Schismatic family
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- This revision was by author genewardsmith and made on 2010-09-20 22:24:40 UTC.
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Original Wikitext content:
The 5-limit parent comma for the schismatic family is the schisma of 32805/32768, which is the amount by which the Pythagorean comma exceeds the Didymus comma (81/80), or alternatively put, the difference between a just major third and a just diminished fourth. Its [[monzo]] is |-15 8 1>, and flipping that yields <<1 -8 -15|| for the [[Wedgies and Multivals|wedgie]]. This tells us the generator is a fifth and that we will need eight fourths in succession to reach the pitch class of a major third. In fact, 10 = (4/3)^8 * 32805/32768.
The 5-limit version of the temperament is a [[Microtempering|microtemperament]], sometimes called helmholtz or schismatic, which flattens the fifth by a fraction of a schisma, but some other members of the family are less accurate. As a 5-limit system, it is far more accurate than meantone but still with manageable complexity. [[53edo]] is a possible tuning for schismatic, but you need [[118edo]] if you want to get the full effect. In exact analogy with 1/4 comma meantone there is also 1/8 schisma schismatic, with pure major thirds and fifths flattened by 1/8 schisma. Since 1/8 of a schisma is 0.244 cents, this falls into the range of microtempering.
==Seven limit children==
The second comma of the [[Normal lists|normal comma list]] defines which 7-limit family member we are looking at. Adding |25 -14 0 -1> gives garibaldi, |-44 26 0 1> grackle, |6 -2 0 -1> schism and |-59 39 0 -1> pontiac; these all have a fifth as generator. Bischismic adds |-69 40 0 2> and has a fifth generator with a half-octave period. Guiron adds 1029/1024 = |-10 1 0 3>, with an 8/7 generator, three of which give the fifth, and term adds |-94 54 0 3> with a 1/3 octave period. Sesquiquartififths adds |-35 15 0 4> and slices the fifth in four.
===Garibaldi===
Commas: {225/224, 3125/3087}
7-limit minimax tuning:
7-limit: [|1 0 0 0>, |5/3 1/15 0 -1/15>,
|5/3 -8/15 0 8/15>, |5/3 -14/15 0 14/15>]
Eigenmonzos: 2, 7/6
9-limit: [|1 0 0 0>, |25/16 1/8 0 -1/16>,
|5/2 -1 0 1/2>, |25/8 -7/4 0 7/8>]
Eigenmonzos: 2, 9/7
11-limit
Commas: {225/224, 385/384, 2200/2187}
Minimax tuning:
[|1 0 0 0 0>, |25/16 1/8 0 -1/16 0>, |5/2 -1 0 1/2 0>,
|25/8 -7/4 0 7/8 0>, |47/16 23/8 0 -23/16 0>]
Eigenmonzos: 2, 9/7
Map: [<1 0 15 25 -33|, <0 1 -8 -14 23|]
Edos: 94, 135
===Pontiac===
Commas: {32805/32768, 4375/4374}
7-limit minimax:
[|1 0 0 0>, |74/47 0 -1/47 1/47>, |113/47 0 8/47 -8/47>,
|113/47 0 -39/47 39/47>]
Eigenmonzos: 2, 7/5
9-limit minimax:
[|1 0 0 0>, |3/2 1/5 -1/10 0>,
|3 -8/5 4/5 0>, |-1/2 39/5 -39/10 0>]
Eigenmonzos: 2, 10/9
Map: [<1 0 15 -59|, <0 1 -8 39|]
Edos: 171, 1079, 1250, 1421
===Grackle===
Commas: 126/125, 32805/32768
7-limit minimax
Eigenmonzos: 2, 7/6
9-limit minimax
Eigenmonzos: 2, 9/7
Map: [<1 0 15 -44|, <0 1 -8 -26|]
Edos: 77, 89
===Bischismic===
Commas: 3136/3125, 32805/32768
7-limit minimax
Eigenmonzos: 2, 7/6
9-limit minimax
Eigenmonzos: 2, 9/7
Map: [<2 0 30 69|, <0 1 -8 -20|]
Edos: 118, 130, 248, 378, 508Original HTML content:
<html><head><title>Schismatic family</title></head><body>The 5-limit parent comma for the schismatic family is the schisma of 32805/32768, which is the amount by which the Pythagorean comma exceeds the Didymus comma (81/80), or alternatively put, the difference between a just major third and a just diminished fourth. Its <a class="wiki_link" href="/monzo">monzo</a> is |-15 8 1>, and flipping that yields <<1 -8 -15|| for the <a class="wiki_link" href="/Wedgies%20and%20Multivals">wedgie</a>. This tells us the generator is a fifth and that we will need eight fourths in succession to reach the pitch class of a major third. In fact, 10 = (4/3)^8 * 32805/32768. <br />
<br />
The 5-limit version of the temperament is a <a class="wiki_link" href="/Microtempering">microtemperament</a>, sometimes called helmholtz or schismatic, which flattens the fifth by a fraction of a schisma, but some other members of the family are less accurate. As a 5-limit system, it is far more accurate than meantone but still with manageable complexity. <a class="wiki_link" href="/53edo">53edo</a> is a possible tuning for schismatic, but you need <a class="wiki_link" href="/118edo">118edo</a> if you want to get the full effect. In exact analogy with 1/4 comma meantone there is also 1/8 schisma schismatic, with pure major thirds and fifths flattened by 1/8 schisma. Since 1/8 of a schisma is 0.244 cents, this falls into the range of microtempering.<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 |25 -14 0 -1> gives garibaldi, |-44 26 0 1> grackle, |6 -2 0 -1> schism and |-59 39 0 -1> pontiac; these all have a fifth as generator. Bischismic adds |-69 40 0 2> and has a fifth generator with a half-octave period. Guiron adds 1029/1024 = |-10 1 0 3>, with an 8/7 generator, three of which give the fifth, and term adds |-94 54 0 3> with a 1/3 octave period. Sesquiquartififths adds |-35 15 0 4> and slices the fifth in four.<br />
<br />
<!-- ws:start:WikiTextHeadingRule:2:<h3> --><h3 id="toc1"><a name="x-Seven limit children-Garibaldi"></a><!-- ws:end:WikiTextHeadingRule:2 -->Garibaldi</h3>
Commas: {225/224, 3125/3087}<br />
<br />
7-limit minimax tuning:<br />
7-limit: [|1 0 0 0>, |5/3 1/15 0 -1/15>,<br />
|5/3 -8/15 0 8/15>, |5/3 -14/15 0 14/15>]<br />
Eigenmonzos: 2, 7/6<br />
<br />
9-limit: [|1 0 0 0>, |25/16 1/8 0 -1/16>, <br />
|5/2 -1 0 1/2>, |25/8 -7/4 0 7/8>]<br />
Eigenmonzos: 2, 9/7<br />
<br />
11-limit<br />
Commas: {225/224, 385/384, 2200/2187}<br />
<br />
Minimax tuning:<br />
[|1 0 0 0 0>, |25/16 1/8 0 -1/16 0>, |5/2 -1 0 1/2 0>,<br />
|25/8 -7/4 0 7/8 0>, |47/16 23/8 0 -23/16 0>]<br />
Eigenmonzos: 2, 9/7<br />
<br />
Map: [<1 0 15 25 -33|, <0 1 -8 -14 23|]<br />
Edos: 94, 135<br />
<br />
<!-- ws:start:WikiTextHeadingRule:4:<h3> --><h3 id="toc2"><a name="x-Seven limit children-Pontiac"></a><!-- ws:end:WikiTextHeadingRule:4 -->Pontiac</h3>
Commas: {32805/32768, 4375/4374}<br />
<br />
7-limit minimax:<br />
[|1 0 0 0>, |74/47 0 -1/47 1/47>, |113/47 0 8/47 -8/47>, <br />
|113/47 0 -39/47 39/47>]<br />
Eigenmonzos: 2, 7/5<br />
<br />
9-limit minimax:<br />
[|1 0 0 0>, |3/2 1/5 -1/10 0>, <br />
|3 -8/5 4/5 0>, |-1/2 39/5 -39/10 0>]<br />
Eigenmonzos: 2, 10/9<br />
<br />
Map: [<1 0 15 -59|, <0 1 -8 39|]<br />
Edos: 171, 1079, 1250, 1421<br />
<br />
<!-- ws:start:WikiTextHeadingRule:6:<h3> --><h3 id="toc3"><a name="x-Seven limit children-Grackle"></a><!-- ws:end:WikiTextHeadingRule:6 -->Grackle</h3>
Commas: 126/125, 32805/32768<br />
<br />
7-limit minimax<br />
Eigenmonzos: 2, 7/6<br />
<br />
9-limit minimax<br />
Eigenmonzos: 2, 9/7<br />
<br />
Map: [<1 0 15 -44|, <0 1 -8 -26|]<br />
Edos: 77, 89<br />
<br />
<!-- ws:start:WikiTextHeadingRule:8:<h3> --><h3 id="toc4"><a name="x-Seven limit children-Bischismic"></a><!-- ws:end:WikiTextHeadingRule:8 -->Bischismic</h3>
Commas: 3136/3125, 32805/32768<br />
<br />
7-limit minimax<br />
Eigenmonzos: 2, 7/6<br />
<br />
9-limit minimax<br />
Eigenmonzos: 2, 9/7<br />
<br />
Map: [<2 0 30 69|, <0 1 -8 -20|]<br />
Edos: 118, 130, 248, 378, 508</body></html>