Schismatic family: Difference between revisions

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=Five limit=

The 5-limit parent comma for the schismatic family is the [[~~schisma|~~schisma]] of 32805/32768, which is the amount by which the Pythagorean comma exceeds the [[~~Didymus_comma|~~Didymus comma]] ([[~~81/80|~~81/80]]), or alternatively put, the difference between a just major third and a Pythagorean diminished fourth. Its [[~~monzo|~~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 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 Pythagorean 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|~~53edo]] is a possible tuning for schismatic, but you need [[~~118edo|~~118edo]] if you want to get the full effect. In exact analogy with 1/4 comma meantone there is also 1/8 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.

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 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.

[[POTE_tuning|POTE generator]]: ~3/2 = 701.736

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Map: [<1 0 15|, <0 1 -8|]

EDOs: [[12edo|12]], [[29edo|29]], [[41edo|41]], [[53edo|53]], [[118edo|118]], [[171edo|171]], [[289edo|289]], 460, 749, 3456bc, 4205bc, 4954bc, 5703bc

EDOs: [[12edo|12]], [[29edo|29]], [[41edo|41]], [[53edo|53]], [[65edo|65]], [[77edo|77]], [[94edo|94]], [[118edo|118]], [[171edo|171]], [[183edo|183]], [[224edo|224]], [[289edo|289]], 407, 460, 749, 3456bc, 4205bc, 4954bc, 5703bc

Badness: 0.00426

=Seven limit children=

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Map: [<1 0 15 25 -33 -28|, <0 1 -8 -14 23 20|]

EDOs: 12, 41, 53, 94, 429cdef, 523cdef

EDOs: 12e, 41, 53, 94, 135, 429cdef, 523cdef

Badness: 0.0207

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Map: [<1 0 15 25 32|, <0 1 -8 -14 -18|]

EDOs: 12, 29, 41~~, 123c~~, 217ce, 258ce

EDOs: 12, 29, 41, 217ce, 258ce

Badness: 0.0236

===13-limit ~~Cassandra~~===

===13-limit===

Commas: 100/99, 105/104, 196/195, 245/242

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Map: [<1 0 15 25 32 37|, <0 1 -8 -14 -18 -21|]

EDOs: 12, 29, 41, 152cdf, 193cdf, 234cdf

EDOs: 12f, 29, 41, 152cdf, 193cdf, 234cdf

Badness: 0.0207

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EDOs: 29, 53, 82e, 135ef, 188cef

Badness: 27.~~464~~

Badness: 0.027464

=Guiron=

[[Comma|Commas]]: 1029/1024, 10976/10935

[[~~Minimax_tuning|~~Minimax tuning]]:

[[Minimax tuning]]:

7+9 limit: [|1 0 0 0>, |15/8 0 -1/8 0>,

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[[Comma|Commas]]: 385/384, 441/440, 10976/10935

[[~~Minimax_tuning|~~Minimax tuning]]:

[[Minimax tuning]]:

[|1 0 0 0 0>, |15/8 0 -1/8 0 0>,

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Map: [<1 1 7 3 -2|, <0 3 -24 -1 28|]

[[EDO|Edos]]: [[41edo|41]], [[77edo|77]], [[118edo|118]], [[159edo|159]], [[200edo|200]], [[~~277~~|277d]]

[[EDO|Edos]]: [[41edo|41]], [[77edo|77]], [[118edo|118]], [[159edo|159]], [[200edo|200]], [[277edo|277d]]

Badness: 0.0266

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Wedgie: <<1 -8 -2 -15 -6 18||

EDOs: 12, 41d, 53d

EDOs: 12, 29d, 41d, 53d

Badness: 0.0566

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[[EDO|Edos]]: [[77edo|77]], [[89edo|89]], [[101edo|101]], [[166edo|166c]], [[243edo|243c]]

Badness: 0.~~070~~

Badness: 0.0704

=Bischismic=

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Wedgie: <<5 -40 24 -75 24 168||

EDOs: 17, ~~26, 43, 60~~, 77, 94, 171

EDOs: 17, 60c, 77, 94, 171

Badness: 0.0307

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Badness: 0.0304

[[Category:definition]]

[[Category:family]]

@@ Line 3: / Line 3: @@
 =Five limit=
-The 5-limit parent comma for the schismatic family is the [[schisma|schisma]] of 32805/32768, which is the amount by which the Pythagorean comma exceeds the [[Didymus_comma|Didymus comma]] ([[81/80|81/80]]), or alternatively put, the difference between a just major third and a Pythagorean diminished fourth. Its [[monzo|monzo]] is |-15 8 1&gt;, and flipping that yields &lt;&lt;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 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 Pythagorean diminished fourth. Its [[monzo]] is |-15 8 1&gt;, and flipping that yields &lt;&lt;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|53edo]] is a possible tuning for schismatic, but you need [[118edo|118edo]] if you want to get the full effect. In exact analogy with 1/4 comma meantone there is also 1/8 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.
+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 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.
 [[POTE_tuning|POTE generator]]: ~3/2 = 701.736
@@ Line 13: / Line 13: @@
 Map: [&lt;1 0 15|, &lt;0 1 -8|]
-EDOs: [[12edo|12]], [[29edo|29]], [[41edo|41]], [[53edo|53]], [[118edo|118]], [[171edo|171]], [[289edo|289]], 460, 749, 3456bc, 4205bc, 4954bc, 5703bc
+EDOs: [[12edo|12]], [[29edo|29]], [[41edo|41]], [[53edo|53]], [[65edo|65]], [[77edo|77]], [[94edo|94]], [[118edo|118]], [[171edo|171]], [[183edo|183]], [[224edo|224]], [[289edo|289]], 407, 460, 749, 3456bc, 4205bc, 4954bc, 5703bc
+Badness: 0.00426
 =Seven limit children=
@@ Line 74: / Line 76: @@
 Map: [&lt;1 0 15 25 -33 -28|, &lt;0 1 -8 -14 23 20|]
-EDOs: 12, 41, 53, 94, 429cdef, 523cdef
+EDOs: 12e, 41, 53, 94, 135, 429cdef, 523cdef
 Badness: 0.0207
@@ Line 87: / Line 89: @@
 Map: [&lt;1 0 15 25 32|, &lt;0 1 -8 -14 -18|]
-EDOs: 12, 29, 41, 123c, 217ce, 258ce
+EDOs: 12, 29, 41, 217ce, 258ce
 Badness: 0.0236
-===13-limit Cassandra===
+===13-limit===
 Commas: 100/99, 105/104, 196/195, 245/242
@@ Line 100: / Line 102: @@
 Map: [&lt;1 0 15 25 32 37|, &lt;0 1 -8 -14 -18 -21|]
-EDOs: 12, 29, 41, 152cdf, 193cdf, 234cdf
+EDOs: 12f, 29, 41, 152cdf, 193cdf, 234cdf
 Badness: 0.0207
@@ Line 154: / Line 156: @@
 EDOs: 29, 53, 82e, 135ef, 188cef
-Badness: 27.464
+Badness: 0.027464
 =Guiron=
 [[Comma|Commas]]: 1029/1024, 10976/10935
-[[Minimax_tuning|Minimax tuning]]:
+[[Minimax tuning]]:
 +9 limit: [|1 0 0 0&gt;, |15/8 0 -1/8 0&gt;,
@@ Line 181: / Line 183: @@
 [[Comma|Commas]]: 385/384, 441/440, 10976/10935
-[[Minimax_tuning|Minimax tuning]]:
+[[Minimax tuning]]:
 [|1 0 0 0 0&gt;, |15/8 0 -1/8 0 0&gt;,
@@ Line 195: / Line 197: @@
 Map: [&lt;1 1 7 3 -2|, &lt;0 3 -24 -1 28|]
-[[EDO|Edos]]: [[41edo|41]], [[77edo|77]], [[118edo|118]], [[159edo|159]], [[200edo|200]], [[277|277d]]
+[[EDO|Edos]]: [[41edo|41]], [[77edo|77]], [[118edo|118]], [[159edo|159]], [[200edo|200]], [[277edo|277d]]
 Badness: 0.0266
@@ Line 315: / Line 317: @@
 Wedgie: &lt;&lt;1 -8 -2 -15 -6 18||
-EDOs: 12, 41d, 53d
+EDOs: 12, 29d, 41d, 53d
 Badness: 0.0566
@@ Line 382: / Line 384: @@
 [[EDO|Edos]]: [[77edo|77]], [[89edo|89]], [[101edo|101]], [[166edo|166c]], [[243edo|243c]]
-Badness: 0.070
+Badness: 0.0704
 =Bischismic=
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 Wedgie: &lt;&lt;5 -40 24 -75 24 168||
-EDOs: 17, 26, 43, 60, 77, 94, 171
+EDOs: 17, 60c, 77, 94, 171
 Badness: 0.0307
@@ Line 688: / Line 690: @@
 Badness: 0.0304
 [[Category:definition]]
 [[Category:family]]