2L 3s: Difference between revisions

Revision as of 21:44, 10 February 2015

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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) ||
||   ||   ||   ||   ||   || 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 12 7 12 7 ||   ||
||   ||   ||   ||   || 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 ||   ||
||   ||   ||   ||   ||   || 15\34 || 529.41 || 141.18 || 247.06 || 105.88 || 11 4 4 11 4 ||   ||
||   ||   ||   ||   ||   ||   || 532.93 || 134.16 || 264.61 || 130.45 || 2.97 1 2.97 1 1 ||= L/s = 3/2^(1/75) ||
||   ||   || 4\9 ||   ||   ||   || 533.33 || 133.33 || 266.67 || 133.33 || 3 1 1 3 1 ||= L/s = 3 ||
||   ||   ||   ||   ||   ||   || 533.74 || 132.51 || 268.72 || 136.2 || 3.03 1 3.03 1 1 ||= <span style="display: block; text-align: center;">L/s = 3*2^(1/75)</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>&quot;Classic&quot; <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>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 12 7 12 7<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>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><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>532.93<br />
</td>
        <td>134.16<br />
</td>
        <td>264.61<br />
</td>
        <td>130.45<br />
</td>
        <td>2.97 1 2.97 1 1<br />
</td>
        <td style="text-align: center;">L/s = 3/2^(1/75)<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>533.74<br />
</td>
        <td>132.51<br />
</td>
        <td>268.72<br />
</td>
        <td>136.2<br />
</td>
        <td>3.03 1 3.03 1 1<br />
</td>
        <td style="text-align: center;"><span style="display: block; text-align: center;">L/s = 3*2^(1/75)</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> (&quot;no-fives <a class="wiki_link" href="/dominant">dominant</a>&quot;).</body></html>

@@ 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:OmegaNine|OmegaNine]] and made on <tt>2014-10-31 20:26:45 UTC</tt>.<br>
+: This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on <tt>2015-02-10 21:44:18 UTC</tt>.<br>
-: The original revision id was <tt>529021124</tt>.<br>
+: The original revision id was <tt>540543036</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 41: / Line 41: @@
 ||   ||   ||   ||   || 11\25 ||   || 528 || 144 || 240 || 96 || 8 3 3 8 3 ||   ||
 ||   ||   ||   ||   ||   || 15\34 || 529.41 || 141.18 || 247.06 || 105.88 || 11 4 4 11 4 ||   ||
+||   ||   ||   ||   ||   ||   || 532.93 || 134.16 || 264.61 || 130.45 || 2.97 1 2.97 1 1 ||= L/s = 3/2^(1/75) ||
 ||   ||   || 4\9 ||   ||   ||   || 533.33 || 133.33 || 266.67 || 133.33 || 3 1 1 3 1 ||= L/s = 3 ||
+||   ||   ||   ||   ||   ||   || 533.74 || 132.51 || 268.72 || 136.2 || 3.03 1 3.03 1 1 ||= &lt;span style="display: block; text-align: center;"&gt;L/s = 3*2^(1/75)&lt;/span&gt; ||
 ||   ||   ||   ||   ||   || 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 ||   ||
@@ Line 48: / Line 50: @@
 ||   ||   ||   ||   ||   || 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 ||   ||   ||
+||   ||   ||   ||   ||   || 7\15 || 560 || 80 || 480 || 400 || 6 1 1 6 1 ||   ||
 || 1\2 ||   ||   ||   ||   ||   || 600 || 0 || 600 || 600 || 1 0 0 1 0 ||=   ||
@@ Line 758: / Line 760: @@
 &lt;/td&gt;
          &lt;td&gt;&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&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;&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;532.93&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;134.16&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;264.61&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;130.45&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;2.97 1 2.97 1 1&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td style="text-align: center;"&gt;L/s = 3/2^(1/75)&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 784: / Line 812: @@
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;L/s = 3&lt;br /&gt;
+&lt;/td&gt;
+    &lt;/tr&gt;
+    &lt;tr&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;&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;533.74&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;132.51&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;268.72&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;136.2&lt;br /&gt;
+&lt;/td&gt;
+        &lt;td&gt;3.03 1 3.03 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 = 3*2^(1/75)&lt;/span&gt;&lt;br /&gt;
 &lt;/td&gt;
      &lt;/tr&gt;
@@ Line 963: / Line 1,017: @@
          &lt;td&gt;400&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td&gt;&lt;br /&gt;
+         &lt;td&gt;6 1 1 6 1&lt;br /&gt;
 &lt;/td&gt;
          &lt;td&gt;&lt;br /&gt;

2L 3s: Difference between revisions

Revision as of 21:44, 10 February 2015

IMPORTED REVISION FROM WIKISPACES

Original Wikitext content:

Original HTML content:

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