@
          &lt;td&gt;147.57&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;pi 1&lt;br /&gt;
+         &lt;td&gt;pi 1 pi 1 1&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;&lt;span style="display: block; text-align: center;"&gt;L/s = pi&lt;/span&gt;&lt;br /&gt;
@@ Line 911: / Line 915: @@
          &lt;td&gt;165.52&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;10 3&lt;br /&gt;
+         &lt;td&gt;10 3 3 10 3&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;&lt;br /&gt;
@@ Line 937: / Line 941: @@
          &lt;td&gt;180&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;7 2&lt;br /&gt;
+         &lt;td&gt;7 2 2 7 2&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;&lt;br /&gt;
@@ Line 963: / Line 967: @@
          &lt;td&gt;193.55&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;11 3&lt;br /&gt;
+         &lt;td&gt;11 3 3 11 3&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;&lt;br /&gt;
@@ Line 989: / Line 993: @@
          &lt;td&gt;218.18&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;4 1&lt;br /&gt;
+         &lt;td&gt;4 1 1 4 1&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;L/s = 4&lt;br /&gt;
@@ Line 1,015: / Line 1,019: @@
          &lt;td&gt;250&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;9 2&lt;br /&gt;
+         &lt;td&gt;9 2 2 9 2&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;&lt;br /&gt;
@@ Line 1,041: / Line 1,045: @@
          &lt;td&gt;276.92&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;5 1&lt;br /&gt;
+         &lt;td&gt;5 1 1 5 1&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;&lt;br /&gt;
@@ Line 1,067: / Line 1,071: @@
          &lt;td&gt;400&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;6 1&lt;br /&gt;
+         &lt;td&gt;6 1 1 6 1&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;&lt;br /&gt;