3L 10s: Difference between revisions

Revision as of 00:31, 6 June 2012

IMPORTED REVISION FROM WIKISPACES

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The original revision id was 343053468.

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Original Wikitext content:

This scale, though it belongs to magic and würschmidt temperaments, is functionally identical to a compromise between the meantone/superpyth diatonic and chromatic scales. However, meantone and superpyth chords are slightly more complex and less accurate because 3125/3072, which is nearly 1/41 of the octave, is tempered out instead of 81/80 or 64/63. As a magic scale, it is uneven enough to fall within the region of optimal melodic tunings for magic[13] (L:s = sqrt(6) to 6-sqrt(6)). Because of this property, magic[13] is a good 19 tone mode for people who want to explore new sounds without having to give up using strongly harmonic fifths (3/2 +/- ~cbrt(145/143)) for building consonant chords.

Of course, magic[13] is still slightly improper and würschmidt[13] is almost totally lopsided (L:s = 6:1 to 8:1). Even so, the excellence of magic temperament as a fit for 13 tone per octave melodic scales and its close affinity to strongly harmonic fifth temperaments make up for the roughness of the major third that 13 tone circles favor multiple times over.

|| 1/3 ||   ||   ||   ||   ||   ||   || 800 ||=   ||
||   ||   ||   ||   ||   ||   || 10/31 || 735.48387 ||=   ||
||   ||   ||   ||   ||   || 9/28 ||   || 728.57143 ||=   ||
||   ||   ||   ||   ||   ||   || 17/53 || 724.5283 ||=   ||
||   ||   ||   ||   || 8/25 ||   ||   || 720 ||=   ||
||   ||   ||   ||   ||   ||   || 23/72 || 716.__6__ ||   ||
||   ||   ||   ||   ||   || 15/47 ||   || 714.89362 ||=   ||
||   ||   ||   ||   ||   ||   || 22/69 || 713.04348 ||   ||
||   ||   ||   || 7/22 ||   ||   ||   || 709.__09__ ||=   ||
||   ||   ||   ||   ||   ||   || 27/85 || 705.88235 ||   ||
||   ||   ||   ||   ||   || 20/63 ||   || 704.761905 ||=   ||
||   ||   ||   ||   ||   ||   || 33/104 || 703.84615 ||   ||
||   ||   ||   ||   || 13/41 ||   ||   || 702.__43902__ ||=   ||
||   ||   ||   ||   ||   ||   || 32/101 || 700.__9900__ ||   ||
||   ||   ||   ||   ||   || 19/60 ||   || 700 ||= Optimal patent val for 5-limit magic ||
||   ||   ||   ||   ||   ||   || 25/79 || 698.73418 ||   ||
||   ||   || 6/19 ||   ||   ||   ||   || 694.73684 ||=   ||
||   ||   ||   ||   ||   ||   || 29/92 || 691.30435 ||   ||
||   ||   ||   ||   ||   || 23/73 ||   || 690.41096 ||=   ||
||   ||   ||   ||   ||   ||   || 40/127 || 689.76378 ||   ||
||   ||   ||   ||   || 17/54 ||   ||   || 688.__8__ ||=   ||
||   ||   ||   ||   ||   ||   || 45/143 || 688.11189 ||   ||
||   ||   ||   ||   ||   || 28/89 ||   || 687.64045 ||=   ||
||   ||   ||   ||   ||   ||   || 39/124 || 687.09677 ||   ||
||   ||   ||   || 11/35 ||   ||   ||   || 685.__714285__ ||=   ||
||   ||   ||   ||   ||   ||   || 38/121 || 684.29752 ||   ||
||   ||   ||   ||   ||   || 27/86 ||   || 683.72093 ||=   ||
||   ||   ||   ||   ||   ||   || 43/137 || 683.21168 ||   ||
||   ||   ||   ||   || 16/51 ||   ||   || 682.35294 ||=   ||
||   ||   ||   ||   ||   ||   || 37/118 || 681.35593 ||   ||
||   ||   ||   ||   ||   || 21/67 ||   || 680.597015 ||=   ||
||   ||   ||   ||   ||   ||   || 26/83 || 679.51807 ||   ||
||   || 5/16 ||   ||   ||   ||   ||   || 675 ||= Boundary of propriety
(Smaller generators are proper) ||
||   ||   ||   ||   ||   ||   || 29/93 || 670.96774 ||   ||
||   ||   ||   ||   ||   || 24/77 ||   || 670.12987 ||=   ||
||   ||   ||   ||   ||   ||   || 43/138 || 669.56522 ||   ||
||   ||   ||   ||   || 19/61 ||   ||   || 668.52459 ||=   ||
||   ||   ||   ||   ||   ||   || 52/167 || 668.26347 ||   ||
||   ||   ||   ||   ||   || 33/106 ||   || 667.92452 ||=   ||
||   ||   ||   ||   ||   ||   || 47/151 || 667.54967 ||   ||
||   ||   ||   || 14/45 ||   ||   ||   || 666.__6__ ||=   ||
||   ||   ||   ||   ||   ||   || 51/164 ||   ||   ||
||   ||   ||   ||   ||   || 37/119 ||   || 665.54622 ||=   ||
||   ||   ||   ||   ||   ||   ||   ||   ||   ||
||   ||   ||   ||   || 23/74 ||   ||   || 664.__864__ ||=   ||
||   ||   ||   ||   ||   ||   ||   ||   ||   ||
||   ||   ||   ||   ||   || 32/103 ||   || 664.07767 ||=   ||
||   ||   ||   ||   ||   ||   ||   ||   ||   ||
||   ||   || 9/29 ||   ||   ||   ||   || 662.068965 ||=   ||
||   ||   ||   || 13/42 ||   ||   ||   || 657.14286 ||=   ||
|| 4/13 ||   ||   ||   ||   ||   ||   || 646.15385 ||=   ||

Original HTML content:

<html><head><title>3L 10s</title></head><body>This scale, though it belongs to magic and würschmidt temperaments, is functionally identical to a compromise between the meantone/superpyth diatonic and chromatic scales. However, meantone and superpyth chords are slightly more complex and less accurate because 3125/3072, which is nearly 1/41 of the octave, is tempered out instead of 81/80 or 64/63. As a magic scale, it is uneven enough to fall within the region of optimal melodic tunings for magic[13] (L:s = sqrt(6) to 6-sqrt(6)). Because of this property, magic[13] is a good 19 tone mode for people who want to explore new sounds without having to give up using strongly harmonic fifths (3/2 +/- ~cbrt(145/143)) for building consonant chords.<br />
<br />
Of course, magic[13] is still slightly improper and würschmidt[13] is almost totally lopsided (L:s = 6:1 to 8:1). Even so, the excellence of magic temperament as a fit for 13 tone per octave melodic scales and its close affinity to strongly harmonic fifth temperaments make up for the roughness of the major third that 13 tone circles favor multiple times over.<br />
<br />


<table class="wiki_table">
    <tr>
        <td>1/3<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>800<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><br />
</td>
        <td>10/31<br />
</td>
        <td>735.48387<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>9/28<br />
</td>
        <td><br />
</td>
        <td>728.57143<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><br />
</td>
        <td>17/53<br />
</td>
        <td>724.5283<br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>8/25<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>720<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><br />
</td>
        <td>23/72<br />
</td>
        <td>716.<u>6</u><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>15/47<br />
</td>
        <td><br />
</td>
        <td>714.89362<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><br />
</td>
        <td>22/69<br />
</td>
        <td>713.04348<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>7/22<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>709.<u>09</u><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><br />
</td>
        <td>27/85<br />
</td>
        <td>705.88235<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>20/63<br />
</td>
        <td><br />
</td>
        <td>704.761905<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><br />
</td>
        <td>33/104<br />
</td>
        <td>703.84615<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>13/41<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>702.<u>43902</u><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><br />
</td>
        <td>32/101<br />
</td>
        <td>700.<u>9900</u><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>19/60<br />
</td>
        <td><br />
</td>
        <td>700<br />
</td>
        <td style="text-align: center;">Optimal patent val for 5-limit magic<br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>25/79<br />
</td>
        <td>698.73418<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td><br />
</td>
        <td>6/19<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>694.73684<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><br />
</td>
        <td>29/92<br />
</td>
        <td>691.30435<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>23/73<br />
</td>
        <td><br />
</td>
        <td>690.41096<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><br />
</td>
        <td>40/127<br />
</td>
        <td>689.76378<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>17/54<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>688.<u>8</u><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><br />
</td>
        <td>45/143<br />
</td>
        <td>688.11189<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>28/89<br />
</td>
        <td><br />
</td>
        <td>687.64045<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><br />
</td>
        <td>39/124<br />
</td>
        <td>687.09677<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>11/35<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>685.<u>714285</u><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><br />
</td>
        <td>38/121<br />
</td>
        <td>684.29752<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>27/86<br />
</td>
        <td><br />
</td>
        <td>683.72093<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><br />
</td>
        <td>43/137<br />
</td>
        <td>683.21168<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>16/51<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>682.35294<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><br />
</td>
        <td>37/118<br />
</td>
        <td>681.35593<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>21/67<br />
</td>
        <td><br />
</td>
        <td>680.597015<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><br />
</td>
        <td>26/83<br />
</td>
        <td>679.51807<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td>5/16<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>675<br />
</td>
        <td style="text-align: center;">Boundary of propriety<br />
(Smaller generators are proper)<br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>29/93<br />
</td>
        <td>670.96774<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>24/77<br />
</td>
        <td><br />
</td>
        <td>670.12987<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><br />
</td>
        <td>43/138<br />
</td>
        <td>669.56522<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>19/61<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>668.52459<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><br />
</td>
        <td>52/167<br />
</td>
        <td>668.26347<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>33/106<br />
</td>
        <td><br />
</td>
        <td>667.92452<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><br />
</td>
        <td>47/151<br />
</td>
        <td>667.54967<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>14/45<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>666.<u>6</u><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><br />
</td>
        <td>51/164<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>37/119<br />
</td>
        <td><br />
</td>
        <td>665.54622<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><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>23/74<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>664.<u>864</u><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><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>32/103<br />
</td>
        <td><br />
</td>
        <td>664.07767<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><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td><br />
</td>
        <td>9/29<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>662.068965<br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>13/42<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>657.14286<br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
    <tr>
        <td>4/13<br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td><br />
</td>
        <td>646.15385<br />
</td>
        <td style="text-align: center;"><br />
</td>
    </tr>
</table>

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3L 10s: Difference between revisions

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