Semicomma family
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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 thirds. 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. [[POTE tuning|POTE generator]]: 271.627 Map: [<1 0 3|, <0 7 -3|] EDOs: 22, 31, 53, 190, 253, 296 ==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=== 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-EDO]] and [[84edo]], 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. The 11-limit version of orwell tempers out 99/98, which means that two of its sharpened 7/6 generators give a flattened 11/8, as well as 121/120, 176/175, 385/384 and 540/539. Despite lowered tuning accuracy, orwell comes into its own in the 11-limit, giving acceptable accuracy and relatively low complexity. Tempering out the orwellisma, 1728/1715, means that orwell interprets three stacked 7/6 generators as an 8/5, and the tempered 1-7/6-11/8-8/5 chord is natural to orwell. 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 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. ===Vital statistics=== [[Comma|Commas]]: 225/224, 1728/1715 7-limit [|1 0 0 0>, |14/11 0 -7/11 7/11>, |27/11 0 3/11 -3/11>, |27/11 0 -8/11 8/11>] [[Fractional monzos|Eigenmonzos]]: 2, 7/5 9-limit [|1 0 0 0>, |21/17 14/17 -7/17 0>, |42/17 -6/17 3/17 0>, |41/17 16/17 -8/17 0>] [[Eigenmonzo|Eigenmonzos]]: 2, 10/9 [[POTE tuning|POTE generator]]: 271.509 Algebraic generators: Sabra3, the real root of 12x^3-7x-48. Map: [<1 0 3 1|, <0 7 -3 8|] EDOs: 22, 31, 53, 84, 137 11-limit [[Comma|Commas]]: 99/98, 121/120, 176/175 [[Minimax tuning]] [|1 0 0 0 0>, |14/11 0 -7/11 7/11 0>, |27/11 0 3/11 -3/11 0>, |27/11 0 -8/11 8/11 0>, |37/11 0 -2/11 2/11 0>] [[Eigenmonzo|Eigenmonzos]]: 2, 7/5 [[POTE tuning|POTE generator]]: 271.426 Map: [<1 0 3 1 3|, <0 7 -3 8 2|] [[edo|Edos]]: [[22edo|22]], [[31edo|31]], [[53edo|53]], [[84edo|84]] ==Music== http://www.archive.org/details/TrioInOrwell [[http://soundclick.com/share?songid=9101705|one drop of rain]], [[http://soundclick.com/share?songid=9101704|i've come with a bucket of roses]], and [[http://soundclick.com/share?songid=8839071|my own house]] by Andrew Heathwaite
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<html><head><title>Semicomma family</title></head><body>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 thirds. Orson, the <a class="wiki_link" href="/5-limit">5-limit</a> temperament tempering it out, has a <a class="wiki_link" href="/generator">generator</a> 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 <a class="wiki_link" href="/53edo">53edo</a> or <a class="wiki_link" href="/84edo">84edo</a>. These give tunings to the generator which are sharp of 7/6 by less than five <a class="wiki_link" href="/cent">cent</a>s, making it hard to treat orson as anything other than an (at least) 7-limit system, leading to orwell.<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 271.627<br /> <br /> Map: [<1 0 3|, <0 7 -3|]<br /> EDOs: 22, 31, 53, 190, 253, 296<br /> <br /> <!-- ws:start:WikiTextHeadingRule:0:<h2> --><h2 id="toc0"><a name="x-Seven limit children"></a><!-- ws:end:WikiTextHeadingRule:0 -->Seven limit children</h2> The second comma of the <a class="wiki_link" href="/Normal%20lists">normal comma list</a> 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||.<br /> <br /> <!-- ws:start:WikiTextHeadingRule:2:<h3> --><h3 id="toc1"><a name="x-Seven limit children-Orwell"></a><!-- ws:end:WikiTextHeadingRule:2 -->Orwell</h3> So called because 19/84 (as a <a class="wiki_link" href="/fraction%20of%20the%20octave">fraction of the octave</a>) 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 <a class="wiki_link" href="/22edo">22</a>, <a class="wiki_link" href="/31edo">31</a>, <a class="wiki_link" href="/53edo">53-EDO</a> and <a class="wiki_link" href="/84edo">84edo</a>, and may be described as the 22&31 temperament, or <<7 -3 8 -21 -7 27||. It's a good system in the <a class="wiki_link" href="/7-limit">7-limit</a> and naturally extends into the <a class="wiki_link" href="/11-limit">11-limit</a>. <a class="wiki_link" href="/84edo">84edo</a>, 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 <a class="wiki_link" href="/POTE%20tuning">POTE tuning</a>, 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. <a class="wiki_link" href="/53edo">53edo</a> 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.<br /> <br /> The 11-limit version of orwell tempers out 99/98, which means that two of its sharpened 7/6 generators give a flattened 11/8, as well as 121/120, 176/175, 385/384 and 540/539. Despite lowered tuning accuracy, orwell comes into its own in the 11-limit, giving acceptable accuracy and relatively low complexity. Tempering out the orwellisma, 1728/1715, means that orwell interprets three stacked 7/6 generators as an 8/5, and the tempered 1-7/6-11/8-8/5 chord is natural to orwell.<br /> <br /> 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 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.<br /> <br /> <!-- ws:start:WikiTextHeadingRule:4:<h3> --><h3 id="toc2"><a name="x-Seven limit children-Vital statistics"></a><!-- ws:end:WikiTextHeadingRule:4 -->Vital statistics</h3> <a class="wiki_link" href="/Comma">Commas</a>: 225/224, 1728/1715<br /> <br /> 7-limit<br /> [|1 0 0 0>, |14/11 0 -7/11 7/11>, <br /> |27/11 0 3/11 -3/11>, |27/11 0 -8/11 8/11>]<br /> <a class="wiki_link" href="/Fractional%20monzos">Eigenmonzos</a>: 2, 7/5<br /> <br /> 9-limit<br /> [|1 0 0 0>, |21/17 14/17 -7/17 0>, <br /> |42/17 -6/17 3/17 0>, |41/17 16/17 -8/17 0>]<br /> <a class="wiki_link" href="/Eigenmonzo">Eigenmonzos</a>: 2, 10/9<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 271.509<br /> Algebraic generators: Sabra3, the real root of 12x^3-7x-48. <br /> <br /> Map: [<1 0 3 1|, <0 7 -3 8|]<br /> EDOs: 22, 31, 53, 84, 137<br /> <br /> 11-limit<br /> <a class="wiki_link" href="/Comma">Commas</a>: 99/98, 121/120, 176/175<br /> <br /> <a class="wiki_link" href="/Minimax%20tuning">Minimax tuning</a><br /> [|1 0 0 0 0>, |14/11 0 -7/11 7/11 0>, |27/11 0 3/11 -3/11 0>,<br /> |27/11 0 -8/11 8/11 0>, |37/11 0 -2/11 2/11 0>]<br /> <a class="wiki_link" href="/Eigenmonzo">Eigenmonzos</a>: 2, 7/5<br /> <br /> <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 271.426<br /> <br /> Map: [<1 0 3 1 3|, <0 7 -3 8 2|]<br /> <a class="wiki_link" href="/edo">Edos</a>: <a class="wiki_link" href="/22edo">22</a>, <a class="wiki_link" href="/31edo">31</a>, <a class="wiki_link" href="/53edo">53</a>, <a class="wiki_link" href="/84edo">84</a><br /> <br /> <!-- ws:start:WikiTextHeadingRule:6:<h2> --><h2 id="toc3"><a name="x-Music"></a><!-- ws:end:WikiTextHeadingRule:6 -->Music</h2> <!-- ws:start:WikiTextUrlRule:88:http://www.archive.org/details/TrioInOrwell --><a class="wiki_link_ext" href="http://www.archive.org/details/TrioInOrwell" rel="nofollow">http://www.archive.org/details/TrioInOrwell</a><!-- ws:end:WikiTextUrlRule:88 --><br /> <a class="wiki_link_ext" href="http://soundclick.com/share?songid=9101705" rel="nofollow">one drop of rain</a>, <a class="wiki_link_ext" href="http://soundclick.com/share?songid=9101704" rel="nofollow">i've come with a bucket of roses</a>, and <a class="wiki_link_ext" href="http://soundclick.com/share?songid=8839071" rel="nofollow">my own house</a> by Andrew Heathwaite</body></html>