4L 2s
IMPORTED REVISION FROM WIKISPACES
This is an imported revision from Wikispaces. The revision metadata is included below for reference:
- This revision was by author JosephRuhf and made on 2015-11-03 14:36:22 UTC.
- The original revision id was 564990321.
- The revision comment was:
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.
Original Wikitext content:
There are three scales with this [[MOSScales|MOS]] pattern that are significant minima of harmonic entropy. The first is [[Chromatic pairs#Antikythera|antikythera]], or no-3's [[Diaschismic family|srutal/pajara]], which is srutal/pajara without any intervals containing 3 in their prime factorization, so it becomes a 2.5.9 or 2.5.7.9 subgroup temperament. This means that the generator is twice that of srutal/pajara (210-220 cents rather than 105-110), since odd numbers of generators are only needed for intervals with 3. So this is a basically "whole tone" scale, but made uneven so some 2-step intervals are 5/4 and others are 9/7. The second is [[Dicot family|decimal]], in which two generators make a 4/3, and the third is [[Jubilismic clan|Doublewide]], in which the generator is 7/6 so the period minus the generator is 6/5. In addition to the true MOS with pattern LLsLLs, all these scales also come in a near-MOS version, LLLsLs, in which the period is the only generic interval that has more than two specific representatives. The true MOS is always proper, but the near-MOS is only proper if the generator is smaller than 2\10 of an octave (240 cents). ||||||||||||||||||||||~ Generator ||~ Cents ||~ Comments || || 1\6 || || || || || || || || || || || 200 ||= || || || || || || || 6\34 || || || || || || 211.76 ||= || || || || || || 5\28 || || || || || || || 214.29 ||= Antikythera is around here || || || || || 4\22 || || || || || || || || 218.18 ||= || || || || 3\16 || || || || || || || || || 225 ||= || || || || || || || || || || || || || 227.56 || || || || || || || 8\42 || || || || || || || 228.57 ||= || || || || || || || || || || || || || 600/(1+phi) ||= Golden lemba || || || || || || || 13\68 || || || || || || 229.41 ||= || || || || || 5\26 || || || || || || || || 230.77 ||= || || || || || || || || || || || || || 232.8 || || || || || || || 7\36 || || || || || || || 233.33 ||= Lemba is around here || || || 2\10 || || || || || || || || || || 240 ||= Boundary of propriety for near-MOS Optimum rank range (L/s=2/1) for MOS || || || || || 5\24 || || || || || || || || 250 ||= Decimal is around here || || || || || || || || || || || || || 251.89 || || || || || || || 8\38 || || || || || || || 252.63 || || || || || || || || || || || || || || 253.39 ||= <span style="display: block; text-align: center;">L/s = e</span> || || || || 3\14 || || || || || || || || || 257.14 ||= L/s = 3 || || || || || || || || || || || || || 258.81 ||= L/s = pi || || || || || 4\18 || || || || || || || || 266.67 ||= L/s = 4 || || || || || || 5\22 || || || || || || || 272.73 ||= || || || || || || || 6\26 || || || || || || 276.92 ||= || || || || || || || || 7\30 || || || || || 280 ||= || || || || || || || || || 8\34 || || || || 282.35 ||= || || || || || || || || || || 9\38 || || || 284.21 ||= || || || || || || || || || || || 10\42 || || 285.71 ||= || || || || || || || || || || || || 11\46 || 286.96 ||= Doublewide is around here || || 1\4 || || || || || || || || || || || 300 ||= ||
Original HTML content:
<html><head><title>4L 2s</title></head><body>There are three scales with this <a class="wiki_link" href="/MOSScales">MOS</a> pattern that are significant minima of harmonic entropy.<br />
<br />
The first is <a class="wiki_link" href="/Chromatic%20pairs#Antikythera">antikythera</a>, or no-3's <a class="wiki_link" href="/Diaschismic%20family">srutal/pajara</a>, which is srutal/pajara without any intervals containing 3 in their prime factorization, so it becomes a 2.5.9 or 2.5.7.9 subgroup temperament. This means that the generator is twice that of srutal/pajara (210-220 cents rather than 105-110), since odd numbers of generators are only needed for intervals with 3. So this is a basically "whole tone" scale, but made uneven so some 2-step intervals are 5/4 and others are 9/7.<br />
<br />
The second is <a class="wiki_link" href="/Dicot%20family">decimal</a>, in which two generators make a 4/3, and the third is <a class="wiki_link" href="/Jubilismic%20clan">Doublewide</a>, in which the generator is 7/6 so the period minus the generator is 6/5.<br />
<br />
In addition to the true MOS with pattern LLsLLs, all these scales also come in a near-MOS version, LLLsLs, in which the period is the only generic interval that has more than two specific representatives. The true MOS is always proper, but the near-MOS is only proper if the generator is smaller than 2\10 of an octave (240 cents).<br />
<br />
<table class="wiki_table">
<tr>
<th colspan="11">Generator<br />
</th>
<th>Cents<br />
</th>
<th>Comments<br />
</th>
</tr>
<tr>
<td>1\6<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>
<td><br />
</td>
<td><br />
</td>
<td>200<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>6\34<br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td>211.76<br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td>5\28<br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td>214.29<br />
</td>
<td style="text-align: center;">Antikythera is around here<br />
</td>
</tr>
<tr>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td>4\22<br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td>218.18<br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td><br />
</td>
<td><br />
</td>
<td>3\16<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>
<td>225<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>
<td><br />
</td>
<td><br />
</td>
<td>227.56<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td>8\42<br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td>228.57<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>
<td><br />
</td>
<td><br />
</td>
<td>600/(1+phi)<br />
</td>
<td style="text-align: center;">Golden lemba<br />
</td>
</tr>
<tr>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td>13\68<br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td>229.41<br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
<tr>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td>5\26<br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td>230.77<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>
<td><br />
</td>
<td><br />
</td>
<td>232.8<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td>7\36<br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td>233.33<br />
</td>
<td style="text-align: center;">Lemba is around here<br />
</td>
</tr>
<tr>
<td><br />
</td>
<td>2\10<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>
<td><br />
</td>
<td>240<br />
</td>
<td style="text-align: center;">Boundary of propriety for near-MOS<br />
Optimum rank range (L/s=2/1) for MOS<br />
</td>
</tr>
<tr>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td>5\24<br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td>250<br />
</td>
<td style="text-align: center;">Decimal 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><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td>251.89<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td>8\38<br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td>252.63<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><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td>253.39<br />
</td>
<td style="text-align: center;"><span style="display: block; text-align: center;">L/s = e</span><br />
</td>
</tr>
<tr>
<td><br />
</td>
<td><br />
</td>
<td>3\14<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>
<td>257.14<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><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td>258.81<br />
</td>
<td style="text-align: center;">L/s = pi<br />
</td>
</tr>
<tr>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td>4\18<br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td>266.67<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>5\22<br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td>272.73<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>6\26<br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td>276.92<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>7\30<br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td>280<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>8\34<br />
</td>
<td><br />
</td>
<td><br />
</td>
<td><br />
</td>
<td>282.35<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>9\38<br />
</td>
<td><br />
</td>
<td><br />
</td>
<td>284.21<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>
<td>10\42<br />
</td>
<td><br />
</td>
<td>285.71<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>
<td><br />
</td>
<td>11\46<br />
</td>
<td>286.96<br />
</td>
<td style="text-align: center;">Doublewide is around here<br />
</td>
</tr>
<tr>
<td>1\4<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>
<td><br />
</td>
<td><br />
</td>
<td>300<br />
</td>
<td style="text-align: center;"><br />
</td>
</tr>
</table>
</body></html>