Semicomma family: Difference between revisions

=Orson=  

The 5-limit parent comma for the **semicomma family** is the semicomma, 2109375/2097152 = |-21 3 7>. This is the amount by which three pure 3/1 twelfths exceed seven pure 8/5 minor sixths. **Orson**, the [[5-limit]] temperament tempering it out, has a [[generator]] of 75/64, which is sharper than 7/6 by 225/224 when untempered, and less sharp than that in any good orson tempering, for example [[53edo]] or [[84edo]]. These give tunings to the generator which are sharp of 7/6 by less than five [[cent]]s, making it hard to treat orson as anything other than an (at least) 7-limit system, leading to orwell.

[[Tuning Ranges of Regular Temperaments|valid range]]: [257.143, 276.923] (14b to 13)

[[POTE tuning|POTE generator]]: ~75/64 = 271.627

==Seven limit children==  

The second comma of the [[Normal lists|normal comma list]] defines which 7-limit family member we are looking at. Adding 65536/64625 leads to orwell, but we could also add 1029/1024, leading to the 31&159 temperament with wedgie <<21 -9 -7 -63 -70 9||, or 67528125/67108864, giving the 31&243 temperament with wedgie <<28 -12 1 -84 -77 36||, or 4375/4374, giving the 53&243 temperament with wedgie <<7 -3 61 -21 77 150||.

=Orwell=  

Main article: [[Orwell]]

So called because 19\84 (as a [[fraction of the octave]]) is a possible generator of this temperament, orwell divides the interval of a twelfth (a tempered 3/1 frequency ratio) into 7 equal steps. It's compatible with [[22edo|22]], [[31edo|31]], [[53edo|53]] and [[84edo|84]] equal, and may be described as the 22&31 temperament, or <<7 -3 8 -21 -7 27||. It's a good system in the [[7-limit]] and naturally extends into the [[11-limit]]. [[84edo]], with the 19\84 generator, provides a good tuning for the 5, 7 and 11 limits, but it does use its second-best 11. However, the 19\84 generator is remarkably close to the 11-limit [[POTE tuning]], as the generator is only 0.0024 cents sharper, and it is a good approximation to the 7-limit POTE generator also; hence 84 may be considered the most recommendable tuning in the 7-limit. [[53edo]] might be preferred in the 5-limit because of its nearly pure fifth and in the 11-limit because of it slightly better 11, though most of its 11-limit harmony is actually worse. Aside from the semicomma and 65625/65536, 7-limit orwell tempers out 2430/2401, the nuwell comma, 1728/1715, the orwellisma, 225/224, the septimal kleisma, and 6144/6125, the porwell comma.

Orwell has MOS of size 9, 13, 22 and 31. The 9-note MOS is small enough to be retained in the mind as a genuine scale, is pleasing melodically, and has [[Retuning 12edo to Orwell9|considerable harmonic resources]] despite its absence of 5-limit triads. The 13 note MOS has those, and of course the 22 and 31 note MOS are very well supplied with everything.

[[Fractional monzos|Eigenmonzos]]: 2, 7/5

[[POTE tuning|POTE generator]]: ~7/6 = 271.509

==11-limit==  

[[Minimax tuning]]

[[POTE tuning|POTE generator]]: ~7/6 = 271.426

[[edo|Edos]]: [[22edo|22]], [[31edo|31]], [[53edo|53]], [[84edo|84e]]

==13-limit==  

[[POTE tuning|POTE generator]]: ~7/6 = 271.546

==Blair==  

==Newspeak==  

==Winston==  

[[POTE tuning|POTE generator]]: ~7/6 = 271.088

=Doublethink=  

=Borwell=  

=Triwell=  

==11-limit==  

Badness: 0.0298</pre></div>

&lt;a class="wiki_link" href="/Tuning%20Ranges%20of%20Regular%20Temperaments"&gt;valid range&lt;/a&gt;: [257.143, 276.923] (14b to 13)&lt;br /&gt;

nice range: [271.229, 271.708]&lt;br /&gt;

Map: [&amp;lt;1 0 3|, &amp;lt;0 7 -3|]&lt;br /&gt;

The second comma of the &lt;a class="wiki_link" href="/Normal%20lists"&gt;normal comma list&lt;/a&gt; defines which 7-limit family member we are looking at. Adding 65536/64625 leads to orwell, but we could also add 1029/1024, leading to the 31&amp;amp;159 temperament with wedgie &amp;lt;&amp;lt;21 -9 -7 -63 -70 9||, or 67528125/67108864, giving the 31&amp;amp;243 temperament with wedgie &amp;lt;&amp;lt;28 -12 1 -84 -77 36||, or 4375/4374, giving the 53&amp;amp;243 temperament with wedgie &amp;lt;&amp;lt;7 -3 61 -21 77 150||.&lt;br /&gt;

|27/11 0 -8/11 8/11 0&amp;gt;, |37/11 0 -2/11 2/11 0&amp;gt;]&lt;br /&gt;

valid range: [270.968, 272.727] (31 to 22)&lt;br /&gt;

nice range: [266.871, 275.659]&lt;br /&gt;

strict range: [270.968, 272.727]&lt;br /&gt;

&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: ~7/6 = 271.426&lt;br /&gt;

Map: [&amp;lt;1 0 3 1 3|, &amp;lt;0 7 -3 8 2|]&lt;br /&gt;

valid range: [270.968, 271.698] (31 to 53)&lt;br /&gt;

nice range: [266.871, 275.659]&lt;br /&gt;

strict range: [270.968, 271.698]&lt;br /&gt;

&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: ~7/6 = 271.546&lt;br /&gt;

Map: [&amp;lt;1 0 3 1 3 8|, &amp;lt;0 7 -3 8 2 -19|]&lt;br /&gt;

valid range: []&lt;br /&gt;

nice range: [265.357, 289.210]&lt;br /&gt;

strict range: []&lt;br /&gt;

POTE generator: ~7/6 = 271.301&lt;br /&gt;

Map: [&amp;lt;1 0 3 1 3 3|, &amp;lt;0 7 -3 8 2 3|]&lt;br /&gt;

valid range: [270.968, 271.698] (31 to 53)&lt;br /&gt;

nice range: [266.871, 272.514]&lt;br /&gt;

strict range: [270.968, 271.698]&lt;br /&gt;

POTE tuning: ~7/6 = 271.288&lt;br /&gt;

Map: [&amp;lt;1 0 3 1 -4|, &amp;lt;0 7 -3 8 33|]&lt;br /&gt;

valid range: [270.968, 272.727] (31 to 22f)&lt;br /&gt;

nice range: [266.871, 281.691]&lt;br /&gt;

strict range: [270.968, 272.727]&lt;br /&gt;

&lt;a class="wiki_link" href="/POTE%20tuning"&gt;POTE generator&lt;/a&gt;: ~7/6 = 271.088&lt;br /&gt;

Map: [&amp;lt;1 0 3 1 3 1|, &amp;lt;0 7 -3 8 2 12|]&lt;br /&gt;

valid range: [135.484, 136.364] (62 to 44)&lt;br /&gt;

nice range: [128.298, 138.573]&lt;br /&gt;

strict range: [135.484, 136.364]&lt;br /&gt;

POTE tuning: ~13/12 = 135.723&lt;br /&gt;

Map: [&amp;lt;1 0 3 1 3 2|, &amp;lt;0 14 -6 16 4 15|]&lt;br /&gt;

Map: [&amp;lt;1 7 0 1 13|, &amp;lt;0 -21 9 7 -37]]&lt;br /&gt;