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:xenwolf|xenwolf]] and made on <tt>2016-11-07 03:11:27 UTC</tt>.

: This revision was by author [[User:xenwolf|xenwolf]] and made on <tt>2016-11-07 03:24:39 UTC</tt>.

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

: The original revision id was <tt>598651052</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 comment was: <tt>JosephRuhf's Trichords merged (as a new column)</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>

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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| ||~ Scale steps ||~ Comments ||

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

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

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

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

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

would be in this region ||

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

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

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

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

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

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

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

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

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

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

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

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

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

|| || || 5\12 || || || || 500 || 200 || 100 || 100 || 3 2 2 3 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 pi 2 2 || ||

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

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

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

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

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

is around here ||

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

generators than this are strictly proper) ||

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

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

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

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

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

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

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

resembles a MOS) would be in this region ||

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

|| 1\2 || || || || || || 600 || 0 || 600 || 600 || 1 0 0 1 0 || 1 0 ||= a degenerated pentatonic scale with only 2 different steps ||

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

Line 79:

</th>

<th>Scale steps

</th>

<th>Trichord

</th>

<th>Comments

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<td>1 1 1 1 1

</td>

<td>1 1

</td>

<td>=

</td>

</tr>

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</td>

<td>6 5 5 6 5

</td>

<td>6 5

</td>

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

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Line 164:

</td>

<td>5 4 4 5 4

</td>

<td>5 4

</td>

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Line 192:

</td>

<td>9 7 7 9 7

</td>

<td>9 7

</td>

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

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Line 220:

</td>

<td>4 3 3 4 3

</td>

<td>4 3

</td>

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</td>

<td>11 8 8 11 8

</td>

<td>11 8

</td>

<td>

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</td>

<td>7 5 5 7 5

</td>

<td>7 5

</td>

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</td>

<td>10 7 7 10 7

</td>

<td>10 7

</td>

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

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</td>

<td>3 2 2 3 2

</td>

<td>3 2

</td>

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

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</td>

<td>pi 2 pi 2 2

</td>

<td>pi 2

</td>

<td>

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</td>

<td>11 7 7 11 7

</td>

<td>11 7

</td>

<td>

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</td>

<td>8 5 5 8 5

</td>

<td>8 5

</td>

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

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</td>

<td>phi 1 1 phi 1

</td>

<td>phi 1

</td>

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

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</td>

<td>13 8 8 13 8

</td>

<td>13 8

</td>

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</td>

<td>5 3 3 5 3

</td>

<td>5 3

</td>

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</td>

<td>12 7 7 12 7

</td>

<td>12 7

</td>

<td>

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Line 558:

</td>

<td>√3 1 √3 1 1

</td>

<td>√3 1

</td>

<td>

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Line 586:

</td>

<td>7 4 4 7 4

</td>

<td>7 4

</td>

<td>

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</td>

<td>9 5 5 9 5

</td>

<td>9 5

</td>

<td>

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</td>

<td>2 1 1 2 1

</td>

<td>2 1

</td>

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

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</td>

<td>9 4 4 9 4

</td>

<td>9 4

</td>

<td>

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</td>

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

</td>

<td>7 3

</td>

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</td>

<td>12 5 5 12 5

</td>

<td>12 5

</td>

<td>

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</td>

<td>5 2 2 5 2

</td>

<td>5 2

</td>

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

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</td>

<td>13 5 5 13 5

</td>

<td>13 5

</td>

<td>

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</td>

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

</td>

<td>phi+1 1

</td>

<td>

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</td>

<td>8 3 3 8 3

</td>

<td>8 3

</td>

<td>

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</td>

<td>e 1 e 1 1

</td>

<td>e 1

</td>

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

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</td>

<td>11 4 4 11 4

</td>

<td>11 4

</td>

<td>

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</td>

<td>3 1 1 3 1

</td>

<td>3 1

</td>

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

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</td>

<td>pi 1 pi 1 1

</td>

<td>pi 1

</td>

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

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</td>

<td>10 3 3 10 3

</td>

<td>10 3

</td>

<td>

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Line 1,008:

</td>

<td>7 2 2 7 2

</td>

<td>7 2

</td>

<td>

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Line 1,036:

</td>

<td>11 3 3 11 3

</td>

<td>11 3

</td>

<td>

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Line 1,064:

</td>

<td>4 1 1 4 1

</td>

<td>4 1

</td>

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

Line 1,020:

Line 1,092:

</td>

<td>9 2 2 9 2

</td>

<td>9 2

</td>

<td>

Line 1,046:

Line 1,120:

</td>

<td>5 1 1 5 1

</td>

<td>5 1

</td>

<td>

Line 1,072:

Line 1,148:

</td>

<td>6 1 1 6 1

</td>

<td>6 1

</td>

<td>

Line 1,099:

Line 1,177:

<td>1 0 0 1 0

</td>

<td>1 0

</td>

<td style="text-align: center;">a degenerated pentatonic scale with only 2 different steps

</td>

</tr>

@@ 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>2016-11-07 03:11:27 UTC</tt>.<br>
+: This revision was by author [[User:xenwolf|xenwolf]] and made on <tt>2016-11-07 03:24:39 UTC</tt>.<br>
-: The original revision id was <tt>598650408</tt>.<br>
+: The original revision id was <tt>598651052</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 comment was: <tt>JosephRuhf's Trichords merged (as a new column)</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| ||~ Scale steps ||~ Comments ||
+||||||||||||~ Generator ||~ Cents ||~ s ||~ L-s ||~ |L-2s| ||~ Scale steps ||~ Trichord ||~ Comments ||
-|| 2\5 ||   ||   ||   ||   ||   || 480 || 240 || 0 || 240 || 1 1 1 1 1 ||=   ||
+|| 2\5 ||   ||   ||   ||   ||   || 480 || 240 || 0 || 240 || 1 1 1 1 1 || 1 1 || =   ||
-||   ||   ||   ||   ||   || 11\27 || 488.89 || 222.22 || 44.44 || 177.78 || 6 5 5 6 5 ||= Slendro (insofar as it resembles a MOS)
+||   ||   ||   ||   ||   || 11\27 || 488.89 || 222.22 || 44.44 || 177.78 || 6 5 5 6 5 || 6 5 ||= Slendro (insofar as it resembles a MOS)
 would be in this region ||
-||   ||   ||   ||   || 9\22 ||   || 490.91 || 218.18 || 54.545 || 163.64 || 5 4 4 5 4 ||=   ||
+||   ||   ||   ||   || 9\22 ||   || 490.91 || 218.18 || 54.545 || 163.64 || 5 4 4 5 4 || 5 4 ||=   ||
-||   ||   ||   ||   ||   || 16\39 || 492.31 || 215.38 || 61.54 || 153.85 || 9 7 7 9 7 ||= No-5's superpyth/dominant is around here ||
+||   ||   ||   ||   ||   || 16\39 || 492.31 || 215.38 || 61.54 || 153.85 || 9 7 7 9 7 || 9 7 ||= No-5's superpyth/dominant is around here ||
-||   ||   ||   || 7\17 ||   ||   || 494.12 || 211.76 || 70.59 || 141.18 || 4 3 3 4 3 ||=   ||
+||   ||   ||   || 7\17 ||   ||   || 494.12 || 211.76 || 70.59 || 141.18 || 4 3 3 4 3 || 4 3 ||=   ||
-||   ||   ||   ||   ||   || 19\46 || 495.65 || 208.7 || 78.26 || 130.435 || 11 8 8 11 8 ||   ||
+||   ||   ||   ||   ||   || 19\46 || 495.65 || 208.7 || 78.26 || 130.435 || 11 8 8 11 8 ||  11 8 ||  ||
-||   ||   ||   ||   || 12\29 ||   || 496.55 || 206.9 || 82.76 || 124.14 || 7 5 5 7 5 ||=   ||
+||   ||   ||   ||   || 12\29 ||   || 496.55 || 206.9 || 82.76 || 124.14 || 7 5 5 7 5 || 7 5 ||=   ||
-||   ||   ||   ||   ||   || 17\41 || 497.56 || 204.88 || 87.8 || 117.07 || 10 7 7 10 7 ||= Pythagorean pentatonic is around here ||
+||   ||   ||   ||   ||   || 17\41 || 497.56 || 204.88 || 87.8 || 117.07 || 10 7 7 10 7 || 10 7 ||= Pythagorean pentatonic is around here ||
-||   ||   || 5\12 ||   ||   ||   || 500 || 200 || 100 || 100 || 3 2 2 3 2 ||= Familiar 12-equal pentatonic
+||   ||   || 5\12 ||   ||   ||   || 500 || 200 || 100 || 100 || 3 2 2 3 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 pi 2 2 ||   ||
+||   ||   ||   ||   ||   ||   || 502.305 || 195.39 || 111.53 || 83.86 || pi 2 pi 2 2 || pi 2 ||  ||
-||   ||   ||   ||   ||   || 18\43 || 502.33 || 195.35 || 111.63 || 83.72 || 11 7 7 11 7 ||   ||
+||   ||   ||   ||   ||   || 18\43 || 502.33 || 195.35 || 111.63 || 83.72 || 11 7 7 11 7 || 11 7 ||  ||
-||   ||   ||   ||   || 13\31 ||   || 503.23 || 193.55 || 116.13 || 77.42 || 8 5 5 8 5 ||= Optimal meantone pentatonic
+||   ||   ||   ||   || 13\31 ||   || 503.23 || 193.55 || 116.13 || 77.42 || 8 5 5 8 5 || 8 5 ||= Optimal meantone pentatonic
 is around here ||
-||   ||   ||   ||   ||   ||   || 1200/(4-phi) || 192.43 || 118.93 || 73.50 || phi 1 1 phi 1 ||= Golden meantone ||
+||   ||   ||   ||   ||   ||   || 1200/(4-phi) || 192.43 || 118.93 || 73.50 || phi 1 1 phi 1 || phi 1 ||= Golden meantone ||
-||   ||   ||   ||   ||   || 21\50 || 504 || 192 || 120 || 72 || 13 8 8 13 8 ||=   ||
+||   ||   ||   ||   ||   || 21\50 || 504 || 192 || 120 || 72 || 13 8 8 13 8 || 13 8 ||=  ||
-||   ||   ||   || 8\19 ||   ||   || 505.26 || 189.47 || 126.32 || 63.16 || 5 3 3 5 3 ||=   ||
+||   ||   ||   || 8\19 ||   ||   || 505.26 || 189.47 || 126.32 || 63.16 || 5 3 3 5 3 || 5 3 ||=   ||
-||   ||   ||   ||   ||   || 19\45 || 506.67 || 186.67 || 133.33 || 53.33 || 12 7 7 12 7 ||   ||
+||   ||   ||   ||   ||   || 19\45 || 506.67 || 186.67 || 133.33 || 53.33 || 12 7 7 12 7 || 12 7 ||  ||
-||   ||   ||   ||   ||   ||   || 507.18 || 185.64 || 135.9 || 49.74 || √3 1 √3 1 1 ||   ||
+||   ||   ||   ||   ||   ||   || 507.18 || 185.64 || 135.9 || 49.74 || √3 1 √3 1 1 || √3 1 ||  ||
-||   ||   ||   ||   || 11\26 ||   || 507.69 || 184.615 || 138.46 || 46.15 || 7 4 4 7 4 ||   ||
+||   ||   ||   ||   || 11\26 ||   || 507.69 || 184.615 || 138.46 || 46.15 || 7 4 4 7 4 || 7 4 ||  ||
-||   ||   ||   ||   ||   || 14\33 || 509.09 || 181.82 || 145.455 || 36.36 || 9 5 5 9 5 ||   ||
+||   ||   ||   ||   ||   || 14\33 || 509.09 || 181.82 || 145.455 || 36.36 || 9 5 5 9 5 || 9 5 ||  ||
-||   || 3\7 ||   ||   ||   ||   || 514.29 || 171.43 || 171.43 || 0 || 2 1 1 2 1 ||= (Boundary of propriety: smaller
+||   || 3\7 ||   ||   ||   ||   || 514.29 || 171.43 || 171.43 || 0 || 2 1 1 2 1 || 2 1 ||= (Boundary of propriety: smaller
 generators than this are strictly proper) ||
-||   ||   ||   ||   ||   || 13\30 || 520 || 160 || 200 || 40 || 9 4 4 9 4 ||   ||
+||   ||   ||   ||   ||   || 13\30 || 520 || 160 || 200 || 40 || 9 4 4 9 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 3 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 || 7 3 ||&lt;   ||
-||   ||   ||   ||   ||   || 17\39 || 523.08 || 153.84 || 215.385 || 61.54 || 12 5 5 12 5 ||   ||
+||   ||   ||   ||   ||   || 17\39 || 523.08 || 153.84 || 215.385 || 61.54 || 12 5 5 12 5 ||  12 5 ||  ||
-||   ||   ||   || 7\16 ||   ||   || 525 || 150 || 225 || 75 || 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 2 ||= 5-note subset of pelog (insofar as it
 resembles a MOS) would be in this region ||
-||   ||   ||   ||   ||   || 18\41 || 526.83 || 146.34 || 234.15 || 87.8 || 13 5 5 13 5 ||   ||
+||   ||   ||   ||   ||   || 18\41 || 526.83 || 146.34 || 234.15 || 87.8 || 13 5 5 13 5 ||  13 5 ||  ||
-||   ||   ||   ||   ||   ||   || 600(25+√5)/31 || 145.7 || 235.75 || 90.05 || phi+1 1 1 phi+1 1 ||   ||
+||   ||   ||   ||   ||   ||   || 600(25+√5)/31 || 145.7 || 235.75 || 90.05 || phi+1 1 1 phi+1 1 || phi+1 1 ||  ||
-||   ||   ||   ||   || 11\25 ||   || 528 || 144 || 240 || 96 || 8 3 3 8 3 ||   ||
+||   ||   ||   ||   || 11\25 ||   || 528 || 144 || 240 || 96 || 8 3 3 8 3 ||  8 3 || ||
-||   ||   ||   ||   ||   ||   || 528.88 || 142.24 || 244.405 || 102.17 || e 1 e 1 1 ||= L/s = e ||
+||   ||   ||   ||   ||   ||   || 528.88 || 142.24 || 244.405 || 102.17 || e 1 e 1 1 || e 1 ||= L/s = e ||
-||   ||   ||   ||   ||   || 15\34 || 529.41 || 141.18 || 247.06 || 105.88 || 11 4 4 11 4 ||   ||
+||   ||   ||   ||   ||   || 15\34 || 529.41 || 141.18 || 247.06 || 105.88 || 11 4 4 11 4 ||  11 4 || ||
-||   ||   || 4\9 ||   ||   ||   || 533.33 || 133.33 || 266.67 || 133.33 || 3 1 1 3 1 ||= L/s = 3 ||
+||   ||   || 4\9 ||   ||   ||   || 533.33 || 133.33 || 266.67 || 133.33 || 3 1 1 3 1 || 3 1 ||= L/s = 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; ||
+||   ||   ||   ||   ||   ||   || 535.36 || 129.26 || 276.835 || 147.57 || pi 1 pi 1 1 || pi 1 ||= &lt;span style="display: block; text-align: center;"&gt;L/s = pi&lt;/span&gt; ||
-||   ||   ||   ||   ||   || 13\29 || 537.93 || 124.14 || 289.655 || 165.52 || 10 3 3 10 3 ||   ||
+||   ||   ||   ||   ||   || 13\29 || 537.93 || 124.14 || 289.655 || 165.52 || 10 3 3 10 3 ||  10 3 || ||
-||   ||   ||   ||   || 9\20 ||   || 540 || 120 || 240 || 180 || 7 2 2 7 2 ||   ||
+||   ||   ||   ||   || 9\20 ||   || 540 || 120 || 240 || 180 || 7 2 2 7 2 ||  7 2 || ||
-||   ||   ||   ||   ||   || 14\31 || 541.935 || 116.13 || 309.68 || 193.55 || 11 3 3 11 3 ||   ||
+||   ||   ||   ||   ||   || 14\31 || 541.935 || 116.13 || 309.68 || 193.55 || 11 3 3 11 3 ||  11 3 || ||
-||   ||   ||   || 5\11 ||   ||   || 545.45 || 109.09 || 327.27 || 218.18 || 4 1 1 4 1 ||= L/s = 4 ||
+||   ||   ||   || 5\11 ||   ||   || 545.45 || 109.09 || 327.27 || 218.18 || 4 1 1 4 1 || 4 1 ||= L/s = 4 ||
-||   ||   ||   ||   ||   || 11\24 || 550 || 100 || 350 || 250 || 9 2 2 9 2 ||   ||
+||   ||   ||   ||   ||   || 11\24 || 550 || 100 || 350 || 250 || 9 2 2 9 2 ||  9 2 ||  ||
-||   ||   ||   ||   || 6\13 ||   || 553.85 || 92.31 || 369.23 || 276.92 || 5 1 1 5 1 ||   ||
+||   ||   ||   ||   || 6\13 ||   || 553.85 || 92.31 || 369.23 || 276.92 || 5 1 1 5 1 || 5 1 ||  ||
-||   ||   ||   ||   ||   || 7\15 || 560 || 80 || 480 || 400 || 6 1 1 6 1 ||   ||
+||   ||   ||   ||   ||   || 7\15 || 560 || 80 || 480 || 400 || 6 1 1 6 1 ||  6 1 ||  ||
-|| 1\2 ||   ||   ||   ||   ||   || 600 || 0 || 600 || 600 || 1 0 0 1 0 ||=   ||
+|| 1\2 ||   ||   ||   ||   ||   || 600 || 0 || 600 || 600 || 1 0 0 1 0 || 1 0 ||=  a degenerated pentatonic scale with only 2 different steps ||
 From a [[3-limit]] perspective, just make a chain of four 4/3's and octave-reduce, and you end up with pentatonic.
@@ Line 79: / Line 79: @@
 &lt;/th&gt;
          &lt;th&gt;Scale steps&lt;br /&gt;
+&lt;/th&gt;
+        &lt;th&gt;Trichord&lt;br /&gt;
 &lt;/th&gt;
          &lt;th&gt;Comments&lt;br /&gt;
@@ Line 106: / Line 108: @@
          &lt;td&gt;1 1 1 1 1&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+         &lt;td&gt;1 1&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;=&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 131: / Line 135: @@
 &lt;/td&gt;
          &lt;td&gt;6 5 5 6 5&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;6 5&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;Slendro (insofar as it resembles a MOS)&lt;br /&gt;
@@ Line 158: / Line 164: @@
 &lt;/td&gt;
          &lt;td&gt;5 4 4 5 4&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;5 4&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;&lt;br /&gt;
@@ Line 184: / Line 192: @@
 &lt;/td&gt;
          &lt;td&gt;9 7 7 9 7&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;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 210: / Line 220: @@
 &lt;/td&gt;
          &lt;td&gt;4 3 3 4 3&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;4 3&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;&lt;br /&gt;
@@ Line 236: / Line 248: @@
 &lt;/td&gt;
          &lt;td&gt;11 8 8 11 8&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;11 8&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;&lt;br /&gt;
@@ Line 262: / Line 276: @@
 &lt;/td&gt;
          &lt;td&gt;7 5 5 7 5&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;7 5&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;&lt;br /&gt;
@@ Line 288: / Line 304: @@
 &lt;/td&gt;
          &lt;td&gt;10 7 7 10 7&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;10 7&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;Pythagorean pentatonic is around here&lt;br /&gt;
@@ Line 314: / Line 332: @@
 &lt;/td&gt;
          &lt;td&gt;3 2 2 3 2&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;3 2&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;Familiar 12-equal pentatonic&lt;br /&gt;
@@ Line 341: / Line 361: @@
 &lt;/td&gt;
          &lt;td&gt;pi 2 pi 2 2&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;pi 2&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;&lt;br /&gt;
@@ Line 367: / Line 389: @@
 &lt;/td&gt;
          &lt;td&gt;11 7 7 11 7&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;11 7&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;&lt;br /&gt;
@@ Line 393: / Line 417: @@
 &lt;/td&gt;
          &lt;td&gt;8 5 5 8 5&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;8 5&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;Optimal meantone pentatonic&lt;br /&gt;
@@ Line 420: / Line 446: @@
 &lt;/td&gt;
          &lt;td&gt;phi 1 1 phi 1&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;phi 1&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;Golden meantone&lt;br /&gt;
@@ Line 446: / Line 474: @@
 &lt;/td&gt;
          &lt;td&gt;13 8 8 13 8&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;13 8&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;&lt;br /&gt;
@@ Line 472: / Line 502: @@
 &lt;/td&gt;
          &lt;td&gt;5 3 3 5 3&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;5 3&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;&lt;br /&gt;
@@ Line 498: / Line 530: @@
 &lt;/td&gt;
          &lt;td&gt;12 7 7 12 7&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;12 7&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;&lt;br /&gt;
@@ Line 524: / Line 558: @@
 &lt;/td&gt;
          &lt;td&gt;√3 1 √3 1 1&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;√3 1&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;&lt;br /&gt;
@@ Line 550: / Line 586: @@
 &lt;/td&gt;
          &lt;td&gt;7 4 4 7 4&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;7 4&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;&lt;br /&gt;
@@ Line 576: / Line 614: @@
 &lt;/td&gt;
          &lt;td&gt;9 5 5 9 5&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;9 5&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;&lt;br /&gt;
@@ Line 602: / Line 642: @@
 &lt;/td&gt;
          &lt;td&gt;2 1 1 2 1&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;2 1&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;(Boundary of propriety: smaller&lt;br /&gt;
@@ Line 629: / Line 671: @@
 &lt;/td&gt;
          &lt;td&gt;9 4 4 9 4&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;9 4&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;&lt;br /&gt;
@@ Line 655: / Line 699: @@
 &lt;/td&gt;
          &lt;td style="text-align: left;"&gt;7 3 3 7 3&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;7 3&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: left;"&gt;&lt;br /&gt;
@@ Line 681: / Line 727: @@
 &lt;/td&gt;
          &lt;td&gt;12 5 5 12 5&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;12 5&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;&lt;br /&gt;
@@ Line 707: / Line 755: @@
 &lt;/td&gt;
          &lt;td&gt;5 2 2 5 2&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;5 2&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;5-note subset of pelog (insofar as it&lt;br /&gt;
@@ Line 734: / Line 784: @@
 &lt;/td&gt;
          &lt;td&gt;13 5 5 13 5&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;13 5&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;&lt;br /&gt;
@@ Line 760: / Line 812: @@
 &lt;/td&gt;
          &lt;td&gt;phi+1 1 1 phi+1 1&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;phi+1 1&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;&lt;br /&gt;
@@ Line 786: / Line 840: @@
 &lt;/td&gt;
          &lt;td&gt;8 3 3 8 3&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;8 3&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;&lt;br /&gt;
@@ Line 812: / Line 868: @@
 &lt;/td&gt;
          &lt;td&gt;e 1 e 1 1&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;e 1&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;L/s = e&lt;br /&gt;
@@ Line 838: / Line 896: @@
 &lt;/td&gt;
          &lt;td&gt;11 4 4 11 4&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;11 4&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;&lt;br /&gt;
@@ Line 864: / Line 924: @@
 &lt;/td&gt;
          &lt;td&gt;3 1 1 3 1&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;3 1&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;L/s = 3&lt;br /&gt;
@@ Line 890: / Line 952: @@
 &lt;/td&gt;
          &lt;td&gt;pi 1 pi 1 1&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;pi 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 916: / Line 980: @@
 &lt;/td&gt;
          &lt;td&gt;10 3 3 10 3&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;10 3&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;&lt;br /&gt;
@@ Line 942: / Line 1,008: @@
 &lt;/td&gt;
          &lt;td&gt;7 2 2 7 2&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;7 2&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;&lt;br /&gt;
@@ Line 968: / Line 1,036: @@
 &lt;/td&gt;
          &lt;td&gt;11 3 3 11 3&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;11 3&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;&lt;br /&gt;
@@ Line 994: / Line 1,064: @@
 &lt;/td&gt;
          &lt;td&gt;4 1 1 4 1&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;4 1&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;L/s = 4&lt;br /&gt;
@@ Line 1,020: / Line 1,092: @@
 &lt;/td&gt;
          &lt;td&gt;9 2 2 9 2&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;9 2&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;&lt;br /&gt;
@@ Line 1,046: / Line 1,120: @@
 &lt;/td&gt;
          &lt;td&gt;5 1 1 5 1&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;5 1&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;&lt;br /&gt;
@@ Line 1,072: / Line 1,148: @@
 &lt;/td&gt;
          &lt;td&gt;6 1 1 6 1&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;6 1&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;&lt;br /&gt;
@@ Line 1,099: / Line 1,177: @@
          &lt;td&gt;1 0 0 1 0&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td style="text-align: center;"&gt;&lt;br /&gt;
+        &lt;td&gt;1 0&lt;br /&gt;
+&lt;/td&gt;
+         &lt;td style="text-align: center;"&gt;a degenerated pentatonic scale with only 2 different steps&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;