2L 3s
IMPORTED REVISION FROM WIKISPACES
This is an imported revision from Wikispaces. The revision metadata is included below for reference:
- This revision was by author xenwolf and made on 2016-11-07 03:11:27 UTC.
- The original revision id was 598650408.
- The revision comment was: 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.
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"Classic" [[pentatonic]]. Perhaps the most common scale in the world.
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 ||
|| 2\5 || || || || || || 480 || 240 || 0 || 240 || 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)
would be in this region ||
|| || || || || 9\22 || || 490.91 || 218.18 || 54.545 || 163.64 || 5 4 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 ||
|| || || || 7\17 || || || 494.12 || 211.76 || 70.59 || 141.18 || 4 3 3 4 3 ||= ||
|| || || || || || 19\46 || 495.65 || 208.7 || 78.26 || 130.435 || 11 8 8 11 8 || ||
|| || || || || 12\29 || || 496.55 || 206.9 || 82.76 || 124.14 || 7 5 5 7 5 ||= ||
|| || || || || || 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 pi 2 2 || ||
|| || || || || || 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 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 ||
|| || || || || || 21\50 || 504 || 192 || 120 || 72 || 13 8 8 13 8 ||= ||
|| || || || 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 7 12 7 || ||
|| || || || || || || 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 4 7 4 || ||
|| || || || || || 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 1 2 1 ||= (Boundary of propriety: smaller
generators than this are strictly proper) ||
|| || || || || || 13\30 || 520 || 160 || 200 || 40 || 9 4 4 9 4 || ||
||< ||< ||< ||< ||< 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 5 12 5 || ||
|| || || || 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 ||
|| || || || || || 18\41 || 526.83 || 146.34 || 234.15 || 87.8 || 13 5 5 13 5 || ||
|| || || || || || || 600(25+√5)/31 || 145.7 || 235.75 || 90.05 || phi+1 1 1 phi+1 1 || ||
|| || || || || 11\25 || || 528 || 144 || 240 || 96 || 8 3 3 8 3 || ||
|| || || || || || || 528.88 || 142.24 || 244.405 || 102.17 || e 1 e 1 1 ||= L/s = e ||
|| || || || || || 15\34 || 529.41 || 141.18 || 247.06 || 105.88 || 11 4 4 11 4 || ||
|| || || 4\9 || || || || 533.33 || 133.33 || 266.67 || 133.33 || 3 1 1 3 1 ||= L/s = 3 ||
|| || || || || || || 535.36 || 129.26 || 276.835 || 147.57 || pi 1 pi 1 1 ||= <span style="display: block; text-align: center;">L/s = pi</span> ||
|| || || || || || 13\29 || 537.93 || 124.14 || 289.655 || 165.52 || 10 3 3 10 3 || ||
|| || || || || 9\20 || || 540 || 120 || 240 || 180 || 7 2 2 7 2 || ||
|| || || || || || 14\31 || 541.935 || 116.13 || 309.68 || 193.55 || 11 3 3 11 3 || ||
|| || || || 5\11 || || || 545.45 || 109.09 || 327.27 || 218.18 || 4 1 1 4 1 ||= L/s = 4 ||
|| || || || || || 11\24 || 550 || 100 || 350 || 250 || 9 2 2 9 2 || ||
|| || || || || 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 ||= ||
From a [[3-limit]] perspective, just make a chain of four 4/3's and octave-reduce, and you end up with pentatonic.
From a [[5-limit]] perspective, the most interesting temperaments with this kind of pentatonic scale are [[meantone]] and [[Pelogic family|mavila]].
There is also the interesting 2.3.7 temperament that tempers out [[64_63|64/63]] ("no-fives [[dominant]]").Original HTML content:
<html><head><title>2L 3s</title></head><body>"Classic" <a class="wiki_link" href="/pentatonic">pentatonic</a>. Perhaps the most common scale in the world.<br />
<br />
The <a class="wiki_link" href="/meantone">meantone</a> 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 <a class="wiki_link" href="/Rothenberg%20propriety">proper</a>.<br />
<table class="wiki_table">
<tr>
<th colspan="6">Generator<br />
</th>
<th>Cents<br />
</th>
<th>s<br />
</th>
<th>L-s<br />
</th>
<th>|L-2s|<br />
</th>
<th>Scale steps<br />
</th>
<th>Comments<br />
</th>
</tr>
<tr>
<td>2\5<br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td>480<br />
</td>
<td>240<br />
</td>
<td>0<br />
</td>
<td>240<br />
</td>
<td>1 1 1 1 1<br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td>11\27<br />
</td>
<td>488.89<br />
</td>
<td>222.22<br />
</td>
<td>44.44<br />
</td>
<td>177.78<br />
</td>
<td>6 5 5 6 5<br />
</td>
<td style="text-align: center;">Slendro (insofar as it resembles a MOS)<br />
would be in this region<br />
</td>
</tr>
<tr>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td>9\22<br />
</td>
<td><br />
</td>
<td>490.91<br />
</td>
<td>218.18<br />
</td>
<td>54.545<br />
</td>
<td>163.64<br />
</td>
<td>5 4 4 5 4<br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td>16\39<br />
</td>
<td>492.31<br />
</td>
<td>215.38<br />
</td>
<td>61.54<br />
</td>
<td>153.85<br />
</td>
<td>9 7 7 9 7<br />
</td>
<td style="text-align: center;">No-5's superpyth/dominant is around here<br />
</td>
</tr>
<tr>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td>7\17<br />
</td>
<td><br />
</td>
<td><br />
</td>
<td>494.12<br />
</td>
<td>211.76<br />
</td>
<td>70.59<br />
</td>
<td>141.18<br />
</td>
<td>4 3 3 4 3<br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td>19\46<br />
</td>
<td>495.65<br />
</td>
<td>208.7<br />
</td>
<td>78.26<br />
</td>
<td>130.435<br />
</td>
<td>11 8 8 11 8<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td>12\29<br />
</td>
<td><br />
</td>
<td>496.55<br />
</td>
<td>206.9<br />
</td>
<td>82.76<br />
</td>
<td>124.14<br />
</td>
<td>7 5 5 7 5<br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td>17\41<br />
</td>
<td>497.56<br />
</td>
<td>204.88<br />
</td>
<td>87.8<br />
</td>
<td>117.07<br />
</td>
<td>10 7 7 10 7<br />
</td>
<td style="text-align: center;">Pythagorean pentatonic is around here<br />
</td>
</tr>
<tr>
<td><br />
</td>
<td><br />
</td>
<td>5\12<br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td>500<br />
</td>
<td>200<br />
</td>
<td>100<br />
</td>
<td>100<br />
</td>
<td>3 2 2 3 2<br />
</td>
<td style="text-align: center;">Familiar 12-equal pentatonic<br />
(also optimum rank range: L/s=3/2)<br />
</td>
</tr>
<tr>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td>502.305<br />
</td>
<td>195.39<br />
</td>
<td>111.53<br />
</td>
<td>83.86<br />
</td>
<td>pi 2 pi 2 2<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td>18\43<br />
</td>
<td>502.33<br />
</td>
<td>195.35<br />
</td>
<td>111.63<br />
</td>
<td>83.72<br />
</td>
<td>11 7 7 11 7<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td>13\31<br />
</td>
<td><br />
</td>
<td>503.23<br />
</td>
<td>193.55<br />
</td>
<td>116.13<br />
</td>
<td>77.42<br />
</td>
<td>8 5 5 8 5<br />
</td>
<td style="text-align: center;">Optimal meantone pentatonic<br />
is around here<br />
</td>
</tr>
<tr>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td>1200/(4-phi)<br />
</td>
<td>192.43<br />
</td>
<td>118.93<br />
</td>
<td>73.50<br />
</td>
<td>phi 1 1 phi 1<br />
</td>
<td style="text-align: center;">Golden meantone<br />
</td>
</tr>
<tr>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td>21\50<br />
</td>
<td>504<br />
</td>
<td>192<br />
</td>
<td>120<br />
</td>
<td>72<br />
</td>
<td>13 8 8 13 8<br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td>8\19<br />
</td>
<td><br />
</td>
<td><br />
</td>
<td>505.26<br />
</td>
<td>189.47<br />
</td>
<td>126.32<br />
</td>
<td>63.16<br />
</td>
<td>5 3 3 5 3<br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td>19\45<br />
</td>
<td>506.67<br />
</td>
<td>186.67<br />
</td>
<td>133.33<br />
</td>
<td>53.33<br />
</td>
<td>12 7 7 12 7<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td>507.18<br />
</td>
<td>185.64<br />
</td>
<td>135.9<br />
</td>
<td>49.74<br />
</td>
<td>√3 1 √3 1 1<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td>11\26<br />
</td>
<td><br />
</td>
<td>507.69<br />
</td>
<td>184.615<br />
</td>
<td>138.46<br />
</td>
<td>46.15<br />
</td>
<td>7 4 4 7 4<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td>14\33<br />
</td>
<td>509.09<br />
</td>
<td>181.82<br />
</td>
<td>145.455<br />
</td>
<td>36.36<br />
</td>
<td>9 5 5 9 5<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td><br />
</td>
<td>3\7<br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td>514.29<br />
</td>
<td>171.43<br />
</td>
<td>171.43<br />
</td>
<td>0<br />
</td>
<td>2 1 1 2 1<br />
</td>
<td style="text-align: center;">(Boundary of propriety: smaller<br />
generators than this are strictly proper)<br />
</td>
</tr>
<tr>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td>13\30<br />
</td>
<td>520<br />
</td>
<td>160<br />
</td>
<td>200<br />
</td>
<td>40<br />
</td>
<td>9 4 4 9 4<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td style="text-align: left;"><br />
</td>
<td style="text-align: left;"><br />
</td>
<td style="text-align: left;"><br />
</td>
<td style="text-align: left;"><br />
</td>
<td style="text-align: left;">10\23<br />
</td>
<td style="text-align: left;"><br />
</td>
<td style="text-align: left;">521.74<br />
</td>
<td style="text-align: left;">156.52<br />
</td>
<td style="text-align: left;">208.7<br />
</td>
<td style="text-align: left;">52.17<br />
</td>
<td style="text-align: left;">7 3 3 7 3<br />
</td>
<td style="text-align: left;"><br />
</td>
</tr>
<tr>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td>17\39<br />
</td>
<td>523.08<br />
</td>
<td>153.84<br />
</td>
<td>215.385<br />
</td>
<td>61.54<br />
</td>
<td>12 5 5 12 5<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td>7\16<br />
</td>
<td><br />
</td>
<td><br />
</td>
<td>525<br />
</td>
<td>150<br />
</td>
<td>225<br />
</td>
<td>75<br />
</td>
<td>5 2 2 5 2<br />
</td>
<td style="text-align: center;">5-note subset of pelog (insofar as it<br />
resembles a MOS) would be in this region<br />
</td>
</tr>
<tr>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td>18\41<br />
</td>
<td>526.83<br />
</td>
<td>146.34<br />
</td>
<td>234.15<br />
</td>
<td>87.8<br />
</td>
<td>13 5 5 13 5<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td>600(25+√5)/31<br />
</td>
<td>145.7<br />
</td>
<td>235.75<br />
</td>
<td>90.05<br />
</td>
<td>phi+1 1 1 phi+1 1<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td>11\25<br />
</td>
<td><br />
</td>
<td>528<br />
</td>
<td>144<br />
</td>
<td>240<br />
</td>
<td>96<br />
</td>
<td>8 3 3 8 3<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td>528.88<br />
</td>
<td>142.24<br />
</td>
<td>244.405<br />
</td>
<td>102.17<br />
</td>
<td>e 1 e 1 1<br />
</td>
<td style="text-align: center;">L/s = e<br />
</td>
</tr>
<tr>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td>15\34<br />
</td>
<td>529.41<br />
</td>
<td>141.18<br />
</td>
<td>247.06<br />
</td>
<td>105.88<br />
</td>
<td>11 4 4 11 4<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td><br />
</td>
<td><br />
</td>
<td>4\9<br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td>533.33<br />
</td>
<td>133.33<br />
</td>
<td>266.67<br />
</td>
<td>133.33<br />
</td>
<td>3 1 1 3 1<br />
</td>
<td style="text-align: center;">L/s = 3<br />
</td>
</tr>
<tr>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td>535.36<br />
</td>
<td>129.26<br />
</td>
<td>276.835<br />
</td>
<td>147.57<br />
</td>
<td>pi 1 pi 1 1<br />
</td>
<td style="text-align: center;"><span style="display: block; text-align: center;">L/s = pi</span><br />
</td>
</tr>
<tr>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td>13\29<br />
</td>
<td>537.93<br />
</td>
<td>124.14<br />
</td>
<td>289.655<br />
</td>
<td>165.52<br />
</td>
<td>10 3 3 10 3<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td>9\20<br />
</td>
<td><br />
</td>
<td>540<br />
</td>
<td>120<br />
</td>
<td>240<br />
</td>
<td>180<br />
</td>
<td>7 2 2 7 2<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td>14\31<br />
</td>
<td>541.935<br />
</td>
<td>116.13<br />
</td>
<td>309.68<br />
</td>
<td>193.55<br />
</td>
<td>11 3 3 11 3<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td>5\11<br />
</td>
<td><br />
</td>
<td><br />
</td>
<td>545.45<br />
</td>
<td>109.09<br />
</td>
<td>327.27<br />
</td>
<td>218.18<br />
</td>
<td>4 1 1 4 1<br />
</td>
<td style="text-align: center;">L/s = 4<br />
</td>
</tr>
<tr>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td>11\24<br />
</td>
<td>550<br />
</td>
<td>100<br />
</td>
<td>350<br />
</td>
<td>250<br />
</td>
<td>9 2 2 9 2<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td>6\13<br />
</td>
<td><br />
</td>
<td>553.85<br />
</td>
<td>92.31<br />
</td>
<td>369.23<br />
</td>
<td>276.92<br />
</td>
<td>5 1 1 5 1<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td>7\15<br />
</td>
<td>560<br />
</td>
<td>80<br />
</td>
<td>480<br />
</td>
<td>400<br />
</td>
<td>6 1 1 6 1<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>1\2<br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td>600<br />
</td>
<td>0<br />
</td>
<td>600<br />
</td>
<td>600<br />
</td>
<td>1 0 0 1 0<br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
</table>
<br />
From a <a class="wiki_link" href="/3-limit">3-limit</a> perspective, just make a chain of four 4/3's and octave-reduce, and you end up with pentatonic.<br />
<br />
From a <a class="wiki_link" href="/5-limit">5-limit</a> perspective, the most interesting temperaments with this kind of pentatonic scale are <a class="wiki_link" href="/meantone">meantone</a> and <a class="wiki_link" href="/Pelogic%20family">mavila</a>.<br />
<br />
There is also the interesting 2.3.7 temperament that tempers out <a class="wiki_link" href="/64_63">64/63</a> ("no-fives <a class="wiki_link" href="/dominant">dominant</a>").</body></html>