3L 6s: Difference between revisions

From Xenharmonic Wiki
Jump to navigation Jump to search
← Older edit
VisualWikitext
Wikispaces>JosephRuhf
**Imported revision 602432596 - Original comment: **
ArrowHead294 (talk | contribs)
14,512 edits
mNo edit summary
 
(24 intermediate revisions by 11 users not shown)
Line 1: Line 1:
<h2>IMPORTED REVISION FROM WIKISPACES</h2>
{{Infobox MOS
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
| Periods = 3
: This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on <tt>2016-12-18 07:27:51 UTC</tt>.<br>
| nLargeSteps = 3
: The original revision id was <tt>602432596</tt>.<br>
| nSmallSteps = 6
: The revision comment was: <tt></tt><br>
| Equalized = 1
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
| Collapsed = 0
<h4>Original Wikitext content:</h4>
| Pattern = LssLssLss
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">This MOS has generators which range between 0 and 133.333 cents and three periods per octave and runs Lss Lss Lss.
| Name = tcherepnin
}}
{{MOS intro}}


||||||||||~ Generator ||~ Cents ||~ Comments ||
== Name ==
|| 0\3 ||  ||  ||  ||  || 0 ||=   ||
[[TAMNAMS]] suggests the temperaement-agnostic name '''tcherepnin''' for this scale. This is named after Alexander Tcherepnin's 9-note scale which, in 12edo, is notated as C Db Eb E F G Ab A B C, forming the step pattern of HWHHWHHWH in standard notation (assuming a root of C).
||  ||  ||  || 1\18 ||  || 66.667 ||= L/s=4 ||
||  ||  ||  ||  ||  || 400/(2+pi) ||  ||
||  ||  || 1\15 ||  ||  || 80 ||= L/s=3 ||
||  ||  ||  ||  ||  || 400/(2+e) ||  ||
||  ||  ||  ||  || 3\42 || 85.714 ||  ||
||  ||  ||  ||  ||  || 400/(3+phi) ||  ||
||  ||  ||  || 2\27 ||  || 88.889 ||= Augene is around here ||
||  || 1\12 ||  ||  ||  || 100 ||= Boundary of propriety (generators
larger than this are proper) ||
||  ||  ||  ||  ||  || 400/(2+sqrt(3)) ||  ||
||  ||  ||  || 3\33 ||  || 109.091 ||= August is around here ||
||  ||  ||  ||  ||  || 400/(2+phi) ||  ||
||  ||  ||  ||  || 5\54 || 111.111 ||  ||
||  ||  ||  ||  ||  || 400/(2+pi/2) ||  ||
||  ||  || 2\21 ||  ||  || 114.286 ||=  ||
||  ||  ||  || 3\30 ||  || 120 ||=  ||
|| 1\9 ||  ||  ||  ||  || 133.333 ||=  ||


From a standard diatonic point of view, an optimized (MOD)MOS pattern of 3L+6s works out to become one of the modes of the Mohajira diatonic or Rast scale extended to a minor tenth, the MOS itself being the tritetrachordal Rast b10 scale, the interesting property of which being that it treats a seventh as a perfect interval rather than a fifth, however far off it may be from a perfect 7/4. As a temperament, it has period ~4/3 and generator up to 1/3 of that, making a subrange of the range between its median and maximum generators represent an ~12/11 (given harmonic entropy as coarse as that of [[Mavila]], 120 cents can be an acceptable 12\11).
== Scale properties ==
{{TAMNAMS use}}


||||||||||~ Minor Tenth ||~ Fourth ||~ Hemififth ||||||||~ &lt;span style="display: block; text-align: center;"&gt;Generator&lt;/span&gt; ||
=== Intervals ===
||||||||||~  ||~  ||~  ||~ &lt;span style="display: block; text-align: center;"&gt;Mean&lt;/span&gt; ||~ &lt;span style="display: block; text-align: center;"&gt;Median&lt;/span&gt; ||~ &lt;span style="display: block; text-align: center;"&gt;Golden&lt;/span&gt; ||~ Maximum ||
{{MOS intervals}}
|| 9\7 ||  ||  ||  ||  || 514.286 || 342.857 || 85.714 || 128.571 || 142.145 || 171.429 ||
||  ||  ||  ||  || 48\38 || 505.263 || 347.368 || 84.2105 || 126.316 || 139.651 || 168.421 ||
||  ||  ||  || 39\31 ||  || 503.226 || 348.387 || 83.871 || 125.806'5 || 139.088 || 167.742 ||
||  ||  ||  ||  || 69\55 || 506.667 || 349.091 || 83.636 || 125.454'5 || 138.699 || 167.273 ||
||  ||  || 30\24 ||  ||  || 500 || 350 || 83.333 || 125 || 138.197 || 166.667 ||
||  ||  ||  ||  || 81\65 || 498.461'5 || 350.769 || 83.076 || 124.615 || 137.771 || 166.154 ||
||  ||  ||  || 51\41 ||  || 497.561 || 351.219'5 || 82.927 || 124.39 || 137.522'5 || 165.854 ||
||  ||  ||  ||  || 72\58 || 496.552 || 351.724 || 82.758 || 124.137 || 137.243'5 || 165.517 ||
||  || 21\17 ||  ||  ||  || 494.116 || 325.941 || 82.352 || 123.529 || 136.571 || 164.706 ||
||  ||  ||  ||  || 75\61 || 491.803 || 354.098 || 81.967 || 122.951 || 135.931 || 163.934 ||
||  ||  ||  || 54\44 ||  || 490.909 || 354.545'5 || 81.818 || 122.727 || 135.684 || 163.636 ||
||  ||  ||  ||  || 87\71 || 490.141 || 354.93 || 81.69 || 122.535 || 135.472 || 163.38 ||
||  ||  || 33\27 ||  ||  || 488.889 || 355.556 || 81.481 || 122.222 || 135.126 || 162.462 ||
||  ||  ||  ||  || 78\64 || 487.5 || 356.25 || 81.25 || 121.875 || 134.742 || 162.5 ||
||  ||  ||  || 45\37 ||  || 486.486'5 || 356.757 || 81.081 || 121.622 || 134.462 || 162.162 ||
||  ||  ||  ||  || 57\47 || 485.106 || 357.447 || 80.851 || 121.277 || 134.08 || 162.702 ||
|| 12\10 ||  ||  ||  ||  || 480 || 360 || 80 || 120 || 132.669 || 160 ||


From an antidiatonic point of view, an optimized (MOD)MOS pattern of 3L+6s works out to become one of the modes of the (Neapolitan/Melodic) antimajor scale extended to a major tenth, the MOS itself being the tritetrachordal Antimixolydian p10 scale, the interesting property of which being that it treats a seventh as a perfect interval rather than a fifth, however far off it may be from a perfect 11\6. As a temperament, it has period ~4/3 and generator up to 1/3 of that, making the range between its median and maximum generators represent an ~12/11 (given that it is an extended mode of [[Mavila]], 178 cents can be an acceptable 12\11).
=== Generator chain ===
{{MOS genchain}}


||||||||||~ Major Tenth ||~ Fourth ||~  ||||||||||~ Generator ||
=== Modes ===
||||||||||~  ||~  ||~ &lt;span style="display: block; text-align: center;"&gt;Mean&lt;/span&gt; ||||~ &lt;span style="display: block; text-align: center;"&gt;Median&lt;/span&gt; ||~ Golden ||~  ||~ Maximum ||
{{MOS mode degrees}}
|| 9\7 ||  ||  ||  ||  || 514.286 || 85.714 |||| 128.571 |||| 142.145 || 171.429 ||
||  ||  ||  ||  || 48\37 || 518.919 || 86.4865 |||| 129.73 |||| 143.426 || 172.973 ||
||  ||  ||  || 39\30 ||  || 520 || 86.667 |||| 130 |||| 143.724'5 || 173.333 ||
||  ||  ||  ||  || 69\53 || 520.755 || 86.7925 |||| 130.189 |||| 143.933 || 173.585 ||
||  ||  || 30\23 ||  ||  || 521.739 || 86.9565 |||| 130.435 |||| 144.205 || 173.913 ||
||  ||  ||  ||  || 81\62 || 522.582 || 87.079 |||| 130.645 |||| 144.438 || 174.194 ||
||  ||  ||  || 51\39 ||  || 523.077 || 87.1795 |||| 130.769 |||| 144.575 || 174.359 ||
||  ||  ||  ||  || 72\55 || 523.636 || 87.273 |||| 130.909 |||| 144.729'5 || 174.545'5 ||
||  || 21\16 ||  ||  ||  || 525 || 87.5 |||| 131.25 |||| 145.106 || 175 ||
||  ||  ||  ||  || 75\57 || 526.316 || 87.719 |||| 131.579 |||| 145.47 || 175.439 ||
||  ||  ||  || 54\41 ||  || 526.829 || 87.805 |||| 131.707 |||| 145.612 || 175.61 ||
||  ||  ||  ||  || 87\66 || 527.273 || 87.879 |||| 131.818 |||| 145.735 || 175.758 ||
||  ||  || 33\25 ||  ||  || 528 || 88 |||| 132 |||| 145.936 || 176 ||
||  ||  ||  ||  || 78\59 || 528.814 || 88.136 |||| 132.203 |||| 146.160'5 || 176.271 ||
||  ||  ||  || 45\34 ||  || 529.412 || 88.235 |||| 132.353 |||| 146.326 || 176.471 ||
||  ||  ||  ||  || 57\43 || 530.233 || 88.372 |||| 132.558 |||| 146.553 || 176.744 ||
|| 12\9 ||  ||  ||  ||  || 533.333 || 88.889 |||| 133.333 |||| 147.41 || 177.778 ||</pre></div>
<h4>Original HTML content:</h4>
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;3L 6s&lt;/title&gt;&lt;/head&gt;&lt;body&gt;This MOS has generators which range between 0 and 133.333 cents and three periods per octave and runs Lss Lss Lss.&lt;br /&gt;
&lt;br /&gt;


== Scale tree ==
{{MOS tuning spectrum
| 9/5 = August
| 12/5 = Augene
| 11/4 = August
| 10/3 = Inflated
| 6/1 = Hemiug, hemiaug&nbsp;↓
}}


&lt;table class="wiki_table"&gt;
== External links ==
    &lt;tr&gt;
* Piano Encyclopedia, on Tcherepnin's 9-note scale: https://pianoencyclopedia.com/scales/tcherepnin-nine-note-mode-i/
        &lt;th colspan="5"&gt;Generator&lt;br /&gt;
&lt;/th&gt;
        &lt;th&gt;Cents&lt;br /&gt;
&lt;/th&gt;
        &lt;th&gt;Comments&lt;br /&gt;
&lt;/th&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;0\3&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;0&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&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;1\18&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;66.667&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;L/s=4&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;400/(2+pi)&lt;br /&gt;
&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;1\15&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;80&lt;br /&gt;
&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;400/(2+e)&lt;br /&gt;
&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;3\42&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;85.714&lt;br /&gt;
&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;400/(3+phi)&lt;br /&gt;
&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;2\27&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;88.889&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Augene is around here&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;1\12&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;100&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;Boundary of propriety (generators&lt;br /&gt;
larger than this are proper)&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;400/(2+sqrt(3))&lt;br /&gt;
&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;3\33&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;109.091&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;August is around here&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;400/(2+phi)&lt;br /&gt;
&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;5\54&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;111.111&lt;br /&gt;
&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;400/(2+pi/2)&lt;br /&gt;
&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;2\21&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;114.286&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&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;3\30&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;120&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;1\9&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;133.333&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
&lt;/table&gt;


&lt;br /&gt;
[[Category:Tcherepnin| ]]
From a standard diatonic point of view, an optimized (MOD)MOS pattern of 3L+6s works out to become one of the modes of the Mohajira diatonic or Rast scale extended to a minor tenth, the MOS itself being the tritetrachordal Rast b10 scale, the interesting property of which being that it treats a seventh as a perfect interval rather than a fifth, however far off it may be from a perfect 7/4. As a temperament, it has period ~4/3 and generator up to 1/3 of that, making a subrange of the range between its median and maximum generators represent an ~12/11 (given harmonic entropy as coarse as that of &lt;a class="wiki_link" href="/Mavila"&gt;Mavila&lt;/a&gt;, 120 cents can be an acceptable 12\11).&lt;br /&gt;
[[Category:9-tone scales]]
&lt;br /&gt;
<!--main article-->
 
 
&lt;table class="wiki_table"&gt;
    &lt;tr&gt;
        &lt;th colspan="5"&gt;Minor Tenth&lt;br /&gt;
&lt;/th&gt;
        &lt;th&gt;Fourth&lt;br /&gt;
&lt;/th&gt;
        &lt;th&gt;Hemififth&lt;br /&gt;
&lt;/th&gt;
        &lt;th colspan="4"&gt;&lt;span style="display: block; text-align: center;"&gt;Generator&lt;/span&gt;&lt;br /&gt;
&lt;/th&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;th colspan="5"&gt;&lt;br /&gt;
&lt;/th&gt;
        &lt;th&gt;&lt;br /&gt;
&lt;/th&gt;
        &lt;th&gt;&lt;br /&gt;
&lt;/th&gt;
        &lt;th&gt;&lt;span style="display: block; text-align: center;"&gt;Mean&lt;/span&gt;&lt;br /&gt;
&lt;/th&gt;
        &lt;th&gt;&lt;span style="display: block; text-align: center;"&gt;Median&lt;/span&gt;&lt;br /&gt;
&lt;/th&gt;
        &lt;th&gt;&lt;span style="display: block; text-align: center;"&gt;Golden&lt;/span&gt;&lt;br /&gt;
&lt;/th&gt;
        &lt;th&gt;Maximum&lt;br /&gt;
&lt;/th&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;9\7&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;514.286&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;342.857&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;85.714&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;128.571&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;142.145&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;171.429&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;48\38&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;505.263&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;347.368&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;84.2105&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;126.316&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;139.651&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;168.421&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;39\31&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;503.226&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;348.387&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;83.871&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;125.806'5&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;139.088&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;167.742&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;69\55&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;506.667&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;349.091&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;83.636&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;125.454'5&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;138.699&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;167.273&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;30\24&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;500&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;350&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;83.333&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;125&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;138.197&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;166.667&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;81\65&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;498.461'5&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;350.769&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;83.076&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;124.615&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;137.771&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;166.154&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;51\41&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;497.561&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;351.219'5&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;82.927&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;124.39&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;137.522'5&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;165.854&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;72\58&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;496.552&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;351.724&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;82.758&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;124.137&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;137.243'5&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;165.517&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;21\17&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;494.116&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;325.941&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;82.352&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;123.529&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;136.571&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;164.706&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;75\61&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;491.803&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;354.098&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;81.967&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;122.951&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;135.931&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;163.934&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;54\44&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;490.909&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;354.545'5&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;81.818&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;122.727&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;135.684&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;163.636&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;87\71&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;490.141&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;354.93&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;81.69&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;122.535&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;135.472&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;163.38&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;33\27&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;488.889&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;355.556&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;81.481&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;122.222&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;135.126&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;162.462&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;78\64&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;487.5&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;356.25&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;81.25&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;121.875&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;134.742&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;162.5&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;45\37&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;486.486'5&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;356.757&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;81.081&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;121.622&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;134.462&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;162.162&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;57\47&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;485.106&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;357.447&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;80.851&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;121.277&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;134.08&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;162.702&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;12\10&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;480&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;360&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;80&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;120&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;132.669&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;160&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
&lt;/table&gt;
 
&lt;br /&gt;
From an antidiatonic point of view, an optimized (MOD)MOS pattern of 3L+6s works out to become one of the modes of the (Neapolitan/Melodic) antimajor scale extended to a major tenth, the MOS itself being the tritetrachordal Antimixolydian p10 scale, the interesting property of which being that it treats a seventh as a perfect interval rather than a fifth, however far off it may be from a perfect 11\6. As a temperament, it has period ~4/3 and generator up to 1/3 of that, making the range between its median and maximum generators represent an ~12/11 (given that it is an extended mode of &lt;a class="wiki_link" href="/Mavila"&gt;Mavila&lt;/a&gt;, 178 cents can be an acceptable 12\11).&lt;br /&gt;
&lt;br /&gt;
 
 
&lt;table class="wiki_table"&gt;
    &lt;tr&gt;
        &lt;th colspan="5"&gt;Major Tenth&lt;br /&gt;
&lt;/th&gt;
        &lt;th&gt;Fourth&lt;br /&gt;
&lt;/th&gt;
        &lt;th&gt;&lt;br /&gt;
&lt;/th&gt;
        &lt;th colspan="5"&gt;Generator&lt;br /&gt;
&lt;/th&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;th colspan="5"&gt;&lt;br /&gt;
&lt;/th&gt;
        &lt;th&gt;&lt;br /&gt;
&lt;/th&gt;
        &lt;th&gt;&lt;span style="display: block; text-align: center;"&gt;Mean&lt;/span&gt;&lt;br /&gt;
&lt;/th&gt;
        &lt;th colspan="2"&gt;&lt;span style="display: block; text-align: center;"&gt;Median&lt;/span&gt;&lt;br /&gt;
&lt;/th&gt;
        &lt;th&gt;Golden&lt;br /&gt;
&lt;/th&gt;
        &lt;th&gt;&lt;br /&gt;
&lt;/th&gt;
        &lt;th&gt;Maximum&lt;br /&gt;
&lt;/th&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;9\7&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;514.286&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;85.714&lt;br /&gt;
&lt;/td&gt;
        &lt;td colspan="2"&gt;128.571&lt;br /&gt;
&lt;/td&gt;
        &lt;td colspan="2"&gt;142.145&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;171.429&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;48\37&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;518.919&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;86.4865&lt;br /&gt;
&lt;/td&gt;
        &lt;td colspan="2"&gt;129.73&lt;br /&gt;
&lt;/td&gt;
        &lt;td colspan="2"&gt;143.426&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;172.973&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;39\30&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;520&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;86.667&lt;br /&gt;
&lt;/td&gt;
        &lt;td colspan="2"&gt;130&lt;br /&gt;
&lt;/td&gt;
        &lt;td colspan="2"&gt;143.724'5&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;173.333&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;69\53&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;520.755&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;86.7925&lt;br /&gt;
&lt;/td&gt;
        &lt;td colspan="2"&gt;130.189&lt;br /&gt;
&lt;/td&gt;
        &lt;td colspan="2"&gt;143.933&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;173.585&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;30\23&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;521.739&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;86.9565&lt;br /&gt;
&lt;/td&gt;
        &lt;td colspan="2"&gt;130.435&lt;br /&gt;
&lt;/td&gt;
        &lt;td colspan="2"&gt;144.205&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;173.913&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;81\62&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;522.582&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;87.079&lt;br /&gt;
&lt;/td&gt;
        &lt;td colspan="2"&gt;130.645&lt;br /&gt;
&lt;/td&gt;
        &lt;td colspan="2"&gt;144.438&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;174.194&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;51\39&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;523.077&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;87.1795&lt;br /&gt;
&lt;/td&gt;
        &lt;td colspan="2"&gt;130.769&lt;br /&gt;
&lt;/td&gt;
        &lt;td colspan="2"&gt;144.575&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;174.359&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;72\55&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;523.636&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;87.273&lt;br /&gt;
&lt;/td&gt;
        &lt;td colspan="2"&gt;130.909&lt;br /&gt;
&lt;/td&gt;
        &lt;td colspan="2"&gt;144.729'5&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;174.545'5&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;21\16&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;525&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;87.5&lt;br /&gt;
&lt;/td&gt;
        &lt;td colspan="2"&gt;131.25&lt;br /&gt;
&lt;/td&gt;
        &lt;td colspan="2"&gt;145.106&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;175&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;75\57&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;526.316&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;87.719&lt;br /&gt;
&lt;/td&gt;
        &lt;td colspan="2"&gt;131.579&lt;br /&gt;
&lt;/td&gt;
        &lt;td colspan="2"&gt;145.47&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;175.439&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;54\41&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;526.829&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;87.805&lt;br /&gt;
&lt;/td&gt;
        &lt;td colspan="2"&gt;131.707&lt;br /&gt;
&lt;/td&gt;
        &lt;td colspan="2"&gt;145.612&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;175.61&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;87\66&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;527.273&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;87.879&lt;br /&gt;
&lt;/td&gt;
        &lt;td colspan="2"&gt;131.818&lt;br /&gt;
&lt;/td&gt;
        &lt;td colspan="2"&gt;145.735&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;175.758&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;33\25&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;528&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;88&lt;br /&gt;
&lt;/td&gt;
        &lt;td colspan="2"&gt;132&lt;br /&gt;
&lt;/td&gt;
        &lt;td colspan="2"&gt;145.936&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;176&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;78\59&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;528.814&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;88.136&lt;br /&gt;
&lt;/td&gt;
        &lt;td colspan="2"&gt;132.203&lt;br /&gt;
&lt;/td&gt;
        &lt;td colspan="2"&gt;146.160'5&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;176.271&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;45\34&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;529.412&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;88.235&lt;br /&gt;
&lt;/td&gt;
        &lt;td colspan="2"&gt;132.353&lt;br /&gt;
&lt;/td&gt;
        &lt;td colspan="2"&gt;146.326&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;176.471&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;57\43&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;530.233&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;88.372&lt;br /&gt;
&lt;/td&gt;
        &lt;td colspan="2"&gt;132.558&lt;br /&gt;
&lt;/td&gt;
        &lt;td colspan="2"&gt;146.553&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;176.744&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;12\9&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;533.333&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;88.889&lt;br /&gt;
&lt;/td&gt;
        &lt;td colspan="2"&gt;133.333&lt;br /&gt;
&lt;/td&gt;
        &lt;td colspan="2"&gt;147.41&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;177.778&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
&lt;/table&gt;
 
&lt;/body&gt;&lt;/html&gt;</pre></div>

Latest revision as of 18:59, 3 March 2025

↖ 2L 5s ↑ 3L 5s 4L 5s ↗
← 2L 6s 3L 6s 4L 6s →
↙ 2L 7s ↓ 3L 7s 4L 7s ↘ 
┌╥┬┬╥┬┬╥┬┬┐
│║││║││║│││
│││││││││││
└┴┴┴┴┴┴┴┴┴┘
Scale structure
Step pattern LssLssLss
ssLssLssL
Equave 2/1 (1200.0 ¢)
Period 1\3 (400.0 ¢)
Generator size
Bright 2\9 to 1\3 (266.7 ¢ to 400.0 ¢)
Dark 0\3 to 1\9 (0.0 ¢ to 133.3 ¢)
TAMNAMS information
Name tcherepnin
Prefix cher-
Abbrev. ch
Related MOS scales
Parent 3L 3s
Sister 6L 3s
Daughters 9L 3s, 3L 9s
Neutralized 6L 3s
2-Flought 12L 6s, 3L 15s
Equal tunings
Equalized (L:s = 1:1) 2\9 (266.7 ¢)
Supersoft (L:s = 4:3) 7\30 (280.0 ¢)
Soft (L:s = 3:2) 5\21 (285.7 ¢)
Semisoft (L:s = 5:3) 8\33 (290.9 ¢)
Basic (L:s = 2:1) 3\12 (300.0 ¢)
Semihard (L:s = 5:2) 7\27 (311.1 ¢)
Hard (L:s = 3:1) 4\15 (320.0 ¢)
Superhard (L:s = 4:1) 5\18 (333.3 ¢)
Collapsed (L:s = 1:0) 1\3 (400.0 ¢)

3L 6s, named tcherepnin in TAMNAMS, is a 2/1-equivalent (octave-equivalent) moment of symmetry scale containing 3 large steps and 6 small steps, with a period of 1 large step and 2 small steps that repeats every 400.0 ¢, or 3 times every octave. Generators that produce this scale range from 266.7 ¢ to 400 ¢, or from 0 ¢ to 133.3 ¢.

Name

TAMNAMS suggests the temperaement-agnostic name tcherepnin for this scale. This is named after Alexander Tcherepnin's 9-note scale which, in 12edo, is notated as C Db Eb E F G Ab A B C, forming the step pattern of HWHHWHHWH in standard notation (assuming a root of C).

Scale properties

This article uses TAMNAMS conventions for the names of this scale's intervals and scale degrees. The use of 1-indexed ordinal names is reserved for interval regions.

Intervals

Intervals of 3L 6s
Intervals Steps
subtended 		Range in cents
Generic Specific Abbrev.
0-cherstep Perfect 0-cherstep P0chs 0 0.0 ¢
1-cherstep Perfect 1-cherstep P1chs s 0.0 ¢ to 133.3 ¢
Augmented 1-cherstep A1chs L 133.3 ¢ to 400.0 ¢
2-cherstep Diminished 2-cherstep d2chs 2s 0.0 ¢ to 266.7 ¢
Perfect 2-cherstep P2chs L + s 266.7 ¢ to 400.0 ¢
3-cherstep Perfect 3-cherstep P3chs L + 2s 400.0 ¢
4-cherstep Perfect 4-cherstep P4chs L + 3s 400.0 ¢ to 533.3 ¢
Augmented 4-cherstep A4chs 2L + 2s 533.3 ¢ to 800.0 ¢
5-cherstep Diminished 5-cherstep d5chs L + 4s 400.0 ¢ to 666.7 ¢
Perfect 5-cherstep P5chs 2L + 3s 666.7 ¢ to 800.0 ¢
6-cherstep Perfect 6-cherstep P6chs 2L + 4s 800.0 ¢
7-cherstep Perfect 7-cherstep P7chs 2L + 5s 800.0 ¢ to 933.3 ¢
Augmented 7-cherstep A7chs 3L + 4s 933.3 ¢ to 1200.0 ¢
8-cherstep Diminished 8-cherstep d8chs 2L + 6s 800.0 ¢ to 1066.7 ¢
Perfect 8-cherstep P8chs 3L + 5s 1066.7 ¢ to 1200.0 ¢
9-cherstep Perfect 9-cherstep P9chs 3L + 6s 1200.0 ¢

Generator chain

Generator chain of 3L 6s
Bright gens Scale degree Abbrev. Scale degree Abbrev. Scale degree Abbrev.
3 Augmented 0-cherdegree A0chd Augmented 3-cherdegree A3chd Augmented 6-cherdegree A6chd
2 Augmented 1-cherdegree A1chd Augmented 4-cherdegree A4chd Augmented 7-cherdegree A7chd
1 Perfect 2-cherdegree P2chd Perfect 5-cherdegree P5chd Perfect 8-cherdegree P8chd
0 Perfect 0-cherdegree
Perfect 3-cherdegree		 P0chd
P3chd		 Perfect 3-cherdegree
Perfect 6-cherdegree		 P3chd
P6chd		 Perfect 6-cherdegree
Perfect 9-cherdegree		 P6chd
P9chd
−1 Perfect 1-cherdegree P1chd Perfect 4-cherdegree P4chd Perfect 7-cherdegree P7chd
−2 Diminished 2-cherdegree d2chd Diminished 5-cherdegree d5chd Diminished 8-cherdegree d8chd
−3 Diminished 3-cherdegree d3chd Diminished 6-cherdegree d6chd Diminished 9-cherdegree d9chd

Modes

Scale degrees of the modes of 3L 6s
UDP Cyclic
order 		Step
pattern 		Scale degree (cherdegree)
0 1 2 3 4 5 6 7 8 9
6|0(3) 1 LssLssLss Perf. Aug. Perf. Perf. Aug. Perf. Perf. Aug. Perf. Perf.
3|3(3) 3 sLssLssLs Perf. Perf. Perf. Perf. Perf. Perf. Perf. Perf. Perf. Perf.
0|6(3) 2 ssLssLssL Perf. Perf. Dim. Perf. Perf. Dim. Perf. Perf. Dim. Perf.

Scale tree

Scale tree and tuning spectrum of 3L 6s
Generator(edo) Cents Step ratio Comments
Bright Dark L:s Hardness
2\9 266.667 133.333 1:1 1.000 Equalized 3L 6s
11\48 275.000 125.000 6:5 1.200
9\39 276.923 123.077 5:4 1.250
16\69 278.261 121.739 9:7 1.286
7\30 280.000 120.000 4:3 1.333 Supersoft 3L 6s
19\81 281.481 118.519 11:8 1.375
12\51 282.353 117.647 7:5 1.400
17\72 283.333 116.667 10:7 1.429
5\21 285.714 114.286 3:2 1.500 Soft 3L 6s
18\75 288.000 112.000 11:7 1.571
13\54 288.889 111.111 8:5 1.600
21\87 289.655 110.345 13:8 1.625
8\33 290.909 109.091 5:3 1.667 Semisoft 3L 6s
19\78 292.308 107.692 12:7 1.714
11\45 293.333 106.667 7:4 1.750
14\57 294.737 105.263 9:5 1.800 August
3\12 300.000 100.000 2:1 2.000 Basic 3L 6s
Scales with tunings softer than this are proper
13\51 305.882 94.118 9:4 2.250
10\39 307.692 92.308 7:3 2.333
17\66 309.091 90.909 12:5 2.400 Augene
7\27 311.111 88.889 5:2 2.500 Semihard 3L 6s
18\69 313.043 86.957 13:5 2.600
11\42 314.286 85.714 8:3 2.667
15\57 315.789 84.211 11:4 2.750 August
4\15 320.000 80.000 3:1 3.000 Hard 3L 6s
13\48 325.000 75.000 10:3 3.333 Inflated
9\33 327.273 72.727 7:2 3.500
14\51 329.412 70.588 11:3 3.667
5\18 333.333 66.667 4:1 4.000 Superhard 3L 6s
11\39 338.462 61.538 9:2 4.500
6\21 342.857 57.143 5:1 5.000
7\24 350.000 50.000 6:1 6.000 Hemiug, hemiaug ↓
1\3 400.000 0.000 1:0 → ∞ Collapsed 3L 6s

External links

Retrieved from "https://en.xen.wiki/index.php?title=3L_6s&oldid=184350"