Meantone family: Difference between revisions
Wikispaces>genewardsmith **Imported revision 179245713 - Original comment: ** |
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<h2>IMPORTED REVISION FROM WIKISPACES</h2> | <h2>IMPORTED REVISION FROM WIKISPACES</h2> | ||
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | ||
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2010-11 | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2010-12-09 11:48:03 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>186777843</tt>.<br> | ||
: The revision comment was: <tt></tt><br> | : The revision comment was: <tt></tt><br> | ||
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br> | The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br> | ||
<h4>Original Wikitext content:</h4> | <h4>Original Wikitext 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;white-space: pre-wrap ! important" class="old-revision-html">The 5-limit parent [[Comma|comma]] of the meantone family is the Didymos or [[Didymos comma|syntonic comma]], 81/80. This is the one they all temper out. The [[Monzos and Interval Space|monzo]] for 81/80 goes |-4 4 -1>, and that can be flipped around to the corresponding [[Wedgies and Multivals|wedgie]], <<1 4 4||, which tells us that the period is an octave, the generator is a fifth, and four fifths go to make up a 5/1 interval. | <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">The 5-limit parent [[Comma|comma]] of the meantone family is the Didymos or [[Didymos comma|syntonic comma]], 81/80. This is the one they all temper out. The [[Monzos and Interval Space|monzo]] for 81/80 goes |-4 4 -1>, and that can be flipped around to the corresponding [[Wedgies and Multivals|wedgie]], <<1 4 4||, which tells us that the period is an octave, the generator is a fifth, and four fifths go to make up a 5/1 interval. | ||
[[POTE tuning|POTE generator]]: 696.239 | |||
==Seven limit children== | ==Seven limit children== | ||
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[|1 0 0 0>, |1 0 1/4 0>, |0 0 1 0>, |-3 0 5/2 0>] | [|1 0 0 0>, |1 0 1/4 0>, |0 0 1 0>, |-3 0 5/2 0>] | ||
Eigenmonzos: 2, 5 | Eigenmonzos: 2, 5 | ||
[[POTE tuning|POTE generator]]: 696.495 | |||
Algebraic generator: Cybozem, the real root of 15x^3-10x^2-18, which comes to 503.4257 cents. The recurrence converges quickly. | Algebraic generator: Cybozem, the real root of 15x^3-10x^2-18, which comes to 503.4257 cents. The recurrence converges quickly. | ||
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|21/8 -5/4 0 0 5/8>, |25/8 -9/4 0 0 9/8>] | |21/8 -5/4 0 0 5/8>, |25/8 -9/4 0 0 9/8>] | ||
Eigenmonzos: 2, 11/9 | Eigenmonzos: 2, 11/9 | ||
[[POTE tuning|POTE generator]]: 696.967 | |||
Algebraic generator: Traverse, the positive real root of x^4+2x-13, or 696.9529 cents. | Algebraic generator: Traverse, the positive real root of x^4+2x-13, or 696.9529 cents. | ||
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|-3 0 5/2 0 0>, |11 0 -13/4 0 0>] | |-3 0 5/2 0 0>, |11 0 -13/4 0 0>] | ||
Eigenmonzos: 2, 5 | Eigenmonzos: 2, 5 | ||
[[POTE tuning|POTE generator]]: 696.434 | |||
Algebraic generator: Cybozem; or else Radieubiz, the real root of 3x^3+6x-19. Unlike Cybozem, the recurrence for Radieubiz does not converge. | Algebraic generator: Cybozem; or else Radieubiz, the real root of 3x^3+6x-19. Unlike Cybozem, the recurrence for Radieubiz does not converge. | ||
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===Flattone=== | ===Flattone=== | ||
Commas: 81/80, 525/512 | |||
Similarly for flattone, the wedgie is <<1 4 -9 4 -17 -32||, which tells us among other things that 9 generator steps of 4/3 get to the interval class for 7, meaning that 7/4 is a diminished minor seventh interval. Other intervals are 7/6, a diminished minor third, and 7/5, a doubly diminshed fifth. Good tunings for flattone are [[26edo]], [[45edo]] and [[64edo]]. | Similarly for flattone, the wedgie is <<1 4 -9 4 -17 -32||, which tells us among other things that 9 generator steps of 4/3 get to the interval class for 7, meaning that 7/4 is a diminished minor seventh interval. Other intervals are 7/6, a diminished minor third, and 7/5, a doubly diminshed fifth. Good tunings for flattone are [[26edo]], [[45edo]] and [[64edo]]. | ||
7-limit minimax | 7-limit minimax | ||
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|24/11 8/11 0 -4/11>, |34/11 -18/11 0 9/11>] | |24/11 8/11 0 -4/11>, |34/11 -18/11 0 9/11>] | ||
Eigenmonzos: 2, 9/7 | Eigenmonzos: 2, 9/7 | ||
[[POTE tuning|POTE generator]]: 693.779 | |||
Algebraic generator: Squarto, the positive root of 8x^2-4x-9, at 506.3239 cents, equal to (1+sqrt(19))/4. | Algebraic generator: Squarto, the positive root of 8x^2-4x-9, at 506.3239 cents, equal to (1+sqrt(19))/4. | ||
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===Dominant=== | ===Dominant=== | ||
The wedgie for dominant is <<1 4 -2 4 -6 -16||. Now the interval class for 7 is obtained from two fourths in succession, so that 7/4 is a minor seventh. The 7/6 interval is, like 6/5, now a minor third, and 7/5 is a diminished fifth. An excellent tuning for dominant is [[12edo]], but it also works well with Pythagorean tuning of pure 3/2 fifths, and with [[29edo]], [[41edo]], or [[53edo]]. | Commas: 36/35, 64/63 | ||
The wedgie for dominant is <<1 4 -2 4 -6 -16||. Now the interval class for 7 is obtained from two fourths in succession, so that 7/4 is a minor seventh. The 7/6 interval is, like 6/5, now a minor third, and 7/5 is a diminished fifth. An excellent tuning for dominant is [[12edo]], but it also works well with the Pythagorean tuning of pure 3/2 fifths, and with [[29edo]], [[41edo]], or [[53edo]]. | |||
[[POTE tuning|POTE generator]]: 701.573 | |||
===Sharptone=== | ===Sharptone=== | ||
Commas: 21/20, 28/27 | |||
Sharptone, with a wedgie <<1 4 3 4 2 -4||, is a low-accuracy temperament tempering out 21/20 and 28/27. In sharptone, a 7/4 is a major sixth, a 7/6 a whole tone, and a 7/5 a fourth. Genuinely septimal sounding harmony therefore cannot be expected, but it can be used to translate, more or less, 7-limit JI into 5-limit meantone. [[12edo]] tuning does sharptone about as well as such a thing can be done. | Sharptone, with a wedgie <<1 4 3 4 2 -4||, is a low-accuracy temperament tempering out 21/20 and 28/27. In sharptone, a 7/4 is a major sixth, a 7/6 a whole tone, and a 7/5 a fourth. Genuinely septimal sounding harmony therefore cannot be expected, but it can be used to translate, more or less, 7-limit JI into 5-limit meantone. [[12edo]] tuning does sharptone about as well as such a thing can be done. | ||
[[POTE tuning|POTE generator]]: 700.140 | |||
===Injera=== | ===Injera=== | ||
Commas: 50/49, 81/80 | |||
The wedgie for injera is <<2 8 8 8 7 -4||, which tells us it has a half-octave period and a generator which can be taken as a fifth or fourth, but also as a 15/14 semitone difference between a half-octave and a perfect fifth. Injera tempers out 50/49, equating 7/5 with 10/7 and giving a tritone of half an octave. A major third up from this tritone is the 7/4. [[38edo]], which is two parallel 19edos, is an excellent tuning for injera. | The wedgie for injera is <<2 8 8 8 7 -4||, which tells us it has a half-octave period and a generator which can be taken as a fifth or fourth, but also as a 15/14 semitone difference between a half-octave and a perfect fifth. Injera tempers out 50/49, equating 7/5 with 10/7 and giving a tritone of half an octave. A major third up from this tritone is the 7/4. [[38edo]], which is two parallel 19edos, is an excellent tuning for injera. | ||
[[POTE tuning|POTE generator]]: 694.375 | |||
===Godzilla=== | ===Godzilla=== | ||
Commas: 49/48, 81/80 | |||
Godzilla has wedgie <<2 8 1 8 -4 -20||, and tempers out 49/48, equating 8/7 with 7/6. Two of the step-and-a-half intervals these represent give a fourth, and so step-and-a-half generators generate godzilla. [[19edo]] is the perfect godzilla tuning, so much so that's there's not much point in looking elsewhere. Hence it can be more or less equated with taking 4/19 as a generator. Godzilla is not well supplied with good [[MOSScales|MOS scales]], though it has a pentatonic scale which could serve as an alternative to [[5edo]], but other options exist for those wanting to explore it. | Godzilla has wedgie <<2 8 1 8 -4 -20||, and tempers out 49/48, equating 8/7 with 7/6. Two of the step-and-a-half intervals these represent give a fourth, and so step-and-a-half generators generate godzilla. [[19edo]] is the perfect godzilla tuning, so much so that's there's not much point in looking elsewhere. Hence it can be more or less equated with taking 4/19 as a generator. Godzilla is not well supplied with good [[MOSScales|MOS scales]], though it has a pentatonic scale which could serve as an alternative to [[5edo]], but other options exist for those wanting to explore it. | ||
[[POTE tuning|POTE generator]]: 347.635 | |||
===Mohajira=== | ===Mohajira=== | ||
Commas: 81/80, 6144/6125 | |||
Mohajira, with wedgie <<2 8 -11 8 -23 -48||, really makes more sense as an 11-limit temperament. It has a generator of a neutral third, two of which make up a fifth, and which can be taken to represent 128/105. Mohajira tempers out 6144/6125, the porwell comma. [[31edo]] makes for an excellent (7-limit) mohajira tuning, with generator 9/31. It has a 7-note MOS with three larger steps and four smaller ones, going sLsLsLs. | Mohajira, with wedgie <<2 8 -11 8 -23 -48||, really makes more sense as an 11-limit temperament. It has a generator of a neutral third, two of which make up a fifth, and which can be taken to represent 128/105. Mohajira tempers out 6144/6125, the porwell comma. [[31edo]] makes for an excellent (7-limit) mohajira tuning, with generator 9/31. It has a 7-note MOS with three larger steps and four smaller ones, going sLsLsLs. | ||
7 and 9 limit minimax 1/4 comma | 7 and 9 limit minimax 1/4 comma | ||
[|1 0 0 0>, |1 0 1/4 0>, |0 0 1 0>, |6 0 -11/8 0>] | [|1 0 0 0>, |1 0 1/4 0>, |0 0 1 0>, |6 0 -11/8 0>] | ||
Eigenmonzos: 2, 5 | Eigenmonzos: 2, 5 | ||
[[POTE tuning|POTE generator]]: 348.415 | |||
Algebraic generator: Mohabis, real root of 3x^3-3x^2-1, 348.6067 cents. Corresponding recurrence converges quickly. | Algebraic generator: Mohabis, real root of 3x^3-3x^2-1, 348.6067 cents. Corresponding recurrence converges quickly. | ||
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|6 0 -11/8 0 0>, |2 0 5/8 0 0>] | |6 0 -11/8 0 0>, |2 0 5/8 0 0>] | ||
Eigenmonzos: 2, 5 | Eigenmonzos: 2, 5 | ||
[[POTE tuning|POTE generator]]: 348.477 | |||
Map: [<1 1 0 6 2|, <0 2 8 -11 5|] | Map: [<1 1 0 6 2|, <0 2 8 -11 5|] | ||
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===Mothra=== | ===Mothra=== | ||
Commas: 81/80, 1029/1024 | |||
Mothra, with wedgie <<3 12 -1 12 -10 -36||, splits the fifth into three 8/7 generators. It uses 1029/1024, the gamelisma, to accomplish this deed and also tempers out 1728/1715, the orwell comma. Using [[31edo]] with a generator of 6/31 is an excellent tuning choice. Once again something other than a MOS should be used as a scale to get the most out of mothra. | Mothra, with wedgie <<3 12 -1 12 -10 -36||, splits the fifth into three 8/7 generators. It uses 1029/1024, the gamelisma, to accomplish this deed and also tempers out 1728/1715, the orwell comma. Using [[31edo]] with a generator of 6/31 is an excellent tuning choice. Once again something other than a MOS should be used as a scale to get the most out of mothra. | ||
7 and 9 limit minimax 1/4 comma | 7 and 9 limit minimax 1/4 comma | ||
[|1 0 0 0>, |1 0 1/4 0>, |0 0 1 0>, |3 0 -1/12 0>] | [|1 0 0 0>, |1 0 1/4 0>, |0 0 1 0>, |3 0 -1/12 0>] | ||
Eigenmonzos: 2, 5 | Eigenmonzos: 2, 5 | ||
[[POTE tuning|POTE generator]]: 232.193 | |||
Algebraic generator: Rabrindanath, largest real root of x^8-3x^2+1, or 232.0774 cents. | Algebraic generator: Rabrindanath, largest real root of x^8-3x^2+1, or 232.0774 cents. | ||
Map: [<1 1 0 3|, <0 3 12 -1|] | Map: [<1 1 0 3|, <0 3 12 -1|] | ||
Generators: 2, 8/7 | Generators: 2, 8/7 | ||
===Squares=== | ===Squares=== | ||
Commas: 81/80, 2410/2400 | |||
Squares, with wedgie <<4 16 9 16 3 -24||, splits the interval of an eleventh, or 8/3, into four supermajor third (9/7) intervals, and uses it for a generator. [[31edo]], with a generator of 11/31, makes for a good squares tuning, with 8, 11, and 14 note MOS available. Squares tempers out 2401/2400, the breedsma, as well as 2430/2401. | Squares, with wedgie <<4 16 9 16 3 -24||, splits the interval of an eleventh, or 8/3, into four supermajor third (9/7) intervals, and uses it for a generator. [[31edo]], with a generator of 11/31, makes for a good squares tuning, with 8, 11, and 14 note MOS available. Squares tempers out 2401/2400, the breedsma, as well as 2430/2401. | ||
7 and 9 limit minimax 1/4 comma | 7 and 9 limit minimax 1/4 comma | ||
[|1 0 0 0>, |1 0 1/4 0>, |0 0 1 0>, |3/2 0 9/16 0>] | [|1 0 0 0>, |1 0 1/4 0>, |0 0 1 0>, |3/2 0 9/16 0>] | ||
Eigenmonzos: 2, 5 | Eigenmonzos: 2, 5 | ||
[[POTE tuning|POTE generator]]: 425.942 | |||
Algebraic generator: Sceptre2, the positive root of 9x^2+x-16, or (sqrt(577)-1)/18, which is 425.9311 cents. | Algebraic generator: Sceptre2, the positive root of 9x^2+x-16, or (sqrt(577)-1)/18, which is 425.9311 cents. | ||
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===Liese=== | ===Liese=== | ||
Commas: 81/80, 686/675 | |||
Liese, with wedgie <<3 12 11 12 9 -8||, splits the twelfth interval of 3/1 into three generators of 10/7, using the comma 1029/1000. It also tempers out 686/675, the senga. [[74edo]] makes for a good liese tuning, though [[19edo]] can be used. The tuning is well-supplied with MOS: 7, 9, 11, 13, 15, 17, 19, 36, 55. | Liese, with wedgie <<3 12 11 12 9 -8||, splits the twelfth interval of 3/1 into three generators of 10/7, using the comma 1029/1000. It also tempers out 686/675, the senga. [[74edo]] makes for a good liese tuning, though [[19edo]] can be used. The tuning is well-supplied with MOS: 7, 9, 11, 13, 15, 17, 19, 36, 55. | ||
7 and 9 limit minimax 1/4 comma | 7 and 9 limit minimax 1/4 comma | ||
[|1 0 0 0>, |1 0 1/4 0>, |0 0 1 0>, |2/3 0 11/12 0>] | [|1 0 0 0>, |1 0 1/4 0>, |0 0 1 0>, |2/3 0 11/12 0>] | ||
Eigenmonzos: 2, 5 | Eigenmonzos: 2, 5 | ||
[[POTE tuning|POTE generator]]: 632.406 | |||
Algebraic generator: Radix, the real root of x^5-2x^4+2x^3-2x^2+2x-2, also a root of x^6-x^5-2. The recurrence converges. | Algebraic generator: Radix, the real root of x^5-2x^4+2x^3-2x^2+2x-2, also a root of x^6-x^5-2. The recurrence converges. | ||
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<h4>Original HTML content:</h4> | <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"><html><head><title>Meantone family</title></head><body>The 5-limit parent <a class="wiki_link" href="/Comma">comma</a> of the meantone family is the Didymos or <a class="wiki_link" href="/Didymos%20comma">syntonic comma</a>, 81/80. This is the one they all temper out. The <a class="wiki_link" href="/Monzos%20and%20Interval%20Space">monzo</a> for 81/80 goes |-4 4 -1&gt;, and that can be flipped around to the corresponding <a class="wiki_link" href="/Wedgies%20and%20Multivals">wedgie</a>, &lt;&lt;1 4 4||, which tells us that the period is an octave, the generator is a fifth, and four fifths go to make up a 5/1 interval.<br /> | <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"><html><head><title>Meantone family</title></head><body>The 5-limit parent <a class="wiki_link" href="/Comma">comma</a> of the meantone family is the Didymos or <a class="wiki_link" href="/Didymos%20comma">syntonic comma</a>, 81/80. This is the one they all temper out. The <a class="wiki_link" href="/Monzos%20and%20Interval%20Space">monzo</a> for 81/80 goes |-4 4 -1&gt;, and that can be flipped around to the corresponding <a class="wiki_link" href="/Wedgies%20and%20Multivals">wedgie</a>, &lt;&lt;1 4 4||, which tells us that the period is an octave, the generator is a fifth, and four fifths go to make up a 5/1 interval.<br /> | ||
<br /> | |||
<a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 696.239<br /> | |||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:0:&lt;h2&gt; --><h2 id="toc0"><a name="x-Seven limit children"></a><!-- ws:end:WikiTextHeadingRule:0 -->Seven limit children</h2> | <!-- ws:start:WikiTextHeadingRule:0:&lt;h2&gt; --><h2 id="toc0"><a name="x-Seven limit children"></a><!-- ws:end:WikiTextHeadingRule:0 -->Seven limit children</h2> | ||
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[|1 0 0 0&gt;, |1 0 1/4 0&gt;, |0 0 1 0&gt;, |-3 0 5/2 0&gt;]<br /> | [|1 0 0 0&gt;, |1 0 1/4 0&gt;, |0 0 1 0&gt;, |-3 0 5/2 0&gt;]<br /> | ||
Eigenmonzos: 2, 5<br /> | Eigenmonzos: 2, 5<br /> | ||
<br /> | |||
<a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 696.495<br /> | |||
<br /> | <br /> | ||
Algebraic generator: Cybozem, the real root of 15x^3-10x^2-18, which comes to 503.4257 cents. The recurrence converges quickly.<br /> | Algebraic generator: Cybozem, the real root of 15x^3-10x^2-18, which comes to 503.4257 cents. The recurrence converges quickly.<br /> | ||
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|21/8 -5/4 0 0 5/8&gt;, |25/8 -9/4 0 0 9/8&gt;]<br /> | |21/8 -5/4 0 0 5/8&gt;, |25/8 -9/4 0 0 9/8&gt;]<br /> | ||
Eigenmonzos: 2, 11/9<br /> | Eigenmonzos: 2, 11/9<br /> | ||
<br /> | |||
<a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 696.967<br /> | |||
<br /> | <br /> | ||
Algebraic generator: Traverse, the positive real root of x^4+2x-13, or 696.9529 cents.<br /> | Algebraic generator: Traverse, the positive real root of x^4+2x-13, or 696.9529 cents.<br /> | ||
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|-3 0 5/2 0 0&gt;, |11 0 -13/4 0 0&gt;]<br /> | |-3 0 5/2 0 0&gt;, |11 0 -13/4 0 0&gt;]<br /> | ||
Eigenmonzos: 2, 5<br /> | Eigenmonzos: 2, 5<br /> | ||
<br /> | |||
<a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 696.434<br /> | |||
<br /> | <br /> | ||
Algebraic generator: Cybozem; or else Radieubiz, the real root of 3x^3+6x-19. Unlike Cybozem, the recurrence for Radieubiz does not converge.<br /> | Algebraic generator: Cybozem; or else Radieubiz, the real root of 3x^3+6x-19. Unlike Cybozem, the recurrence for Radieubiz does not converge.<br /> | ||
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<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:8:&lt;h3&gt; --><h3 id="toc4"><a name="x-Seven limit children-Flattone"></a><!-- ws:end:WikiTextHeadingRule:8 -->Flattone</h3> | <!-- ws:start:WikiTextHeadingRule:8:&lt;h3&gt; --><h3 id="toc4"><a name="x-Seven limit children-Flattone"></a><!-- ws:end:WikiTextHeadingRule:8 -->Flattone</h3> | ||
Commas: 81/80, 525/512<br /> | |||
<br /> | |||
Similarly for flattone, the wedgie is &lt;&lt;1 4 -9 4 -17 -32||, which tells us among other things that 9 generator steps of 4/3 get to the interval class for 7, meaning that 7/4 is a diminished minor seventh interval. Other intervals are 7/6, a diminished minor third, and 7/5, a doubly diminshed fifth. Good tunings for flattone are <a class="wiki_link" href="/26edo">26edo</a>, <a class="wiki_link" href="/45edo">45edo</a> and <a class="wiki_link" href="/64edo">64edo</a>.<br /> | Similarly for flattone, the wedgie is &lt;&lt;1 4 -9 4 -17 -32||, which tells us among other things that 9 generator steps of 4/3 get to the interval class for 7, meaning that 7/4 is a diminished minor seventh interval. Other intervals are 7/6, a diminished minor third, and 7/5, a doubly diminshed fifth. Good tunings for flattone are <a class="wiki_link" href="/26edo">26edo</a>, <a class="wiki_link" href="/45edo">45edo</a> and <a class="wiki_link" href="/64edo">64edo</a>.<br /> | ||
<br /> | <br /> | ||
7-limit minimax<br /> | 7-limit minimax<br /> | ||
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|24/11 8/11 0 -4/11&gt;, |34/11 -18/11 0 9/11&gt;]<br /> | |24/11 8/11 0 -4/11&gt;, |34/11 -18/11 0 9/11&gt;]<br /> | ||
Eigenmonzos: 2, 9/7<br /> | Eigenmonzos: 2, 9/7<br /> | ||
<br /> | |||
<a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 693.779<br /> | |||
<br /> | <br /> | ||
Algebraic generator: Squarto, the positive root of 8x^2-4x-9, at 506.3239 cents, equal to (1+sqrt(19))/4.<br /> | Algebraic generator: Squarto, the positive root of 8x^2-4x-9, at 506.3239 cents, equal to (1+sqrt(19))/4.<br /> | ||
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<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:10:&lt;h3&gt; --><h3 id="toc5"><a name="x-Seven limit children-Dominant"></a><!-- ws:end:WikiTextHeadingRule:10 -->Dominant</h3> | <!-- ws:start:WikiTextHeadingRule:10:&lt;h3&gt; --><h3 id="toc5"><a name="x-Seven limit children-Dominant"></a><!-- ws:end:WikiTextHeadingRule:10 -->Dominant</h3> | ||
The wedgie for dominant is &lt;&lt;1 4 -2 4 -6 -16||. Now the interval class for 7 is obtained from two fourths in succession, so that 7/4 is a minor seventh. The 7/6 interval is, like 6/5, now a minor third, and 7/5 is a diminished fifth. An excellent tuning for dominant is <a class="wiki_link" href="/12edo">12edo</a>, but it also works well with Pythagorean tuning of pure 3/2 fifths, and with <a class="wiki_link" href="/29edo">29edo</a>, <a class="wiki_link" href="/41edo">41edo</a>, or <a class="wiki_link" href="/53edo">53edo</a>.<br /> | Commas: 36/35, 64/63<br /> | ||
<br /> | |||
The wedgie for dominant is &lt;&lt;1 4 -2 4 -6 -16||. Now the interval class for 7 is obtained from two fourths in succession, so that 7/4 is a minor seventh. The 7/6 interval is, like 6/5, now a minor third, and 7/5 is a diminished fifth. An excellent tuning for dominant is <a class="wiki_link" href="/12edo">12edo</a>, but it also works well with the Pythagorean tuning of pure 3/2 fifths, and with <a class="wiki_link" href="/29edo">29edo</a>, <a class="wiki_link" href="/41edo">41edo</a>, or <a class="wiki_link" href="/53edo">53edo</a>.<br /> | |||
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<a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 701.573<br /> | |||
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<!-- ws:start:WikiTextHeadingRule:12:&lt;h3&gt; --><h3 id="toc6"><a name="x-Seven limit children-Sharptone"></a><!-- ws:end:WikiTextHeadingRule:12 -->Sharptone</h3> | <!-- ws:start:WikiTextHeadingRule:12:&lt;h3&gt; --><h3 id="toc6"><a name="x-Seven limit children-Sharptone"></a><!-- ws:end:WikiTextHeadingRule:12 -->Sharptone</h3> | ||
Commas: 21/20, 28/27<br /> | |||
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Sharptone, with a wedgie &lt;&lt;1 4 3 4 2 -4||, is a low-accuracy temperament tempering out 21/20 and 28/27. In sharptone, a 7/4 is a major sixth, a 7/6 a whole tone, and a 7/5 a fourth. Genuinely septimal sounding harmony therefore cannot be expected, but it can be used to translate, more or less, 7-limit JI into 5-limit meantone. <a class="wiki_link" href="/12edo">12edo</a> tuning does sharptone about as well as such a thing can be done.<br /> | Sharptone, with a wedgie &lt;&lt;1 4 3 4 2 -4||, is a low-accuracy temperament tempering out 21/20 and 28/27. In sharptone, a 7/4 is a major sixth, a 7/6 a whole tone, and a 7/5 a fourth. Genuinely septimal sounding harmony therefore cannot be expected, but it can be used to translate, more or less, 7-limit JI into 5-limit meantone. <a class="wiki_link" href="/12edo">12edo</a> tuning does sharptone about as well as such a thing can be done.<br /> | ||
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<a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 700.140<br /> | |||
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<!-- ws:start:WikiTextHeadingRule:14:&lt;h3&gt; --><h3 id="toc7"><a name="x-Seven limit children-Injera"></a><!-- ws:end:WikiTextHeadingRule:14 -->Injera</h3> | <!-- ws:start:WikiTextHeadingRule:14:&lt;h3&gt; --><h3 id="toc7"><a name="x-Seven limit children-Injera"></a><!-- ws:end:WikiTextHeadingRule:14 -->Injera</h3> | ||
The wedgie for injera is &lt;&lt;2 8 8 8 7 -4||, which tells us it has a half-octave period and a generator which can be taken as a fifth or fourth, but also as a 15/14 semitone difference between a half-octave and a perfect fifth. Injera tempers out 50/49, equating 7/5 with 10/7 and giving a tritone of half an octave. A major third up from this tritone is the 7/4. <a class="wiki_link" href="/38edo">38edo</a>, which is two parallel 19edos, is an excellent tuning for injera.<br /> | Commas: 50/49, 81/80<br /> | ||
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The wedgie for injera is &lt;&lt;2 8 8 8 7 -4||, which tells us it has a half-octave period and a generator which can be taken as a fifth or fourth, but also as a 15/14 semitone difference between a half-octave and a perfect fifth. Injera tempers out 50/49, equating 7/5 with 10/7 and giving a tritone of half an octave. A major third up from this tritone is the 7/4. <a class="wiki_link" href="/38edo">38edo</a>, which is two parallel 19edos, is an excellent tuning for injera.<br /> | |||
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<a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 694.375<br /> | |||
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<!-- ws:start:WikiTextHeadingRule:16:&lt;h3&gt; --><h3 id="toc8"><a name="x-Seven limit children-Godzilla"></a><!-- ws:end:WikiTextHeadingRule:16 -->Godzilla</h3> | <!-- ws:start:WikiTextHeadingRule:16:&lt;h3&gt; --><h3 id="toc8"><a name="x-Seven limit children-Godzilla"></a><!-- ws:end:WikiTextHeadingRule:16 -->Godzilla</h3> | ||
Commas: 49/48, 81/80<br /> | |||
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Godzilla has wedgie &lt;&lt;2 8 1 8 -4 -20||, and tempers out 49/48, equating 8/7 with 7/6. Two of the step-and-a-half intervals these represent give a fourth, and so step-and-a-half generators generate godzilla. <a class="wiki_link" href="/19edo">19edo</a> is the perfect godzilla tuning, so much so that's there's not much point in looking elsewhere. Hence it can be more or less equated with taking 4/19 as a generator. Godzilla is not well supplied with good <a class="wiki_link" href="/MOSScales">MOS scales</a>, though it has a pentatonic scale which could serve as an alternative to <a class="wiki_link" href="/5edo">5edo</a>, but other options exist for those wanting to explore it.<br /> | Godzilla has wedgie &lt;&lt;2 8 1 8 -4 -20||, and tempers out 49/48, equating 8/7 with 7/6. Two of the step-and-a-half intervals these represent give a fourth, and so step-and-a-half generators generate godzilla. <a class="wiki_link" href="/19edo">19edo</a> is the perfect godzilla tuning, so much so that's there's not much point in looking elsewhere. Hence it can be more or less equated with taking 4/19 as a generator. Godzilla is not well supplied with good <a class="wiki_link" href="/MOSScales">MOS scales</a>, though it has a pentatonic scale which could serve as an alternative to <a class="wiki_link" href="/5edo">5edo</a>, but other options exist for those wanting to explore it.<br /> | ||
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<a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 347.635<br /> | |||
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<!-- ws:start:WikiTextHeadingRule:18:&lt;h3&gt; --><h3 id="toc9"><a name="x-Seven limit children-Mohajira"></a><!-- ws:end:WikiTextHeadingRule:18 -->Mohajira</h3> | <!-- ws:start:WikiTextHeadingRule:18:&lt;h3&gt; --><h3 id="toc9"><a name="x-Seven limit children-Mohajira"></a><!-- ws:end:WikiTextHeadingRule:18 -->Mohajira</h3> | ||
Commas: 81/80, 6144/6125<br /> | |||
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Mohajira, with wedgie &lt;&lt;2 8 -11 8 -23 -48||, really makes more sense as an 11-limit temperament. It has a generator of a neutral third, two of which make up a fifth, and which can be taken to represent 128/105. Mohajira tempers out 6144/6125, the porwell comma. <a class="wiki_link" href="/31edo">31edo</a> makes for an excellent (7-limit) mohajira tuning, with generator 9/31. It has a 7-note MOS with three larger steps and four smaller ones, going sLsLsLs.<br /> | Mohajira, with wedgie &lt;&lt;2 8 -11 8 -23 -48||, really makes more sense as an 11-limit temperament. It has a generator of a neutral third, two of which make up a fifth, and which can be taken to represent 128/105. Mohajira tempers out 6144/6125, the porwell comma. <a class="wiki_link" href="/31edo">31edo</a> makes for an excellent (7-limit) mohajira tuning, with generator 9/31. It has a 7-note MOS with three larger steps and four smaller ones, going sLsLsLs.<br /> | ||
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7 and 9 limit minimax 1/4 comma<br /> | 7 and 9 limit minimax 1/4 comma<br /> | ||
[|1 0 0 0&gt;, |1 0 1/4 0&gt;, |0 0 1 0&gt;, |6 0 -11/8 0&gt;]<br /> | [|1 0 0 0&gt;, |1 0 1/4 0&gt;, |0 0 1 0&gt;, |6 0 -11/8 0&gt;]<br /> | ||
Eigenmonzos: 2, 5<br /> | Eigenmonzos: 2, 5<br /> | ||
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<a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 348.415<br /> | |||
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Algebraic generator: Mohabis, real root of 3x^3-3x^2-1, 348.6067 cents. Corresponding recurrence converges quickly.<br /> | Algebraic generator: Mohabis, real root of 3x^3-3x^2-1, 348.6067 cents. Corresponding recurrence converges quickly.<br /> | ||
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|6 0 -11/8 0 0&gt;, |2 0 5/8 0 0&gt;]<br /> | |6 0 -11/8 0 0&gt;, |2 0 5/8 0 0&gt;]<br /> | ||
Eigenmonzos: 2, 5<br /> | Eigenmonzos: 2, 5<br /> | ||
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<a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 348.477<br /> | |||
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Map: [&lt;1 1 0 6 2|, &lt;0 2 8 -11 5|]<br /> | Map: [&lt;1 1 0 6 2|, &lt;0 2 8 -11 5|]<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:22:&lt;h3&gt; --><h3 id="toc11"><a name="x-Seven limit children-Mothra"></a><!-- ws:end:WikiTextHeadingRule:22 -->Mothra</h3> | <!-- ws:start:WikiTextHeadingRule:22:&lt;h3&gt; --><h3 id="toc11"><a name="x-Seven limit children-Mothra"></a><!-- ws:end:WikiTextHeadingRule:22 -->Mothra</h3> | ||
Commas: 81/80, 1029/1024<br /> | |||
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Mothra, with wedgie &lt;&lt;3 12 -1 12 -10 -36||, splits the fifth into three 8/7 generators. It uses 1029/1024, the gamelisma, to accomplish this deed and also tempers out 1728/1715, the orwell comma. Using <a class="wiki_link" href="/31edo">31edo</a> with a generator of 6/31 is an excellent tuning choice. Once again something other than a MOS should be used as a scale to get the most out of mothra.<br /> | Mothra, with wedgie &lt;&lt;3 12 -1 12 -10 -36||, splits the fifth into three 8/7 generators. It uses 1029/1024, the gamelisma, to accomplish this deed and also tempers out 1728/1715, the orwell comma. Using <a class="wiki_link" href="/31edo">31edo</a> with a generator of 6/31 is an excellent tuning choice. Once again something other than a MOS should be used as a scale to get the most out of mothra.<br /> | ||
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7 and 9 limit minimax 1/4 comma <br /> | 7 and 9 limit minimax 1/4 comma <br /> | ||
[|1 0 0 0&gt;, |1 0 1/4 0&gt;, |0 0 1 0&gt;, |3 0 -1/12 0&gt;]<br /> | [|1 0 0 0&gt;, |1 0 1/4 0&gt;, |0 0 1 0&gt;, |3 0 -1/12 0&gt;]<br /> | ||
Eigenmonzos: 2, 5<br /> | Eigenmonzos: 2, 5<br /> | ||
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<a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 232.193<br /> | |||
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Algebraic generator: Rabrindanath, largest real root of x^8-3x^2+1, or 232.0774 cents.<br /> | Algebraic generator: Rabrindanath, largest real root of x^8-3x^2+1, or 232.0774 cents.<br /> | ||
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Map: [&lt;1 1 0 3|, &lt;0 3 12 -1|]<br /> | Map: [&lt;1 1 0 3|, &lt;0 3 12 -1|]<br /> | ||
Generators: 2, 8/7<br /> | Generators: 2, 8/7<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:24:&lt;h3&gt; --><h3 id="toc12"><a name="x-Seven limit children-Squares"></a><!-- ws:end:WikiTextHeadingRule:24 -->Squares</h3> | <!-- ws:start:WikiTextHeadingRule:24:&lt;h3&gt; --><h3 id="toc12"><a name="x-Seven limit children-Squares"></a><!-- ws:end:WikiTextHeadingRule:24 -->Squares</h3> | ||
Commas: 81/80, 2410/2400<br /> | |||
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Squares, with wedgie &lt;&lt;4 16 9 16 3 -24||, splits the interval of an eleventh, or 8/3, into four supermajor third (9/7) intervals, and uses it for a generator. <a class="wiki_link" href="/31edo">31edo</a>, with a generator of 11/31, makes for a good squares tuning, with 8, 11, and 14 note MOS available. Squares tempers out 2401/2400, the breedsma, as well as 2430/2401.<br /> | Squares, with wedgie &lt;&lt;4 16 9 16 3 -24||, splits the interval of an eleventh, or 8/3, into four supermajor third (9/7) intervals, and uses it for a generator. <a class="wiki_link" href="/31edo">31edo</a>, with a generator of 11/31, makes for a good squares tuning, with 8, 11, and 14 note MOS available. Squares tempers out 2401/2400, the breedsma, as well as 2430/2401.<br /> | ||
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7 and 9 limit minimax 1/4 comma<br /> | 7 and 9 limit minimax 1/4 comma<br /> | ||
[|1 0 0 0&gt;, |1 0 1/4 0&gt;, |0 0 1 0&gt;, |3/2 0 9/16 0&gt;]<br /> | [|1 0 0 0&gt;, |1 0 1/4 0&gt;, |0 0 1 0&gt;, |3/2 0 9/16 0&gt;]<br /> | ||
Eigenmonzos: 2, 5<br /> | Eigenmonzos: 2, 5<br /> | ||
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<a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 425.942<br /> | |||
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Algebraic generator: Sceptre2, the positive root of 9x^2+x-16, or (sqrt(577)-1)/18, which is 425.9311 cents.<br /> | Algebraic generator: Sceptre2, the positive root of 9x^2+x-16, or (sqrt(577)-1)/18, which is 425.9311 cents.<br /> | ||
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<!-- ws:start:WikiTextHeadingRule:26:&lt;h3&gt; --><h3 id="toc13"><a name="x-Seven limit children-Liese"></a><!-- ws:end:WikiTextHeadingRule:26 -->Liese</h3> | <!-- ws:start:WikiTextHeadingRule:26:&lt;h3&gt; --><h3 id="toc13"><a name="x-Seven limit children-Liese"></a><!-- ws:end:WikiTextHeadingRule:26 -->Liese</h3> | ||
Commas: 81/80, 686/675<br /> | |||
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Liese, with wedgie &lt;&lt;3 12 11 12 9 -8||, splits the twelfth interval of 3/1 into three generators of 10/7, using the comma 1029/1000. It also tempers out 686/675, the senga. <a class="wiki_link" href="/74edo">74edo</a> makes for a good liese tuning, though <a class="wiki_link" href="/19edo">19edo</a> can be used. The tuning is well-supplied with MOS: 7, 9, 11, 13, 15, 17, 19, 36, 55.<br /> | Liese, with wedgie &lt;&lt;3 12 11 12 9 -8||, splits the twelfth interval of 3/1 into three generators of 10/7, using the comma 1029/1000. It also tempers out 686/675, the senga. <a class="wiki_link" href="/74edo">74edo</a> makes for a good liese tuning, though <a class="wiki_link" href="/19edo">19edo</a> can be used. The tuning is well-supplied with MOS: 7, 9, 11, 13, 15, 17, 19, 36, 55.<br /> | ||
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7 and 9 limit minimax 1/4 comma<br /> | 7 and 9 limit minimax 1/4 comma<br /> | ||
[|1 0 0 0&gt;, |1 0 1/4 0&gt;, |0 0 1 0&gt;, |2/3 0 11/12 0&gt;]<br /> | [|1 0 0 0&gt;, |1 0 1/4 0&gt;, |0 0 1 0&gt;, |2/3 0 11/12 0&gt;]<br /> | ||
Eigenmonzos: 2, 5<br /> | Eigenmonzos: 2, 5<br /> | ||
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<a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 632.406<br /> | |||
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Algebraic generator: Radix, the real root of x^5-2x^4+2x^3-2x^2+2x-2, also a root of x^6-x^5-2. The recurrence converges.<br /> | Algebraic generator: Radix, the real root of x^5-2x^4+2x^3-2x^2+2x-2, also a root of x^6-x^5-2. The recurrence converges.<br /> | ||