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| <h2>IMPORTED REVISION FROM WIKISPACES</h2>
| | __FORCETOC__ |
| This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
| | <span style="display: block; text-align: right;">Other languages: [[:de:Schismatische_Temperaturen Deutsch]]</span> |
| : This revision was by author [[User:xenwolf|xenwolf]] and made on <tt>2016-09-03 07:24:39 UTC</tt>.<br>
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| : The original revision id was <tt>590996938</tt>.<br>
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| : The revision comment was: <tt>Reverted to Aug 31, 2016 7:12 am: removal of DE link seems to be mistakenly</tt><br>
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| 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>
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| <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">[[toc]] | |
| <span style="display: block; text-align: right;">Other languages: [[xenharmonie/Schismatische Temperaturen|Deutsch]]
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| </span>
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| =Five limit=
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| 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|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.
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| 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. | | =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. |
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| | 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. |
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| | [[POTE_tuning|POTE generator]]: ~3/2 = 701.736 |
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| [[POTE tuning|POTE generator]]: ~3/2 = 701.736
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| Mapping generator: ~3 | | Mapping generator: ~3 |
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| Map: [<1 0 15|, <0 1 -8|] | | Map: [<1 0 15|, <0 1 -8|] |
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| 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]], [[118edo|118]], [[171edo|171]], [[289edo|289]], 460, 749, 3456bc, 4205bc, 4954bc, 5703bc |
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| =Seven limit children= | | =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. | | 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. |
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| =Garibaldi= | | =Garibaldi= |
| Commas: 225/224, 3125/3087 | | Commas: 225/224, 3125/3087 |
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| 7-limit minimax tuning: | | 7-limit minimax tuning: |
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| 7-limit: [|1 0 0 0>, |5/3 1/15 0 -1/15>, | | 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>] | | |5/3 -8/15 0 8/15>, |5/3 -14/15 0 14/15>] |
| [[Eigenmonzo]]s: 2, 7/6 | | |
| | [[Eigenmonzo|Eigenmonzo]]s: 2, 7/6 |
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| 9-limit: [|1 0 0 0>, |25/16 1/8 0 -1/16>, | | 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>] | | |5/2 -1 0 1/2>, |25/8 -7/4 0 7/8>] |
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| Eigenmonzos: 2, 9/7 | | Eigenmonzos: 2, 9/7 |
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| [[POTE tuning|POTE generator]]: ~3/2 = 702.085 | | [[POTE_tuning|POTE generator]]: ~3/2 = 702.085 |
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| Mapping generator: ~3 | | Mapping generator: ~3 |
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| Map: [<1 0 15 25|, <0 1 -8 -14|] | | Map: [<1 0 15 25|, <0 1 -8 -14|] |
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| Wedgie: <<1 -8 -14 -15 -25 -10|| | | Wedgie: <<1 -8 -14 -15 -25 -10|| |
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| EDOs: 12, 29, 41, 53, [[94edo|94]], 241c, 335cd, 576cd | | EDOs: 12, 29, 41, 53, [[94edo|94]], 241c, 335cd, 576cd |
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| Badness: 0.0216 | | Badness: 0.0216 |
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| ==11-limit== | | ==11-limit== |
| Commas: 225/224, 385/384, 2200/2187 | | Commas: 225/224, 385/384, 2200/2187 |
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| Minimax tuning: | | Minimax tuning: |
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| [|1 0 0 0 0>, |25/16 1/8 0 -1/16 0>, |5/2 -1 0 1/2 0>, | | [|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>] | | |25/8 -7/4 0 7/8 0>, |47/16 23/8 0 -23/16 0>] |
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| Eigenmonzos: 2, 9/7 | | Eigenmonzos: 2, 9/7 |
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| [[POTE tuning|POTE generator]]: ~3/2 = 702.157 | | [[POTE_tuning|POTE generator]]: ~3/2 = 702.157 |
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| Mapping generator: ~3 | | Mapping generator: ~3 |
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| Map: [<1 0 15 25 -33|, <0 1 -8 -14 23|] | | Map: [<1 0 15 25 -33|, <0 1 -8 -14 23|] |
| [[edo|Edos]]: [[41edo|41]], [[53edo|53]], [[94edo|94]], [[135edo|135]], 229c, 323c, 417ce | | |
| | [[EDO|Edos]]: [[41edo|41]], [[53edo|53]], [[94edo|94]], [[135edo|135]], 229c, 323c, 417ce |
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| Badness: 0.0274 | | Badness: 0.0274 |
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| ===13-limit=== | | ===13-limit=== |
| Commas: 225/224, 275/273, 325/324, 385/384 | | Commas: 225/224, 275/273, 325/324, 385/384 |
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| [[POTE tuning|POTE generator]]: ~3/2 = 702.113 | | [[POTE_tuning|POTE generator]]: ~3/2 = 702.113 |
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| Mapping generator: ~3 | | Mapping generator: ~3 |
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| Map: [<1 0 15 25 -33 -28|, <0 1 -8 -14 23 20|] | | Map: [<1 0 15 25 -33 -28|, <0 1 -8 -14 23 20|] |
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| EDOs: 12, 41, 53, 94, 429cdef, 523cdef | | EDOs: 12, 41, 53, 94, 429cdef, 523cdef |
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| Badness: 0.0207 | | Badness: 0.0207 |
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| ==Cassandra== | | ==Cassandra== |
| Commas: 100/99, 225/224, 245/242 | | Commas: 100/99, 225/224, 245/242 |
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| [[POTE tuning|POTE generator]]: ~3/2 = 702.321 | | [[POTE_tuning|POTE generator]]: ~3/2 = 702.321 |
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| Mapping generator: ~3 | | Mapping generator: ~3 |
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| Map: [<1 0 15 25 32|, <0 1 -8 -14 -18|] | | Map: [<1 0 15 25 32|, <0 1 -8 -14 -18|] |
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| EDOs: 12, 29, 41, 123c, 217ce, 258ce | | EDOs: 12, 29, 41, 123c, 217ce, 258ce |
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| Badness: 0.0236 | | Badness: 0.0236 |
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| ===13-limit Cassandra=== | | ===13-limit Cassandra=== |
| Commas: 100/99, 105/104, 196/195, 245/242 | | Commas: 100/99, 105/104, 196/195, 245/242 |
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| [[POTE tuning|POTE generator]]: ~3/2 = 702.559 | | [[POTE_tuning|POTE generator]]: ~3/2 = 702.559 |
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| Mapping generator: ~3 | | Mapping generator: ~3 |
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| Map: [<1 0 15 25 32 37|, <0 1 -8 -14 -18 -21|] | | Map: [<1 0 15 25 32 37|, <0 1 -8 -14 -18 -21|] |
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| EDOs: 12, 29, 41, 152cdf, 193cdf, 234cdf | | EDOs: 12, 29, 41, 152cdf, 193cdf, 234cdf |
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| Badness: 0.0207 | | Badness: 0.0207 |
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| ==Helenus== | | ==Helenus== |
| Commas: 99/98 176/175 3125/3087 | | Commas: 99/98 176/175 3125/3087 |
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| POTE generator: ~3/2 = 701.725 | | POTE generator: ~3/2 = 701.725 |
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| Mapping generator: ~3 | | Mapping generator: ~3 |
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| Map: [<1 0 15 25 51|, <0 1 -8 -14 -30|] | | Map: [<1 0 15 25 51|, <0 1 -8 -14 -30|] |
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| EDOs: 12, 53, 118d, 171de | | EDOs: 12, 53, 118d, 171de |
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| Badness: 0.0356 | | Badness: 0.0356 |
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| ===13-limit Helenus=== | | ===13-limit Helenus=== |
| Commas: 99/98, 176/175, 275/273, 847/845 | | Commas: 99/98, 176/175, 275/273, 847/845 |
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| POTE generator: ~3/2 = 701.747 | | POTE generator: ~3/2 = 701.747 |
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| Mapping generator: ~3 | | Mapping generator: ~3 |
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| Map: [<1 0 15 25 51 56|, <0 1 -8 -14 -30 -33|] | | Map: [<1 0 15 25 51 56|, <0 1 -8 -14 -30 -33|] |
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| EDOs: 53, 118d, 171de | | EDOs: 53, 118d, 171de |
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| Badness: 0.0263 | | Badness: 0.0263 |
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| =Hemigari= | | =Hemigari= |
| Commas: 121/120, 225/224, 3125/3087 | | Commas: 121/120, 225/224, 3125/3087 |
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| POTE generator: ~63/55 = 248.918 | | POTE generator: ~63/55 = 248.918 |
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| Mapping generator: ~110/63 | | Mapping generator: ~110/63 |
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| Map: [<1 0 15 25 9|, <0 2 -16 -28 -7|] | | Map: [<1 0 15 25 9|, <0 2 -16 -28 -7|] |
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| EDOs: 29, 53, 82e, 135e, 188ce | | EDOs: 29, 53, 82e, 135e, 188ce |
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| Badness: 0.0507 | | Badness: 0.0507 |
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| ==13-limit== | | ==13-limit== |
| Commas: 121/120, 169/168, 225/224, 275/273 | | Commas: 121/120, 169/168, 225/224, 275/273 |
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| POTE generator: ~15/13 = 248.918 | | POTE generator: ~15/13 = 248.918 |
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| Mapping generator: ~26/15 | | Mapping generator: ~26/15 |
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| Map: [<1 0 15 25 9 14|, <0 2 -16 -28 -7 -13|] | | Map: [<1 0 15 25 9 14|, <0 2 -16 -28 -7 -13|] |
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| EDOs: 29, 53, 82e, 135ef, 188cef | | EDOs: 29, 53, 82e, 135ef, 188cef |
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| Badness: 27.464 | | Badness: 27.464 |
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| =Guiron= | | =Guiron= |
| [[Comma|Commas]]: 1029/1024, 10976/10935 | | [[Comma|Commas]]: 1029/1024, 10976/10935 |
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| [[Minimax tuning]]: | | [[Minimax_tuning|Minimax tuning]]: |
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| 7+9 limit: [|1 0 0 0>, |15/8 0 -1/8 0>, | | 7+9 limit: [|1 0 0 0>, |15/8 0 -1/8 0>, |
| |0 0 1 0>, |65/24 0 1/24 0>] | | |0 0 1 0>, |65/24 0 1/24 0>] |
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| [[Eigenmonzo|Eigenmonzos]]: 2, 5/4 | | [[Eigenmonzo|Eigenmonzos]]: 2, 5/4 |
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| [[POTE tuning|POTE generator]]: ~8/7 = 233.930 | | [[POTE_tuning|POTE generator]]: ~8/7 = 233.930 |
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| Mapping generator: ~8/7 | | Mapping generator: ~8/7 |
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| Map: [<1 1 7 3|, <0 3 -24 -1|] | | Map: [<1 1 7 3|, <0 3 -24 -1|] |
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| Wedgie: <<3 -24 -1 -45 -10 65|| | | Wedgie: <<3 -24 -1 -45 -10 65|| |
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| EDOs: 36, 41, [[77edo|77]], 118, 277d | | EDOs: 36, 41, [[77edo|77]], 118, 277d |
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| Badness: 0.0475 | | Badness: 0.0475 |
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| ==11-limit== | | ==11-limit== |
| [[Comma|Commas]]: 385/384, 441/440, 10976/10935 | | [[Comma|Commas]]: 385/384, 441/440, 10976/10935 |
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| [[Minimax tuning]]: | | [[Minimax_tuning|Minimax tuning]]: |
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| [|1 0 0 0 0>, |15/8 0 -1/8 0 0>, | | [|1 0 0 0 0>, |15/8 0 -1/8 0 0>, |
| |0 0 1 0 0>, |65/24 0 1/24 0 0>, | | |0 0 1 0 0>, |65/24 0 1/24 0 0>, |
| |37/6 0 -7/6 0 0>] | | |37/6 0 -7/6 0 0>] |
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| [[Eigenmonzo|Eigenmonzos]]: 2, 5/4 | | [[Eigenmonzo|Eigenmonzos]]: 2, 5/4 |
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| [[POTE tuning|POTE generator]]: ~8/7 = 233.931 | | [[POTE_tuning|POTE generator]]: ~8/7 = 233.931 |
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| Mapping generator: ~8/7 | | Mapping generator: ~8/7 |
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| Map: [<1 1 7 3 -2|, <0 3 -24 -1 28|] | | 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]], [[277|277d]] |
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| Badness: 0.0266 | | Badness: 0.0266 |
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| ==13-limit== | | ==13-limit== |
| Commas: 196/195, 352/351, 385/384, 729/728 | | Commas: 196/195, 352/351, 385/384, 729/728 |
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| POTE generator: ~8/7 = 233.890 | | POTE generator: ~8/7 = 233.890 |
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| Mapping generator: ~8/7 | | Mapping generator: ~8/7 |
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| Map: [<1 1 7 3 -2 0|, <0 3 -24 -1 28 19|] | | Map: [<1 1 7 3 -2 0|, <0 3 -24 -1 28 19|] |
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| EDOs: 41, 77, 118 | | EDOs: 41, 77, 118 |
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| Badness: 0.0284 | | Badness: 0.0284 |
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| =Pogo= | | =Pogo= |
| Commas: 32805/32768, 118098/117649 | | Commas: 32805/32768, 118098/117649 |
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| POTE generator: ~9/7 = 433.901 | | POTE generator: ~9/7 = 433.901 |
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| Mapping generator: ~9/7 | | Mapping generator: ~9/7 |
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| Map: [<2 1 22 2|, <0 3 -24 5|] | | Map: [<2 1 22 2|, <0 3 -24 5|] |
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| EDOs: 36, 94, 130, 171, 183, 224, 354 | | EDOs: 36, 94, 130, 171, 183, 224, 354 |
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| Badness: 0.0796 | | Badness: 0.0796 |
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| ==11-limit== | | ==11-limit== |
| Commas: 540/539, 4000/3993, 32805/32768 | | Commas: 540/539, 4000/3993, 32805/32768 |
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| POTE generator: ~9/7 = 433.911 | | POTE generator: ~9/7 = 433.911 |
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| Mapping generator: ~9/7 | | Mapping generator: ~9/7 |
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| Map: [<2 1 22 2 25|, <0 3 -24 5 -25|] | | Map: [<2 1 22 2 25|, <0 3 -24 5 -25|] |
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| EDOs: 36, 94, 130, 224, 354, 578 | | EDOs: 36, 94, 130, 224, 354, 578 |
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| Badness: 0.0319 | | Badness: 0.0319 |
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| ==13-limit== | | ==13-limit== |
| Commas: 540/539, 729/728, 4000/3993, 4225/4224 | | Commas: 540/539, 729/728, 4000/3993, 4225/4224 |
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| Line 196: |
Line 244: |
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| Map: [<2 1 22 2 25 -2|, <0 3 -24 5 -25 13|] | | Map: [<2 1 22 2 25 -2|, <0 3 -24 5 -25 13|] |
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| EDOs: 36, 94, 130, 224, 354, 578 | | EDOs: 36, 94, 130, 224, 354, 578 |
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| Badness: 0.0175 | | Badness: 0.0175 |
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| =Sanjaab= | | =Sanjaab= |
| Commas: 225/224, 3125/3087, 1331/1323 | | Commas: 225/224, 3125/3087, 1331/1323 |
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| Line 205: |
Line 255: |
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| Map: [<1 2 -1 -3 0|, <0 -3 24 42 25|] | | Map: [<1 2 -1 -3 0|, <0 -3 24 42 25|] |
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| EDOs: 29, 65d, 94, 441cde, 535cde, 629cde | | EDOs: 29, 65d, 94, 441cde, 535cde, 629cde |
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| Badness: 0.0580 | | Badness: 0.0580 |
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| ==13-limit== | | ==13-limit== |
| Commas: 225/224, 275/273, 847/845, 1331/1323 | | Commas: 225/224, 275/273, 847/845, 1331/1323 |
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| Line 214: |
Line 266: |
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| Map: [<1 2 -1 -3 0 -1|, <0 -3 24 42 25 34|] | | Map: [<1 2 -1 -3 0 -1|, <0 -3 24 42 25 34|] |
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| EDOs: 29, 65d, 94 | | EDOs: 29, 65d, 94 |
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| Badness: 0.0338 | | Badness: 0.0338 |
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| =Squirrel= | | =Squirrel= |
| Commmas: 686/675, 32805/32768 | | Commmas: 686/675, 32805/32768 |
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| Line 223: |
Line 277: |
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| Map: [<1 2 -1 1|, <0 -3 24 13|] | | Map: [<1 2 -1 1|, <0 -3 24 13|] |
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| EDOs: 29, 36, 65 | | EDOs: 29, 36, 65 |
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| Badness: 0.1747 | | Badness: 0.1747 |
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| ==11-limit== | | ==11-limit== |
| Commas: 245/242, 686/675, 896/891 | | Commas: 245/242, 686/675, 896/891 |
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| Line 232: |
Line 288: |
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| Map: [<1 2 -1 1 0|, <0 -3 24 13 25|] | | Map: [<1 2 -1 1 0|, <0 -3 24 13 25|] |
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| EDOs: 29, 36, 65 | | EDOs: 29, 36, 65 |
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| Badness: 0.0683 | | Badness: 0.0683 |
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| ==13-limit== | | ==13-limit== |
| Commas: 91/90, 169/168, 245/242, 896/891 | | Commas: 91/90, 169/168, 245/242, 896/891 |
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| Line 241: |
Line 299: |
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| Map: [<1 2 -1 1 0 3|, <0 -3 24 13 25 5|] | | Map: [<1 2 -1 1 0 3|, <0 -3 24 13 25 5|] |
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| EDOs: 29, 36, 65f, 94df, 159df | | EDOs: 29, 36, 65f, 94df, 159df |
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| Badness: 0.0437 | | Badness: 0.0437 |
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| =Schism= | | =Schism= |
| Commas: 64/63, 360/343 | | Commas: 64/63, 360/343 |
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| POTE generator: ~3/2 = 701.556 | | POTE generator: ~3/2 = 701.556 |
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| Mapping generator: ~3 | | Mapping generator: ~3 |
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| Map: [<1 0 15 6|, <0 1 -8 -2|] | | Map: [<1 0 15 6|, <0 1 -8 -2|] |
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| Wedgie: <<1 -8 -2 -15 -6 18|| | | Wedgie: <<1 -8 -2 -15 -6 18|| |
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| EDOs: 12, 41d, 53d | | EDOs: 12, 41d, 53d |
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| Badness: 0.0566 | | Badness: 0.0566 |
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|
| ==11-limit== | | ==11-limit== |
| Commas: 45/44, 64/63, 99/98 | | Commas: 45/44, 64/63, 99/98 |
|
| |
|
| POTE generator ~3/2 = 702.136 | | POTE generator ~3/2 = 702.136 |
| | |
| Mapping generator: ~3 | | Mapping generator: ~3 |
|
| |
|
| Map: [<1 0 15 6 13|, <0 1 -8 -2 -6|] | | Map: [<1 0 15 6 13|, <0 1 -8 -2 -6|] |
| | |
| EDOs: 12, 29de, 41de | | EDOs: 12, 29de, 41de |
| | |
| Badness: 0.0375 | | Badness: 0.0375 |
|
| |
|
| =Pontiac= | | =Pontiac= |
| Commas: 32805/32768, 4375/4374 | | Commas: 32805/32768, 4375/4374 |
|
| |
|
| 7-limit minimax: | | 7-limit minimax: |
| | |
| [|1 0 0 0>, |74/47 0 -1/47 1/47>, |113/47 0 8/47 -8/47>, | | [|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>] | | |113/47 0 -39/47 39/47>] |
| | |
| Eigenmonzos: 2, 7/5 | | Eigenmonzos: 2, 7/5 |
|
| |
|
| 9-limit minimax: | | 9-limit minimax: |
| | |
| [|1 0 0 0>, |3/2 1/5 -1/10 0>, | | [|1 0 0 0>, |3/2 1/5 -1/10 0>, |
| |3 -8/5 4/5 0>, |-1/2 39/5 -39/10 0>] | | |3 -8/5 4/5 0>, |-1/2 39/5 -39/10 0>] |
| | |
| Eigenmonzos: 2, 10/9 | | Eigenmonzos: 2, 10/9 |
|
| |
|
| [[POTE tuning|POTE generator]]: 701.757 | | [[POTE_tuning|POTE generator]]: 701.757 |
| | |
| Mapping generator: ~3 | | Mapping generator: ~3 |
|
| |
|
| Map: [<1 0 15 -59|, <0 1 -8 39|] | | Map: [<1 0 15 -59|, <0 1 -8 39|] |
| | |
| Wedgie: <<1 -8 39 -15 59 113|| | | Wedgie: <<1 -8 39 -15 59 113|| |
| [[edo|Edos]]: [[53edo|53]], [[118edo|118]], [[171edo|171]], [[1079edo|1079]], [[1250edo|1250]], [[1421edo|1421]] | | |
| | [[EDO|Edos]]: [[53edo|53]], [[118edo|118]], [[171edo|171]], [[1079edo|1079]], [[1250edo|1250]], [[1421edo|1421]] |
| | |
| Badness: 0.0141 | | Badness: 0.0141 |
|
| |
|
| =Grackle= | | =Grackle= |
| Commas: 126/125, 32805/32768 | | Commas: 126/125, 32805/32768 |
|
| |
|
| 7-limit minimax | | 7-limit minimax |
| | |
| Eigenmonzos: 2, 7/6 | | Eigenmonzos: 2, 7/6 |
|
| |
|
| 9-limit minimax | | 9-limit minimax |
| | |
| Eigenmonzos: 2, 9/7 | | Eigenmonzos: 2, 9/7 |
|
| |
|
| [[POTE tuning|POTE generator]]: 701.239 | | [[POTE_tuning|POTE generator]]: 701.239 |
| | |
| Mapping generator: ~3 | | Mapping generator: ~3 |
|
| |
|
| Map: [<1 0 15 -44|, <0 1 -8 -26|] | | Map: [<1 0 15 -44|, <0 1 -8 -26|] |
| | |
| Wedgie: <<1 -8 -26 -15 -44 -38|| | | Wedgie: <<1 -8 -26 -15 -44 -38|| |
| [[edo|Edos]]: [[77edo|77]], [[89edo|89]], [[101edo|101]], [[166edo|166c]], [[243edo|243c]] | | |
| | [[EDO|Edos]]: [[77edo|77]], [[89edo|89]], [[101edo|101]], [[166edo|166c]], [[243edo|243c]] |
| | |
| Badness: 0.070 | | Badness: 0.070 |
|
| |
|
| =Bischismic= | | =Bischismic= |
| Commas: 3136/3125, 32805/32768 | | Commas: 3136/3125, 32805/32768 |
|
| |
|
| 7-limit minimax | | 7-limit minimax |
| | |
| Eigenmonzos: 2, 7/6 | | Eigenmonzos: 2, 7/6 |
|
| |
|
| 9-limit minimax | | 9-limit minimax |
| | |
| Eigenmonzos: 2, 9/7 | | Eigenmonzos: 2, 9/7 |
|
| |
|
| [[POTE tuning|POTE generator]]: 701.592 | | [[POTE_tuning|POTE generator]]: 701.592 |
| | |
| Mapping generator: ~3 | | Mapping generator: ~3 |
|
| |
|
| Map: [<2 0 30 69|, <0 1 -8 -20|] | | Map: [<2 0 30 69|, <0 1 -8 -20|] |
| | |
| Wedgie: <<2 -16 -40 -30 -69 -48|| | | Wedgie: <<2 -16 -40 -30 -69 -48|| |
| [[edo|Edos]]: [[118edo|118]], [[130edo|130]], [[248edo|248]], [[378edo|378]], [[508edo|508]] | | |
| | [[EDO|Edos]]: [[118edo|118]], [[130edo|130]], [[248edo|248]], [[378edo|378]], [[508edo|508]] |
| | |
| Badness: 0.0547 | | Badness: 0.0547 |
|
| |
|
| ==11-limit== | | ==11-limit== |
| Commas: 441/440, 3136/3125, 8019/8000 | | Commas: 441/440, 3136/3125, 8019/8000 |
|
| |
|
| POTE generator: ~3/2 = 701.612 | | POTE generator: ~3/2 = 701.612 |
| | |
| Mapping generator: ~3 | | Mapping generator: ~3 |
|
| |
|
| Map: [<2 0 30 69 102|, <0 1 -8 -20 -30|] | | Map: [<2 0 30 69 102|, <0 1 -8 -20 -30|] |
| | |
| EDOs: 12, 118, 130, 248 | | EDOs: 12, 118, 130, 248 |
| | |
| Badness: 0.0282 | | Badness: 0.0282 |
|
| |
|
| ==13-limit== | | ==13-limit== |
| Commas: 441/440, 729/728, 1001/1000, 3136/3125 | | Commas: 441/440, 729/728, 1001/1000, 3136/3125 |
|
| |
|
| POTE generator: ~3/2 = 701.590 | | POTE generator: ~3/2 = 701.590 |
| | |
| Mapping generator: ~3 | | Mapping generator: ~3 |
|
| |
|
| Map: [<2 0 30 69 102 -75|, <0 1 -8 -20 -30 26|] | | Map: [<2 0 30 69 102 -75|, <0 1 -8 -20 -30 26|] |
| | |
| EDOs: 12, 118, 130, 248, 378 | | EDOs: 12, 118, 130, 248, 378 |
| | |
| Badness: 0.0287 | | Badness: 0.0287 |
|
| |
|
| =Kleischismic= | | =Kleischismic= |
| Commas: 32805/32768, 1500625/1492992 | | Commas: 32805/32768, 1500625/1492992 |
|
| |
|
| POTE generator: ~36/35 = 50.920 | | POTE generator: ~36/35 = 50.920 |
| | |
| Mapping generator: ~35/24 | | Mapping generator: ~35/24 |
|
| |
|
| Map: [<2 1 22 -15|, <0 2 -16 19|] | | Map: [<2 1 22 -15|, <0 2 -16 19|] |
| | |
| Wedgie: <<4 -32 38 -60 49 178|| | | Wedgie: <<4 -32 38 -60 49 178|| |
| | |
| EDOs: 24, 94, 118, 212, 330, 542d, 872cd | | EDOs: 24, 94, 118, 212, 330, 542d, 872cd |
| | |
| Badness: 0.1106 | | Badness: 0.1106 |
|
| |
|
| ==11-limit== | | ==11-limit== |
| Commas: 385/384, 9801/9800, 14641/14580 | | Commas: 385/384, 9801/9800, 14641/14580 |
|
| |
|
| POTE generator: ~36/35 = 50.918 | | POTE generator: ~36/35 = 50.918 |
| | |
| Mapping generator: ~16/11 | | Mapping generator: ~16/11 |
|
| |
|
| Map: [<2 1 22 -15 8|, <0 2 -16 19 -1|] | | Map: [<2 1 22 -15 8|, <0 2 -16 19 -1|] |
| | |
| EDOs: 24, 94, 118, 212, 330e, 542de | | EDOs: 24, 94, 118, 212, 330e, 542de |
| | |
| Badness: 0.0367 | | Badness: 0.0367 |
|
| |
|
| ==13-limit== | | ==13-limit== |
| Commas: 352/351, 385/384, 729/728, 1575/1573 | | Commas: 352/351, 385/384, 729/728, 1575/1573 |
|
| |
|
| POTE generator: ~36/35 = 50.938 | | POTE generator: ~36/35 = 50.938 |
| | |
| Mapping generator: ~16/11 | | Mapping generator: ~16/11 |
|
| |
|
| Map: [<2 1 22 -15 8 15|, <0 2 -16 19 -1 -7|] | | Map: [<2 1 22 -15 8 15|, <0 2 -16 19 -1 -7|] |
| | |
| EDOs: 24, 94, 118, 212f | | EDOs: 24, 94, 118, 212f |
| | |
| Badness: 0.0376 | | Badness: 0.0376 |
|
| |
|
| =Term= | | =Term= |
| Commas: 32805/32768, 250047/250000 | | Commas: 32805/32768, 250047/250000 |
|
| |
|
| 7-limit minimax | | 7-limit minimax |
| | |
| Eigenmonzos: 2, 6/5 | | Eigenmonzos: 2, 6/5 |
|
| |
|
| 9-limit minimax | | 9-limit minimax |
| | |
| Eigenmonzos: 2, 9/7 | | Eigenmonzos: 2, 9/7 |
|
| |
|
| [[POTE tuning|POTE generator]]: ~3/2 = 701.742 | | [[POTE_tuning|POTE generator]]: ~3/2 = 701.742 |
| | |
| Mapping generator: ~3 | | Mapping generator: ~3 |
|
| |
|
| Map: [<3 0 45 94|, <0 1 -8 -18|] | | Map: [<3 0 45 94|, <0 1 -8 -18|] |
| | |
| Wedgie: <<3 -24 -54 -45 -94 -58|| | | Wedgie: <<3 -24 -54 -45 -94 -58|| |
| [[edo|Edos]]: [[171edo|171]], [[1038edo|1038]], [[1209edo|1209]] | | |
| | [[EDO|Edos]]: [[171edo|171]], [[1038edo|1038]], [[1209edo|1209]] |
| | |
| Badness: 0.0200 | | Badness: 0.0200 |
|
| |
|
| =Sesquiquartififths= | | =Sesquiquartififths= |
| Commas: 2401/2400, 32805/32768 | | Commas: 2401/2400, 32805/32768 |
|
| |
|
| 7-limit minimax | | 7-limit minimax |
| | |
| Eigenmonzos: 2, 7/6 | | Eigenmonzos: 2, 7/6 |
|
| |
|
| 9-limit minimax | | 9-limit minimax |
| | |
| Eigenmonzos: 2, 9/7 | | Eigenmonzos: 2, 9/7 |
|
| |
|
| [[POTE tuning|POTE generator]]: ~448/405 = 175.434 | | [[POTE_tuning|POTE generator]]: ~448/405 = 175.434 |
| | |
| Mapping generator: ~448/405 | | Mapping generator: ~448/405 |
|
| |
|
| Map: [<1 1 7 5|, <0 4 -32 -15|] | | Map: [<1 1 7 5|, <0 4 -32 -15|] |
| | |
| Wedgie: <<4 -32 -15 -60 -35 55|| | | Wedgie: <<4 -32 -15 -60 -35 55|| |
| [[edo|Edos]]: [[171edo|171]], [[643edo|643]], [[2100edo]] | | |
| | [[EDO|Edos]]: [[171edo|171]], [[643edo|643]], [[2100edo|2100edo]] |
| | |
| Badness: 0.0112 | | Badness: 0.0112 |
|
| |
|
| ==Sesquart== | | ==Sesquart== |
| Commas: 243/242, 441/440, 16384/16335 | | Commas: 243/242, 441/440, 16384/16335 |
|
| |
|
| POTE generator: ~256/231 = 175.406 | | POTE generator: ~256/231 = 175.406 |
| | |
| Mapping generator: ~256/231 | | Mapping generator: ~256/231 |
|
| |
|
| Map: [<1 1 7 5 2|, <0 4 -32 -15 10|] | | Map: [<1 1 7 5 2|, <0 4 -32 -15 10|] |
| | |
| EDOs: 41, 89, 130 | | EDOs: 41, 89, 130 |
| | |
| Badness: 0.0293 | | Badness: 0.0293 |
|
| |
|
| ===13-limit=== | | ===13-limit=== |
| Commas: 243/242, 364/363, 441/440, 105644/105625 | | Commas: 243/242, 364/363, 441/440, 105644/105625 |
|
| |
|
| Line 421: |
Line 539: |
|
| |
|
| Map: [<1 1 7 5 2 -2|, <0 4 -32 -15 10 39|] | | Map: [<1 1 7 5 2 -2|, <0 4 -32 -15 10 39|] |
| | |
| EDOs: 41, 89, 130, 301e, 431e | | EDOs: 41, 89, 130, 301e, 431e |
| | |
| Badness: 0.0224 | | Badness: 0.0224 |
|
| |
|
| ==Bisesqui== | | ==Bisesqui== |
| Commas: 2401/2400, 9801/9800, 32805/32768 | | Commas: 2401/2400, 9801/9800, 32805/32768 |
|
| |
|
| [[POTE tuning|POTE generator]]: ~448/405 = 175.435 | | [[POTE_tuning|POTE generator]]: ~448/405 = 175.435 |
|
| |
|
| Map: [<2 2 14 10 23|, <0 4 -32 -15 -55|] | | Map: [<2 2 14 10 23|, <0 4 -32 -15 -55|] |
| | |
| EDOs: 130, 212, 342, 1156, 1498 | | EDOs: 130, 212, 342, 1156, 1498 |
| | |
| Badness: 0.0170 | | Badness: 0.0170 |
|
| |
|
| =Sextilififths= | | =Sextilififths= |
| Commas: 32768/32805, 235298/234375 | | Commas: 32768/32805, 235298/234375 |
|
| |
|
| POTE generator: ~21/20 = 83.053 | | POTE generator: ~21/20 = 83.053 |
| | |
| Mapping generator: ~21/20 | | Mapping generator: ~21/20 |
|
| |
|
| Map: [<1 2 -1 -1|, <0 -6 48 55|] | | Map: [<1 2 -1 -1|, <0 -6 48 55|] |
| | |
| EDOs: 29, 101, 130, 289, 419 | | EDOs: 29, 101, 130, 289, 419 |
| | |
| Badness: 0.1088 | | Badness: 0.1088 |
|
| |
|
| ==11-limit== | | ==11-limit== |
| Commas: 441/440, 4000/3993, 235298/234375 | | Commas: 441/440, 4000/3993, 235298/234375 |
|
| |
|
| POTE generator: ~21/20 = 83.049 | | POTE generator: ~21/20 = 83.049 |
| | |
| Mapping generator: ~21/20 | | Mapping generator: ~21/20 |
|
| |
|
| Map: [<1 2 -1 -1 0|, <0 -6 48 55 50|] | | Map: [<1 2 -1 -1 0|, <0 -6 48 55 50|] |
| | |
| EDOs: 29, 130, 289 | | EDOs: 29, 130, 289 |
| | |
| Badness: 0.0455 | | Badness: 0.0455 |
|
| |
|
| ==13-limit== | | ==13-limit== |
| Commas: 363/363, 441/440, 676/675, 10985/10976 | | Commas: 363/363, 441/440, 676/675, 10985/10976 |
|
| |
|
| POTE generator: ~21/20 = 83.049 | | POTE generator: ~21/20 = 83.049 |
| | |
| Mapping generator: ~21/20 | | Mapping generator: ~21/20 |
|
| |
|
| Map: [<1 2 -1 -1 0 1|, <0 -6 48 55 50 39|] | | Map: [<1 2 -1 -1 0 1|, <0 -6 48 55 50 39|] |
| | |
| EDOs: 29, 130, 289 | | EDOs: 29, 130, 289 |
| | |
| Badness: 0.0253 | | Badness: 0.0253 |
|
| |
|
| =Tsaharuk= | | =Tsaharuk= |
| Commas: 32805/32768, 420175/419904 | | Commas: 32805/32768, 420175/419904 |
|
| |
|
| POTE generator: ~243/224 = 140.350 | | POTE generator: ~243/224 = 140.350 |
| | |
| Mapping generator: ~243/224 | | Mapping generator: ~243/224 |
|
| |
|
| Map: [<1 1 7 0|, <0 5 -40 24|] | | Map: [<1 1 7 0|, <0 5 -40 24|] |
| | |
| Wedgie: <<5 -40 24 -75 24 168|| | | Wedgie: <<5 -40 24 -75 24 168|| |
| | |
| EDOs: 17, 26, 43, 60, 77, 94, 171 | | EDOs: 17, 26, 43, 60, 77, 94, 171 |
| | |
| Badness: 0.0307 | | Badness: 0.0307 |
|
| |
|
| =Quanharuk= | | =Quanharuk= |
| Commas: 16875/16807, 32805/32768 | | Commas: 16875/16807, 32805/32768 |
|
| |
|
| POTE generator: ~56/45 = 380.355 | | POTE generator: ~56/45 = 380.355 |
| | |
| Mapping generator: ~56/45 | | Mapping generator: ~56/45 |
|
| |
|
| Map: [<1 0 15 12|, <0 5 -40 -29|] | | Map: [<1 0 15 12|, <0 5 -40 -29|] |
| | |
| Wedgie: <<5 -40 -29 -75 -60 45|| | | Wedgie: <<5 -40 -29 -75 -60 45|| |
| | |
| EDOs: 41, 142, 183, 224, 1303d, 1527cd, 1751cd, 1975cd | | EDOs: 41, 142, 183, 224, 1303d, 1527cd, 1751cd, 1975cd |
| | |
| Badness: 0.0720 | | Badness: 0.0720 |
|
| |
|
| ==11-limit== | | ==11-limit== |
| Commas: 540/539, 1375/1372, 32805/32768 | | Commas: 540/539, 1375/1372, 32805/32768 |
|
| |
|
| POTE generator: ~56/45 = 380.352 | | POTE generator: ~56/45 = 380.352 |
| | |
| Mapping generator: ~56/45 | | Mapping generator: ~56/45 |
|
| |
|
| Map: [<1 0 15 12 -7|, <0 5 -40 -29 33|] | | Map: [<1 0 15 12 -7|, <0 5 -40 -29 33|] |
| | |
| EDOs: 41, 142, 183, 224, 631d, 855d, 1079d | | EDOs: 41, 142, 183, 224, 631d, 855d, 1079d |
| | |
| Badness: 0.0315 | | Badness: 0.0315 |
|
| |
|
| ==13-limit== | | ==13-limit== |
| Commas: 540/539, 1375/1372, 4096/4095, 6656/6655 | | Commas: 540/539, 1375/1372, 4096/4095, 6656/6655 |
|
| |
|
| Line 501: |
Line 643: |
|
| |
|
| Map: [<1 0 15 12 -7 -15|, <0 5 -40 -29 33 59|] | | Map: [<1 0 15 12 -7 -15|, <0 5 -40 -29 33 59|] |
| | |
| EDOs: 41, 142, 183, 224, 631d, 855d | | EDOs: 41, 142, 183, 224, 631d, 855d |
| | |
| Badness: 0.0214 | | Badness: 0.0214 |
|
| |
|
| =Octant= | | =Octant= |
| Commas: 32805/32768, 2259436291848/2251875390625 | | Commas: 32805/32768, 2259436291848/2251875390625 |
|
| |
|
| POTE generator: ~3/2 = 701.713 | | POTE generator: ~3/2 = 701.713 |
| | |
| Mapping generator: ~3 | | Mapping generator: ~3 |
|
| |
|
| Map: [<8 0 120 -117|, <0 1 -8 11|] | | Map: [<8 0 120 -117|, <0 1 -8 11|] |
| | |
| Wedgie: <<8 -64 88 -120 117 384|| | | Wedgie: <<8 -64 88 -120 117 384|| |
| | |
| EDOs: 24, 224, 472, 696, 1168 | | EDOs: 24, 224, 472, 696, 1168 |
| | |
| Badness: 0.1572 | | Badness: 0.1572 |
|
| |
|
| ==11-limit== | | ==11-limit== |
| Commas: 9801/9800, 32805/32768, 46656/46585 | | Commas: 9801/9800, 32805/32768, 46656/46585 |
|
| |
|
| POTE generator: ~3/2 = 701.713 | | POTE generator: ~3/2 = 701.713 |
| | |
| Mapping generator: ~3 | | Mapping generator: ~3 |
|
| |
|
| Map: [<8 0 120 -117 15|, <0 1 -8 11 1|] | | Map: [<8 0 120 -117 15|, <0 1 -8 11 1|] |
| | |
| EDOs: 24, 224, 472, 696, 1168 | | EDOs: 24, 224, 472, 696, 1168 |
| | |
| Badness: 0.0448 | | Badness: 0.0448 |
|
| |
|
| ==13-limit== | | ==13-limit== |
| Commas: 729/728, 1575/1573, 2200/2197, 6656/6655 | | Commas: 729/728, 1575/1573, 2200/2197, 6656/6655 |
|
| |
|
| POTE generator: ~3/2 = 701.725 | | POTE generator: ~3/2 = 701.725 |
| | |
| Mapping generator: ~3 | | Mapping generator: ~3 |
|
| |
|
| Map: [<8 0 120 -117 15 93|, <0 1 -8 11 1 -5|] | | Map: [<8 0 120 -117 15 93|, <0 1 -8 11 1 -5|] |
| | |
| EDOs: 24, 224, 472, 696 | | EDOs: 24, 224, 472, 696 |
| Badness: 0.0304</pre></div> | | |
| <h4>Original HTML content:</h4>
| | Badness: 0.0304 |
| <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>Schismatic family</title></head><body><!-- ws:start:WikiTextTocRule:94:&lt;img id=&quot;wikitext@@toc@@normal&quot; class=&quot;WikiMedia WikiMediaToc&quot; title=&quot;Table of Contents&quot; src=&quot;/site/embedthumbnail/toc/normal?w=225&amp;h=100&quot;/&gt; --><div id="toc"><h1 class="nopad">Table of Contents</h1><!-- ws:end:WikiTextTocRule:94 --><!-- ws:start:WikiTextTocRule:95: --><div style="margin-left: 1em;"><a href="#Five limit">Five limit</a></div>
| | [[Category:definition]] |
| <!-- ws:end:WikiTextTocRule:95 --><!-- ws:start:WikiTextTocRule:96: --><div style="margin-left: 1em;"><a href="#Seven limit children">Seven limit children</a></div>
| | [[Category:family]] |
| <!-- ws:end:WikiTextTocRule:96 --><!-- ws:start:WikiTextTocRule:97: --><div style="margin-left: 1em;"><a href="#Garibaldi">Garibaldi</a></div>
| | [[Category:index]] |
| <!-- ws:end:WikiTextTocRule:97 --><!-- ws:start:WikiTextTocRule:98: --><div style="margin-left: 2em;"><a href="#Garibaldi-11-limit">11-limit</a></div>
| | [[Category:list]] |
| <!-- ws:end:WikiTextTocRule:98 --><!-- ws:start:WikiTextTocRule:99: --><div style="margin-left: 3em;"><a href="#Garibaldi-11-limit-13-limit">13-limit</a></div>
| | [[Category:overview]] |
| <!-- ws:end:WikiTextTocRule:99 --><!-- ws:start:WikiTextTocRule:100: --><div style="margin-left: 2em;"><a href="#Garibaldi-Cassandra">Cassandra</a></div>
| | [[Category:schismatic]] |
| <!-- ws:end:WikiTextTocRule:100 --><!-- ws:start:WikiTextTocRule:101: --><div style="margin-left: 3em;"><a href="#Garibaldi-Cassandra-13-limit Cassandra">13-limit Cassandra</a></div>
| | [[Category:theory]] |
| <!-- ws:end:WikiTextTocRule:101 --><!-- ws:start:WikiTextTocRule:102: --><div style="margin-left: 2em;"><a href="#Garibaldi-Helenus">Helenus</a></div>
| |
| <!-- ws:end:WikiTextTocRule:102 --><!-- ws:start:WikiTextTocRule:103: --><div style="margin-left: 3em;"><a href="#Garibaldi-Helenus-13-limit Helenus">13-limit Helenus</a></div>
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| <!-- ws:end:WikiTextTocRule:103 --><!-- ws:start:WikiTextTocRule:104: --><div style="margin-left: 1em;"><a href="#Hemigari">Hemigari</a></div>
| |
| <!-- ws:end:WikiTextTocRule:104 --><!-- ws:start:WikiTextTocRule:105: --><div style="margin-left: 2em;"><a href="#Hemigari-13-limit">13-limit</a></div>
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| <!-- ws:end:WikiTextTocRule:105 --><!-- ws:start:WikiTextTocRule:106: --><div style="margin-left: 1em;"><a href="#Guiron">Guiron</a></div>
| |
| <!-- ws:end:WikiTextTocRule:106 --><!-- ws:start:WikiTextTocRule:107: --><div style="margin-left: 2em;"><a href="#Guiron-11-limit">11-limit</a></div>
| |
| <!-- ws:end:WikiTextTocRule:107 --><!-- ws:start:WikiTextTocRule:108: --><div style="margin-left: 2em;"><a href="#Guiron-13-limit">13-limit</a></div>
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| <!-- ws:end:WikiTextTocRule:108 --><!-- ws:start:WikiTextTocRule:109: --><div style="margin-left: 1em;"><a href="#Pogo">Pogo</a></div>
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| <!-- ws:end:WikiTextTocRule:109 --><!-- ws:start:WikiTextTocRule:110: --><div style="margin-left: 2em;"><a href="#Pogo-11-limit">11-limit</a></div>
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| <!-- ws:end:WikiTextTocRule:110 --><!-- ws:start:WikiTextTocRule:111: --><div style="margin-left: 2em;"><a href="#Pogo-13-limit">13-limit</a></div>
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| <!-- ws:end:WikiTextTocRule:111 --><!-- ws:start:WikiTextTocRule:112: --><div style="margin-left: 1em;"><a href="#Sanjaab">Sanjaab</a></div>
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| <!-- ws:end:WikiTextTocRule:112 --><!-- ws:start:WikiTextTocRule:113: --><div style="margin-left: 2em;"><a href="#Sanjaab-13-limit">13-limit</a></div>
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| <!-- ws:end:WikiTextTocRule:113 --><!-- ws:start:WikiTextTocRule:114: --><div style="margin-left: 1em;"><a href="#Squirrel">Squirrel</a></div>
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| <!-- ws:end:WikiTextTocRule:114 --><!-- ws:start:WikiTextTocRule:115: --><div style="margin-left: 2em;"><a href="#Squirrel-11-limit">11-limit</a></div>
| |
| <!-- ws:end:WikiTextTocRule:115 --><!-- ws:start:WikiTextTocRule:116: --><div style="margin-left: 2em;"><a href="#Squirrel-13-limit">13-limit</a></div>
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| <!-- ws:end:WikiTextTocRule:116 --><!-- ws:start:WikiTextTocRule:117: --><div style="margin-left: 1em;"><a href="#Schism">Schism</a></div>
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| <!-- ws:end:WikiTextTocRule:117 --><!-- ws:start:WikiTextTocRule:118: --><div style="margin-left: 2em;"><a href="#Schism-11-limit">11-limit</a></div>
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| <!-- ws:end:WikiTextTocRule:118 --><!-- ws:start:WikiTextTocRule:119: --><div style="margin-left: 1em;"><a href="#Pontiac">Pontiac</a></div>
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| <!-- ws:end:WikiTextTocRule:119 --><!-- ws:start:WikiTextTocRule:120: --><div style="margin-left: 1em;"><a href="#Grackle">Grackle</a></div>
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| <!-- ws:end:WikiTextTocRule:120 --><!-- ws:start:WikiTextTocRule:121: --><div style="margin-left: 1em;"><a href="#Bischismic">Bischismic</a></div>
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| <!-- ws:end:WikiTextTocRule:121 --><!-- ws:start:WikiTextTocRule:122: --><div style="margin-left: 2em;"><a href="#Bischismic-11-limit">11-limit</a></div>
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| <!-- ws:end:WikiTextTocRule:122 --><!-- ws:start:WikiTextTocRule:123: --><div style="margin-left: 2em;"><a href="#Bischismic-13-limit">13-limit</a></div>
| |
| <!-- ws:end:WikiTextTocRule:123 --><!-- ws:start:WikiTextTocRule:124: --><div style="margin-left: 1em;"><a href="#Kleischismic">Kleischismic</a></div>
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| <!-- ws:end:WikiTextTocRule:124 --><!-- ws:start:WikiTextTocRule:125: --><div style="margin-left: 2em;"><a href="#Kleischismic-11-limit">11-limit</a></div>
| |
| <!-- ws:end:WikiTextTocRule:125 --><!-- ws:start:WikiTextTocRule:126: --><div style="margin-left: 2em;"><a href="#Kleischismic-13-limit">13-limit</a></div>
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| <!-- ws:end:WikiTextTocRule:126 --><!-- ws:start:WikiTextTocRule:127: --><div style="margin-left: 1em;"><a href="#Term">Term</a></div>
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| <!-- ws:end:WikiTextTocRule:127 --><!-- ws:start:WikiTextTocRule:128: --><div style="margin-left: 1em;"><a href="#Sesquiquartififths">Sesquiquartififths</a></div>
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| <!-- ws:end:WikiTextTocRule:128 --><!-- ws:start:WikiTextTocRule:129: --><div style="margin-left: 2em;"><a href="#Sesquiquartififths-Sesquart">Sesquart</a></div>
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| <!-- ws:end:WikiTextTocRule:129 --><!-- ws:start:WikiTextTocRule:130: --><div style="margin-left: 3em;"><a href="#Sesquiquartififths-Sesquart-13-limit">13-limit</a></div>
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| <!-- ws:end:WikiTextTocRule:130 --><!-- ws:start:WikiTextTocRule:131: --><div style="margin-left: 2em;"><a href="#Sesquiquartififths-Bisesqui">Bisesqui</a></div>
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| <!-- ws:end:WikiTextTocRule:131 --><!-- ws:start:WikiTextTocRule:132: --><div style="margin-left: 1em;"><a href="#Sextilififths">Sextilififths</a></div>
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| <!-- ws:end:WikiTextTocRule:132 --><!-- ws:start:WikiTextTocRule:133: --><div style="margin-left: 2em;"><a href="#Sextilififths-11-limit">11-limit</a></div>
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| <!-- ws:end:WikiTextTocRule:133 --><!-- ws:start:WikiTextTocRule:134: --><div style="margin-left: 2em;"><a href="#Sextilififths-13-limit">13-limit</a></div>
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| <!-- ws:end:WikiTextTocRule:134 --><!-- ws:start:WikiTextTocRule:135: --><div style="margin-left: 1em;"><a href="#Tsaharuk">Tsaharuk</a></div>
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| <!-- ws:end:WikiTextTocRule:135 --><!-- ws:start:WikiTextTocRule:136: --><div style="margin-left: 1em;"><a href="#Quanharuk">Quanharuk</a></div>
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| <!-- ws:end:WikiTextTocRule:136 --><!-- ws:start:WikiTextTocRule:137: --><div style="margin-left: 2em;"><a href="#Quanharuk-11-limit">11-limit</a></div>
| |
| <!-- ws:end:WikiTextTocRule:137 --><!-- ws:start:WikiTextTocRule:138: --><div style="margin-left: 2em;"><a href="#Quanharuk-13-limit">13-limit</a></div>
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| <!-- ws:end:WikiTextTocRule:138 --><!-- ws:start:WikiTextTocRule:139: --><div style="margin-left: 1em;"><a href="#Octant">Octant</a></div>
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| <!-- ws:end:WikiTextTocRule:139 --><!-- ws:start:WikiTextTocRule:140: --><div style="margin-left: 2em;"><a href="#Octant-11-limit">11-limit</a></div>
| |
| <!-- ws:end:WikiTextTocRule:140 --><!-- ws:start:WikiTextTocRule:141: --><div style="margin-left: 2em;"><a href="#Octant-13-limit">13-limit</a></div>
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| <!-- ws:end:WikiTextTocRule:141 --><!-- ws:start:WikiTextTocRule:142: --></div>
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| <!-- ws:end:WikiTextTocRule:142 --><span style="display: block; text-align: right;">Other languages: <a class="wiki_link" href="http://xenharmonie.wikispaces.com/Schismatische%20Temperaturen">Deutsch</a><br />
| |
| </span><br />
| |
| <!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Five limit"></a><!-- ws:end:WikiTextHeadingRule:0 -->Five limit</h1>
| |
| The 5-limit parent comma for the schismatic family is the <a class="wiki_link" href="/schisma">schisma</a> of 32805/32768, which is the amount by which the Pythagorean comma exceeds the <a class="wiki_link" href="/Didymus%20comma">Didymus comma</a> (<a class="wiki_link" href="/81_80">81/80</a>), or alternatively put, the difference between a just major third and a Pythagorean diminished fourth. Its <a class="wiki_link" href="/monzo">monzo</a> is |-15 8 1&gt;, and flipping that yields &lt;&lt;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 <strong>Helmholtz</strong> 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 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 />
| |
| <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: ~3/2 = 701.736<br />
| |
| Mapping generator: ~3<br />
| |
| <br />
| |
| Map: [&lt;1 0 15|, &lt;0 1 -8|]<br />
| |
| EDOs: <a class="wiki_link" href="/12edo">12</a>, <a class="wiki_link" href="/29edo">29</a>, <a class="wiki_link" href="/41edo">41</a>, <a class="wiki_link" href="/53edo">53</a>, <a class="wiki_link" href="/118edo">118</a>, <a class="wiki_link" href="/171edo">171</a>, <a class="wiki_link" href="/289edo">289</a>, 460, 749, 3456bc, 4205bc, 4954bc, 5703bc<br />
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| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:2:&lt;h1&gt; --><h1 id="toc1"><a name="Seven limit children"></a><!-- ws:end:WikiTextHeadingRule:2 -->Seven limit children</h1>
| |
| 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&gt; gives garibaldi, |-44 26 0 1&gt; grackle, |6 -2 0 -1&gt; schism and |-59 39 0 -1&gt; pontiac; these all have a fifth as generator. Bischismic adds |-69 40 0 2&gt; and has a fifth generator with a half-octave period. Guiron adds 1029/1024 = |-10 1 0 3&gt;, with an 8/7 generator, three of which give the fifth, and term adds |-94 54 0 3&gt; with a 1/3 octave period. Sesquiquartififths adds |-35 15 0 4&gt; and slices the fifth in four.<br />
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| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:4:&lt;h1&gt; --><h1 id="toc2"><a name="Garibaldi"></a><!-- ws:end:WikiTextHeadingRule:4 -->Garibaldi</h1>
| |
| Commas: 225/224, 3125/3087<br />
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| <br />
| |
| 7-limit minimax tuning:<br />
| |
| 7-limit: [|1 0 0 0&gt;, |5/3 1/15 0 -1/15&gt;,<br />
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| |5/3 -8/15 0 8/15&gt;, |5/3 -14/15 0 14/15&gt;]<br />
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| <a class="wiki_link" href="/Eigenmonzo">Eigenmonzo</a>s: 2, 7/6<br />
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| <br />
| |
| 9-limit: [|1 0 0 0&gt;, |25/16 1/8 0 -1/16&gt;,<br />
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| |5/2 -1 0 1/2&gt;, |25/8 -7/4 0 7/8&gt;]<br />
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| Eigenmonzos: 2, 9/7<br />
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| <br />
| |
| <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: ~3/2 = 702.085<br />
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| Mapping generator: ~3<br />
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| <br />
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| Map: [&lt;1 0 15 25|, &lt;0 1 -8 -14|]<br />
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| Wedgie: &lt;&lt;1 -8 -14 -15 -25 -10||<br />
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| EDOs: 12, 29, 41, 53, <a class="wiki_link" href="/94edo">94</a>, 241c, 335cd, 576cd<br />
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| Badness: 0.0216<br />
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| <br />
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| <!-- ws:start:WikiTextHeadingRule:6:&lt;h2&gt; --><h2 id="toc3"><a name="Garibaldi-11-limit"></a><!-- ws:end:WikiTextHeadingRule:6 -->11-limit</h2>
| |
| Commas: 225/224, 385/384, 2200/2187<br />
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| <br />
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| Minimax tuning:<br />
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| [|1 0 0 0 0&gt;, |25/16 1/8 0 -1/16 0&gt;, |5/2 -1 0 1/2 0&gt;,<br /> | |
| |25/8 -7/4 0 7/8 0&gt;, |47/16 23/8 0 -23/16 0&gt;]<br />
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| Eigenmonzos: 2, 9/7<br />
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| <br />
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| <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: ~3/2 = 702.157<br />
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| Mapping generator: ~3<br />
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| <br />
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| Map: [&lt;1 0 15 25 -33|, &lt;0 1 -8 -14 23|]<br />
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| <a class="wiki_link" href="/edo">Edos</a>: <a class="wiki_link" href="/41edo">41</a>, <a class="wiki_link" href="/53edo">53</a>, <a class="wiki_link" href="/94edo">94</a>, <a class="wiki_link" href="/135edo">135</a>, 229c, 323c, 417ce<br />
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| Badness: 0.0274<br />
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| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:8:&lt;h3&gt; --><h3 id="toc4"><a name="Garibaldi-11-limit-13-limit"></a><!-- ws:end:WikiTextHeadingRule:8 -->13-limit</h3>
| |
| Commas: 225/224, 275/273, 325/324, 385/384<br />
| |
| <br />
| |
| <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: ~3/2 = 702.113<br />
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| Mapping generator: ~3<br />
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| <br />
| |
| Map: [&lt;1 0 15 25 -33 -28|, &lt;0 1 -8 -14 23 20|]<br />
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| EDOs: 12, 41, 53, 94, 429cdef, 523cdef<br />
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| Badness: 0.0207<br />
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| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:10:&lt;h2&gt; --><h2 id="toc5"><a name="Garibaldi-Cassandra"></a><!-- ws:end:WikiTextHeadingRule:10 -->Cassandra</h2>
| |
| Commas: 100/99, 225/224, 245/242<br />
| |
| <br />
| |
| <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: ~3/2 = 702.321<br />
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| Mapping generator: ~3<br />
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| <br />
| |
| Map: [&lt;1 0 15 25 32|, &lt;0 1 -8 -14 -18|]<br />
| |
| EDOs: 12, 29, 41, 123c, 217ce, 258ce<br />
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| Badness: 0.0236<br />
| |
| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:12:&lt;h3&gt; --><h3 id="toc6"><a name="Garibaldi-Cassandra-13-limit Cassandra"></a><!-- ws:end:WikiTextHeadingRule:12 -->13-limit Cassandra</h3>
| |
| Commas: 100/99, 105/104, 196/195, 245/242<br />
| |
| <br />
| |
| <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: ~3/2 = 702.559<br />
| |
| Mapping generator: ~3<br />
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| <br />
| |
| Map: [&lt;1 0 15 25 32 37|, &lt;0 1 -8 -14 -18 -21|]<br />
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| EDOs: 12, 29, 41, 152cdf, 193cdf, 234cdf<br />
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| Badness: 0.0207<br />
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| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:14:&lt;h2&gt; --><h2 id="toc7"><a name="Garibaldi-Helenus"></a><!-- ws:end:WikiTextHeadingRule:14 -->Helenus</h2>
| |
| Commas: 99/98 176/175 3125/3087<br />
| |
| <br />
| |
| POTE generator: ~3/2 = 701.725<br />
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| Mapping generator: ~3<br />
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| <br />
| |
| Map: [&lt;1 0 15 25 51|, &lt;0 1 -8 -14 -30|]<br />
| |
| EDOs: 12, 53, 118d, 171de<br />
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| Badness: 0.0356<br />
| |
| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:16:&lt;h3&gt; --><h3 id="toc8"><a name="Garibaldi-Helenus-13-limit Helenus"></a><!-- ws:end:WikiTextHeadingRule:16 -->13-limit Helenus</h3>
| |
| Commas: 99/98, 176/175, 275/273, 847/845<br />
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| <br />
| |
| POTE generator: ~3/2 = 701.747<br />
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| Mapping generator: ~3<br />
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| <br />
| |
| Map: [&lt;1 0 15 25 51 56|, &lt;0 1 -8 -14 -30 -33|]<br />
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| EDOs: 53, 118d, 171de<br />
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| Badness: 0.0263<br />
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| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:18:&lt;h1&gt; --><h1 id="toc9"><a name="Hemigari"></a><!-- ws:end:WikiTextHeadingRule:18 -->Hemigari</h1>
| |
| Commas: 121/120, 225/224, 3125/3087<br />
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| <br />
| |
| POTE generator: ~63/55 = 248.918<br />
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| Mapping generator: ~110/63<br />
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| <br />
| |
| Map: [&lt;1 0 15 25 9|, &lt;0 2 -16 -28 -7|]<br />
| |
| EDOs: 29, 53, 82e, 135e, 188ce<br />
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| Badness: 0.0507<br />
| |
| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:20:&lt;h2&gt; --><h2 id="toc10"><a name="Hemigari-13-limit"></a><!-- ws:end:WikiTextHeadingRule:20 -->13-limit</h2>
| |
| Commas: 121/120, 169/168, 225/224, 275/273<br />
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| <br />
| |
| POTE generator: ~15/13 = 248.918<br />
| |
| Mapping generator: ~26/15<br />
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| <br />
| |
| Map: [&lt;1 0 15 25 9 14|, &lt;0 2 -16 -28 -7 -13|]<br />
| |
| EDOs: 29, 53, 82e, 135ef, 188cef<br />
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| Badness: 27.464<br />
| |
| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:22:&lt;h1&gt; --><h1 id="toc11"><a name="Guiron"></a><!-- ws:end:WikiTextHeadingRule:22 -->Guiron</h1>
| |
| <a class="wiki_link" href="/Comma">Commas</a>: 1029/1024, 10976/10935<br />
| |
| <br />
| |
| <a class="wiki_link" href="/Minimax%20tuning">Minimax tuning</a>:<br />
| |
| 7+9 limit: [|1 0 0 0&gt;, |15/8 0 -1/8 0&gt;,<br />
| |
| |0 0 1 0&gt;, |65/24 0 1/24 0&gt;]<br />
| |
| <a class="wiki_link" href="/Eigenmonzo">Eigenmonzos</a>: 2, 5/4<br />
| |
| <br />
| |
| <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: ~8/7 = 233.930<br />
| |
| Mapping generator: ~8/7<br />
| |
| <br />
| |
| Map: [&lt;1 1 7 3|, &lt;0 3 -24 -1|]<br />
| |
| Wedgie: &lt;&lt;3 -24 -1 -45 -10 65||<br />
| |
| EDOs: 36, 41, <a class="wiki_link" href="/77edo">77</a>, 118, 277d<br />
| |
| Badness: 0.0475<br />
| |
| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:24:&lt;h2&gt; --><h2 id="toc12"><a name="Guiron-11-limit"></a><!-- ws:end:WikiTextHeadingRule:24 -->11-limit</h2>
| |
| <a class="wiki_link" href="/Comma">Commas</a>: 385/384, 441/440, 10976/10935<br />
| |
| <br />
| |
| <a class="wiki_link" href="/Minimax%20tuning">Minimax tuning</a>:<br />
| |
| [|1 0 0 0 0&gt;, |15/8 0 -1/8 0 0&gt;,<br />
| |
| |0 0 1 0 0&gt;, |65/24 0 1/24 0 0&gt;,<br />
| |
| |37/6 0 -7/6 0 0&gt;]<br />
| |
| <a class="wiki_link" href="/Eigenmonzo">Eigenmonzos</a>: 2, 5/4<br />
| |
| <br />
| |
| <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: ~8/7 = 233.931<br />
| |
| Mapping generator: ~8/7<br />
| |
| <br />
| |
| Map: [&lt;1 1 7 3 -2|, &lt;0 3 -24 -1 28|]<br />
| |
| <a class="wiki_link" href="/edo">Edos</a>: <a class="wiki_link" href="/41edo">41</a>, <a class="wiki_link" href="/77edo">77</a>, <a class="wiki_link" href="/118edo">118</a>, <a class="wiki_link" href="/159edo">159</a>, <a class="wiki_link" href="/200edo">200</a>, <a class="wiki_link" href="/277">277d</a><br />
| |
| Badness: 0.0266<br />
| |
| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:26:&lt;h2&gt; --><h2 id="toc13"><a name="Guiron-13-limit"></a><!-- ws:end:WikiTextHeadingRule:26 -->13-limit</h2>
| |
| Commas: 196/195, 352/351, 385/384, 729/728<br />
| |
| <br />
| |
| POTE generator: ~8/7 = 233.890<br />
| |
| Mapping generator: ~8/7<br />
| |
| <br />
| |
| Map: [&lt;1 1 7 3 -2 0|, &lt;0 3 -24 -1 28 19|]<br />
| |
| EDOs: 41, 77, 118<br />
| |
| Badness: 0.0284<br />
| |
| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:28:&lt;h1&gt; --><h1 id="toc14"><a name="Pogo"></a><!-- ws:end:WikiTextHeadingRule:28 -->Pogo</h1>
| |
| Commas: 32805/32768, 118098/117649<br />
| |
| <br />
| |
| POTE generator: ~9/7 = 433.901<br />
| |
| Mapping generator: ~9/7<br />
| |
| <br />
| |
| Map: [&lt;2 1 22 2|, &lt;0 3 -24 5|]<br />
| |
| EDOs: 36, 94, 130, 171, 183, 224, 354<br />
| |
| Badness: 0.0796<br />
| |
| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:30:&lt;h2&gt; --><h2 id="toc15"><a name="Pogo-11-limit"></a><!-- ws:end:WikiTextHeadingRule:30 -->11-limit</h2>
| |
| Commas: 540/539, 4000/3993, 32805/32768<br />
| |
| <br />
| |
| POTE generator: ~9/7 = 433.911<br />
| |
| Mapping generator: ~9/7<br />
| |
| <br />
| |
| Map: [&lt;2 1 22 2 25|, &lt;0 3 -24 5 -25|]<br />
| |
| EDOs: 36, 94, 130, 224, 354, 578<br />
| |
| Badness: 0.0319<br />
| |
| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:32:&lt;h2&gt; --><h2 id="toc16"><a name="Pogo-13-limit"></a><!-- ws:end:WikiTextHeadingRule:32 -->13-limit</h2>
| |
| Commas: 540/539, 729/728, 4000/3993, 4225/4224<br />
| |
| <br />
| |
| POTE generator: ~9/7 = 433.911<br />
| |
| <br />
| |
| Map: [&lt;2 1 22 2 25 -2|, &lt;0 3 -24 5 -25 13|]<br />
| |
| EDOs: 36, 94, 130, 224, 354, 578<br />
| |
| Badness: 0.0175<br />
| |
| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:34:&lt;h1&gt; --><h1 id="toc17"><a name="Sanjaab"></a><!-- ws:end:WikiTextHeadingRule:34 -->Sanjaab</h1>
| |
| Commas: 225/224, 3125/3087, 1331/1323<br />
| |
| <br />
| |
| POTE generator: ~11/10 = 165.974<br />
| |
| <br />
| |
| Map: [&lt;1 2 -1 -3 0|, &lt;0 -3 24 42 25|]<br />
| |
| EDOs: 29, 65d, 94, 441cde, 535cde, 629cde<br />
| |
| Badness: 0.0580<br />
| |
| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:36:&lt;h2&gt; --><h2 id="toc18"><a name="Sanjaab-13-limit"></a><!-- ws:end:WikiTextHeadingRule:36 -->13-limit</h2>
| |
| Commas: 225/224, 275/273, 847/845, 1331/1323<br />
| |
| <br />
| |
| POTE generator: ~11/10 = 165.963<br />
| |
| <br />
| |
| Map: [&lt;1 2 -1 -3 0 -1|, &lt;0 -3 24 42 25 34|]<br />
| |
| EDOs: 29, 65d, 94<br />
| |
| Badness: 0.0338<br />
| |
| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:38:&lt;h1&gt; --><h1 id="toc19"><a name="Squirrel"></a><!-- ws:end:WikiTextHeadingRule:38 -->Squirrel</h1>
| |
| Commmas: 686/675, 32805/32768<br />
| |
| <br />
| |
| POTE generator: ~160/147 = 166.140<br />
| |
| <br />
| |
| Map: [&lt;1 2 -1 1|, &lt;0 -3 24 13|]<br />
| |
| EDOs: 29, 36, 65<br />
| |
| Badness: 0.1747<br />
| |
| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:40:&lt;h2&gt; --><h2 id="toc20"><a name="Squirrel-11-limit"></a><!-- ws:end:WikiTextHeadingRule:40 -->11-limit</h2>
| |
| Commas: 245/242, 686/675, 896/891<br />
| |
| <br />
| |
| POTE generator: ~11/10 = 166.097<br />
| |
| <br />
| |
| Map: [&lt;1 2 -1 1 0|, &lt;0 -3 24 13 25|]<br />
| |
| EDOs: 29, 36, 65<br />
| |
| Badness: 0.0683<br />
| |
| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:42:&lt;h2&gt; --><h2 id="toc21"><a name="Squirrel-13-limit"></a><!-- ws:end:WikiTextHeadingRule:42 -->13-limit</h2>
| |
| Commas: 91/90, 169/168, 245/242, 896/891<br />
| |
| <br />
| |
| POTE generator: ~11/10 = 166.054<br />
| |
| <br />
| |
| Map: [&lt;1 2 -1 1 0 3|, &lt;0 -3 24 13 25 5|]<br />
| |
| EDOs: 29, 36, 65f, 94df, 159df<br />
| |
| Badness: 0.0437<br />
| |
| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:44:&lt;h1&gt; --><h1 id="toc22"><a name="Schism"></a><!-- ws:end:WikiTextHeadingRule:44 -->Schism</h1>
| |
| Commas: 64/63, 360/343<br />
| |
| <br />
| |
| POTE generator: ~3/2 = 701.556<br />
| |
| Mapping generator: ~3<br />
| |
| <br />
| |
| Map: [&lt;1 0 15 6|, &lt;0 1 -8 -2|]<br />
| |
| Wedgie: &lt;&lt;1 -8 -2 -15 -6 18||<br />
| |
| EDOs: 12, 41d, 53d<br />
| |
| Badness: 0.0566<br />
| |
| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:46:&lt;h2&gt; --><h2 id="toc23"><a name="Schism-11-limit"></a><!-- ws:end:WikiTextHeadingRule:46 -->11-limit</h2>
| |
| Commas: 45/44, 64/63, 99/98<br />
| |
| <br />
| |
| POTE generator ~3/2 = 702.136<br />
| |
| Mapping generator: ~3<br />
| |
| <br />
| |
| Map: [&lt;1 0 15 6 13|, &lt;0 1 -8 -2 -6|]<br />
| |
| EDOs: 12, 29de, 41de<br />
| |
| Badness: 0.0375<br />
| |
| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:48:&lt;h1&gt; --><h1 id="toc24"><a name="Pontiac"></a><!-- ws:end:WikiTextHeadingRule:48 -->Pontiac</h1>
| |
| Commas: 32805/32768, 4375/4374<br />
| |
| <br />
| |
| 7-limit minimax:<br />
| |
| [|1 0 0 0&gt;, |74/47 0 -1/47 1/47&gt;, |113/47 0 8/47 -8/47&gt;,<br />
| |
| |113/47 0 -39/47 39/47&gt;]<br />
| |
| Eigenmonzos: 2, 7/5<br />
| |
| <br />
| |
| 9-limit minimax:<br />
| |
| [|1 0 0 0&gt;, |3/2 1/5 -1/10 0&gt;,<br />
| |
| |3 -8/5 4/5 0&gt;, |-1/2 39/5 -39/10 0&gt;]<br />
| |
| Eigenmonzos: 2, 10/9<br />
| |
| <br />
| |
| <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 701.757<br />
| |
| Mapping generator: ~3<br />
| |
| <br />
| |
| Map: [&lt;1 0 15 -59|, &lt;0 1 -8 39|]<br />
| |
| Wedgie: &lt;&lt;1 -8 39 -15 59 113||<br />
| |
| <a class="wiki_link" href="/edo">Edos</a>: <a class="wiki_link" href="/53edo">53</a>, <a class="wiki_link" href="/118edo">118</a>, <a class="wiki_link" href="/171edo">171</a>, <a class="wiki_link" href="/1079edo">1079</a>, <a class="wiki_link" href="/1250edo">1250</a>, <a class="wiki_link" href="/1421edo">1421</a><br />
| |
| Badness: 0.0141<br />
| |
| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:50:&lt;h1&gt; --><h1 id="toc25"><a name="Grackle"></a><!-- ws:end:WikiTextHeadingRule:50 -->Grackle</h1>
| |
| 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 />
| |
| <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 701.239<br />
| |
| Mapping generator: ~3<br />
| |
| <br />
| |
| Map: [&lt;1 0 15 -44|, &lt;0 1 -8 -26|]<br />
| |
| Wedgie: &lt;&lt;1 -8 -26 -15 -44 -38||<br />
| |
| <a class="wiki_link" href="/edo">Edos</a>: <a class="wiki_link" href="/77edo">77</a>, <a class="wiki_link" href="/89edo">89</a>, <a class="wiki_link" href="/101edo">101</a>, <a class="wiki_link" href="/166edo">166c</a>, <a class="wiki_link" href="/243edo">243c</a><br />
| |
| Badness: 0.070<br />
| |
| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:52:&lt;h1&gt; --><h1 id="toc26"><a name="Bischismic"></a><!-- ws:end:WikiTextHeadingRule:52 -->Bischismic</h1>
| |
| 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 />
| |
| <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: 701.592<br />
| |
| Mapping generator: ~3<br />
| |
| <br />
| |
| Map: [&lt;2 0 30 69|, &lt;0 1 -8 -20|]<br />
| |
| Wedgie: &lt;&lt;2 -16 -40 -30 -69 -48||<br />
| |
| <a class="wiki_link" href="/edo">Edos</a>: <a class="wiki_link" href="/118edo">118</a>, <a class="wiki_link" href="/130edo">130</a>, <a class="wiki_link" href="/248edo">248</a>, <a class="wiki_link" href="/378edo">378</a>, <a class="wiki_link" href="/508edo">508</a><br />
| |
| Badness: 0.0547<br />
| |
| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:54:&lt;h2&gt; --><h2 id="toc27"><a name="Bischismic-11-limit"></a><!-- ws:end:WikiTextHeadingRule:54 -->11-limit</h2>
| |
| Commas: 441/440, 3136/3125, 8019/8000<br />
| |
| <br />
| |
| POTE generator: ~3/2 = 701.612<br />
| |
| Mapping generator: ~3<br />
| |
| <br />
| |
| Map: [&lt;2 0 30 69 102|, &lt;0 1 -8 -20 -30|]<br />
| |
| EDOs: 12, 118, 130, 248<br />
| |
| Badness: 0.0282<br />
| |
| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:56:&lt;h2&gt; --><h2 id="toc28"><a name="Bischismic-13-limit"></a><!-- ws:end:WikiTextHeadingRule:56 -->13-limit</h2>
| |
| Commas: 441/440, 729/728, 1001/1000, 3136/3125<br />
| |
| <br />
| |
| POTE generator: ~3/2 = 701.590<br />
| |
| Mapping generator: ~3<br />
| |
| <br />
| |
| Map: [&lt;2 0 30 69 102 -75|, &lt;0 1 -8 -20 -30 26|]<br />
| |
| EDOs: 12, 118, 130, 248, 378<br />
| |
| Badness: 0.0287<br />
| |
| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:58:&lt;h1&gt; --><h1 id="toc29"><a name="Kleischismic"></a><!-- ws:end:WikiTextHeadingRule:58 -->Kleischismic</h1>
| |
| Commas: 32805/32768, 1500625/1492992<br />
| |
| <br />
| |
| POTE generator: ~36/35 = 50.920<br />
| |
| Mapping generator: ~35/24<br />
| |
| <br />
| |
| Map: [&lt;2 1 22 -15|, &lt;0 2 -16 19|]<br />
| |
| Wedgie: &lt;&lt;4 -32 38 -60 49 178||<br />
| |
| EDOs: 24, 94, 118, 212, 330, 542d, 872cd<br />
| |
| Badness: 0.1106<br />
| |
| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:60:&lt;h2&gt; --><h2 id="toc30"><a name="Kleischismic-11-limit"></a><!-- ws:end:WikiTextHeadingRule:60 -->11-limit</h2>
| |
| Commas: 385/384, 9801/9800, 14641/14580<br />
| |
| <br />
| |
| POTE generator: ~36/35 = 50.918<br />
| |
| Mapping generator: ~16/11<br />
| |
| <br />
| |
| Map: [&lt;2 1 22 -15 8|, &lt;0 2 -16 19 -1|]<br />
| |
| EDOs: 24, 94, 118, 212, 330e, 542de<br />
| |
| Badness: 0.0367<br />
| |
| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:62:&lt;h2&gt; --><h2 id="toc31"><a name="Kleischismic-13-limit"></a><!-- ws:end:WikiTextHeadingRule:62 -->13-limit</h2>
| |
| Commas: 352/351, 385/384, 729/728, 1575/1573<br />
| |
| <br />
| |
| POTE generator: ~36/35 = 50.938<br />
| |
| Mapping generator: ~16/11<br />
| |
| <br />
| |
| Map: [&lt;2 1 22 -15 8 15|, &lt;0 2 -16 19 -1 -7|]<br />
| |
| EDOs: 24, 94, 118, 212f<br />
| |
| Badness: 0.0376<br />
| |
| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:64:&lt;h1&gt; --><h1 id="toc32"><a name="Term"></a><!-- ws:end:WikiTextHeadingRule:64 -->Term</h1>
| |
| Commas: 32805/32768, 250047/250000<br />
| |
| <br />
| |
| 7-limit minimax<br />
| |
| Eigenmonzos: 2, 6/5<br />
| |
| <br />
| |
| 9-limit minimax<br />
| |
| Eigenmonzos: 2, 9/7<br />
| |
| <br />
| |
| <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: ~3/2 = 701.742<br />
| |
| Mapping generator: ~3<br />
| |
| <br />
| |
| Map: [&lt;3 0 45 94|, &lt;0 1 -8 -18|]<br />
| |
| Wedgie: &lt;&lt;3 -24 -54 -45 -94 -58||<br />
| |
| <a class="wiki_link" href="/edo">Edos</a>: <a class="wiki_link" href="/171edo">171</a>, <a class="wiki_link" href="/1038edo">1038</a>, <a class="wiki_link" href="/1209edo">1209</a><br />
| |
| Badness: 0.0200<br />
| |
| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:66:&lt;h1&gt; --><h1 id="toc33"><a name="Sesquiquartififths"></a><!-- ws:end:WikiTextHeadingRule:66 -->Sesquiquartififths</h1>
| |
| Commas: 2401/2400, 32805/32768<br />
| |
| <br />
| |
| 7-limit minimax<br />
| |
| Eigenmonzos: 2, 7/6<br />
| |
| <br />
| |
| 9-limit minimax<br />
| |
| Eigenmonzos: 2, 9/7<br />
| |
| <br />
| |
| <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: ~448/405 = 175.434<br />
| |
| Mapping generator: ~448/405<br />
| |
| <br />
| |
| Map: [&lt;1 1 7 5|, &lt;0 4 -32 -15|]<br />
| |
| Wedgie: &lt;&lt;4 -32 -15 -60 -35 55||<br />
| |
| <a class="wiki_link" href="/edo">Edos</a>: <a class="wiki_link" href="/171edo">171</a>, <a class="wiki_link" href="/643edo">643</a>, <a class="wiki_link" href="/2100edo">2100edo</a><br />
| |
| Badness: 0.0112<br />
| |
| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:68:&lt;h2&gt; --><h2 id="toc34"><a name="Sesquiquartififths-Sesquart"></a><!-- ws:end:WikiTextHeadingRule:68 -->Sesquart</h2>
| |
| Commas: 243/242, 441/440, 16384/16335<br />
| |
| <br />
| |
| POTE generator: ~256/231 = 175.406<br />
| |
| Mapping generator: ~256/231<br />
| |
| <br />
| |
| Map: [&lt;1 1 7 5 2|, &lt;0 4 -32 -15 10|]<br />
| |
| EDOs: 41, 89, 130<br />
| |
| Badness: 0.0293<br />
| |
| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:70:&lt;h3&gt; --><h3 id="toc35"><a name="Sesquiquartififths-Sesquart-13-limit"></a><!-- ws:end:WikiTextHeadingRule:70 -->13-limit</h3>
| |
| Commas: 243/242, 364/363, 441/440, 105644/105625<br />
| |
| <br />
| |
| POTE generator: ~256/231 = 175.409<br />
| |
| <br />
| |
| Map: [&lt;1 1 7 5 2 -2|, &lt;0 4 -32 -15 10 39|]<br />
| |
| EDOs: 41, 89, 130, 301e, 431e<br />
| |
| Badness: 0.0224<br />
| |
| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:72:&lt;h2&gt; --><h2 id="toc36"><a name="Sesquiquartififths-Bisesqui"></a><!-- ws:end:WikiTextHeadingRule:72 -->Bisesqui</h2>
| |
| Commas: 2401/2400, 9801/9800, 32805/32768<br />
| |
| <br />
| |
| <a class="wiki_link" href="/POTE%20tuning">POTE generator</a>: ~448/405 = 175.435<br />
| |
| <br />
| |
| Map: [&lt;2 2 14 10 23|, &lt;0 4 -32 -15 -55|]<br />
| |
| EDOs: 130, 212, 342, 1156, 1498<br />
| |
| Badness: 0.0170<br />
| |
| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:74:&lt;h1&gt; --><h1 id="toc37"><a name="Sextilififths"></a><!-- ws:end:WikiTextHeadingRule:74 -->Sextilififths</h1>
| |
| Commas: 32768/32805, 235298/234375<br />
| |
| <br />
| |
| POTE generator: ~21/20 = 83.053<br />
| |
| Mapping generator: ~21/20<br />
| |
| <br />
| |
| Map: [&lt;1 2 -1 -1|, &lt;0 -6 48 55|]<br />
| |
| EDOs: 29, 101, 130, 289, 419<br />
| |
| Badness: 0.1088<br />
| |
| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:76:&lt;h2&gt; --><h2 id="toc38"><a name="Sextilififths-11-limit"></a><!-- ws:end:WikiTextHeadingRule:76 -->11-limit</h2>
| |
| Commas: 441/440, 4000/3993, 235298/234375<br />
| |
| <br />
| |
| POTE generator: ~21/20 = 83.049<br />
| |
| Mapping generator: ~21/20<br />
| |
| <br />
| |
| Map: [&lt;1 2 -1 -1 0|, &lt;0 -6 48 55 50|]<br />
| |
| EDOs: 29, 130, 289<br />
| |
| Badness: 0.0455<br />
| |
| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:78:&lt;h2&gt; --><h2 id="toc39"><a name="Sextilififths-13-limit"></a><!-- ws:end:WikiTextHeadingRule:78 -->13-limit</h2>
| |
| Commas: 363/363, 441/440, 676/675, 10985/10976<br />
| |
| <br />
| |
| POTE generator: ~21/20 = 83.049<br />
| |
| Mapping generator: ~21/20<br />
| |
| <br />
| |
| Map: [&lt;1 2 -1 -1 0 1|, &lt;0 -6 48 55 50 39|]<br />
| |
| EDOs: 29, 130, 289<br />
| |
| Badness: 0.0253<br />
| |
| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:80:&lt;h1&gt; --><h1 id="toc40"><a name="Tsaharuk"></a><!-- ws:end:WikiTextHeadingRule:80 -->Tsaharuk</h1>
| |
| Commas: 32805/32768, 420175/419904<br />
| |
| <br />
| |
| POTE generator: ~243/224 = 140.350<br />
| |
| Mapping generator: ~243/224<br />
| |
| <br />
| |
| Map: [&lt;1 1 7 0|, &lt;0 5 -40 24|]<br />
| |
| Wedgie: &lt;&lt;5 -40 24 -75 24 168||<br />
| |
| EDOs: 17, 26, 43, 60, 77, 94, 171<br />
| |
| Badness: 0.0307<br />
| |
| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:82:&lt;h1&gt; --><h1 id="toc41"><a name="Quanharuk"></a><!-- ws:end:WikiTextHeadingRule:82 -->Quanharuk</h1>
| |
| Commas: 16875/16807, 32805/32768<br />
| |
| <br />
| |
| POTE generator: ~56/45 = 380.355<br />
| |
| Mapping generator: ~56/45<br />
| |
| <br />
| |
| Map: [&lt;1 0 15 12|, &lt;0 5 -40 -29|]<br />
| |
| Wedgie: &lt;&lt;5 -40 -29 -75 -60 45||<br />
| |
| EDOs: 41, 142, 183, 224, 1303d, 1527cd, 1751cd, 1975cd<br />
| |
| Badness: 0.0720<br />
| |
| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:84:&lt;h2&gt; --><h2 id="toc42"><a name="Quanharuk-11-limit"></a><!-- ws:end:WikiTextHeadingRule:84 -->11-limit</h2>
| |
| Commas: 540/539, 1375/1372, 32805/32768<br />
| |
| <br />
| |
| POTE generator: ~56/45 = 380.352<br />
| |
| Mapping generator: ~56/45<br />
| |
| <br />
| |
| Map: [&lt;1 0 15 12 -7|, &lt;0 5 -40 -29 33|]<br />
| |
| EDOs: 41, 142, 183, 224, 631d, 855d, 1079d<br />
| |
| Badness: 0.0315<br />
| |
| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:86:&lt;h2&gt; --><h2 id="toc43"><a name="Quanharuk-13-limit"></a><!-- ws:end:WikiTextHeadingRule:86 -->13-limit</h2>
| |
| Commas: 540/539, 1375/1372, 4096/4095, 6656/6655<br />
| |
| <br />
| |
| POTE generator: ~56/45 = 380.351<br />
| |
| <br />
| |
| Map: [&lt;1 0 15 12 -7 -15|, &lt;0 5 -40 -29 33 59|]<br />
| |
| EDOs: 41, 142, 183, 224, 631d, 855d<br />
| |
| Badness: 0.0214<br />
| |
| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:88:&lt;h1&gt; --><h1 id="toc44"><a name="Octant"></a><!-- ws:end:WikiTextHeadingRule:88 -->Octant</h1>
| |
| Commas: 32805/32768, 2259436291848/2251875390625<br />
| |
| <br />
| |
| POTE generator: ~3/2 = 701.713<br />
| |
| Mapping generator: ~3<br />
| |
| <br />
| |
| Map: [&lt;8 0 120 -117|, &lt;0 1 -8 11|]<br />
| |
| Wedgie: &lt;&lt;8 -64 88 -120 117 384||<br />
| |
| EDOs: 24, 224, 472, 696, 1168<br />
| |
| Badness: 0.1572<br />
| |
| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:90:&lt;h2&gt; --><h2 id="toc45"><a name="Octant-11-limit"></a><!-- ws:end:WikiTextHeadingRule:90 -->11-limit</h2>
| |
| Commas: 9801/9800, 32805/32768, 46656/46585<br />
| |
| <br />
| |
| POTE generator: ~3/2 = 701.713<br />
| |
| Mapping generator: ~3<br />
| |
| <br />
| |
| Map: [&lt;8 0 120 -117 15|, &lt;0 1 -8 11 1|]<br />
| |
| EDOs: 24, 224, 472, 696, 1168<br />
| |
| Badness: 0.0448<br />
| |
| <br />
| |
| <!-- ws:start:WikiTextHeadingRule:92:&lt;h2&gt; --><h2 id="toc46"><a name="Octant-13-limit"></a><!-- ws:end:WikiTextHeadingRule:92 -->13-limit</h2>
| |
| Commas: 729/728, 1575/1573, 2200/2197, 6656/6655<br />
| |
| <br />
| |
| POTE generator: ~3/2 = 701.725<br />
| |
| Mapping generator: ~3<br />
| |
| <br />
| |
| Map: [&lt;8 0 120 -117 15 93|, &lt;0 1 -8 11 1 -5|]<br />
| |
| EDOs: 24, 224, 472, 696<br />
| |
| Badness: 0.0304</body></html></pre></div>
| |
Other languages: de:Schismatische_Temperaturen Deutsch
Five limit
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 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 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 generator: ~3/2 = 701.736
Mapping generator: ~3
Map: [<1 0 15|, <0 1 -8|]
EDOs: 12, 29, 41, 53, 118, 171, 289, 460, 749, 3456bc, 4205bc, 4954bc, 5703bc
Seven limit children
The second comma of the 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
POTE generator: ~3/2 = 702.085
Mapping generator: ~3
Map: [<1 0 15 25|, <0 1 -8 -14|]
Wedgie: <<1 -8 -14 -15 -25 -10||
EDOs: 12, 29, 41, 53, 94, 241c, 335cd, 576cd
Badness: 0.0216
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
POTE generator: ~3/2 = 702.157
Mapping generator: ~3
Map: [<1 0 15 25 -33|, <0 1 -8 -14 23|]
Edos: 41, 53, 94, 135, 229c, 323c, 417ce
Badness: 0.0274
13-limit
Commas: 225/224, 275/273, 325/324, 385/384
POTE generator: ~3/2 = 702.113
Mapping generator: ~3
Map: [<1 0 15 25 -33 -28|, <0 1 -8 -14 23 20|]
EDOs: 12, 41, 53, 94, 429cdef, 523cdef
Badness: 0.0207
Cassandra
Commas: 100/99, 225/224, 245/242
POTE generator: ~3/2 = 702.321
Mapping generator: ~3
Map: [<1 0 15 25 32|, <0 1 -8 -14 -18|]
EDOs: 12, 29, 41, 123c, 217ce, 258ce
Badness: 0.0236
13-limit Cassandra
Commas: 100/99, 105/104, 196/195, 245/242
POTE generator: ~3/2 = 702.559
Mapping generator: ~3
Map: [<1 0 15 25 32 37|, <0 1 -8 -14 -18 -21|]
EDOs: 12, 29, 41, 152cdf, 193cdf, 234cdf
Badness: 0.0207
Helenus
Commas: 99/98 176/175 3125/3087
POTE generator: ~3/2 = 701.725
Mapping generator: ~3
Map: [<1 0 15 25 51|, <0 1 -8 -14 -30|]
EDOs: 12, 53, 118d, 171de
Badness: 0.0356
13-limit Helenus
Commas: 99/98, 176/175, 275/273, 847/845
POTE generator: ~3/2 = 701.747
Mapping generator: ~3
Map: [<1 0 15 25 51 56|, <0 1 -8 -14 -30 -33|]
EDOs: 53, 118d, 171de
Badness: 0.0263
Hemigari
Commas: 121/120, 225/224, 3125/3087
POTE generator: ~63/55 = 248.918
Mapping generator: ~110/63
Map: [<1 0 15 25 9|, <0 2 -16 -28 -7|]
EDOs: 29, 53, 82e, 135e, 188ce
Badness: 0.0507
13-limit
Commas: 121/120, 169/168, 225/224, 275/273
POTE generator: ~15/13 = 248.918
Mapping generator: ~26/15
Map: [<1 0 15 25 9 14|, <0 2 -16 -28 -7 -13|]
EDOs: 29, 53, 82e, 135ef, 188cef
Badness: 27.464
Guiron
Commas: 1029/1024, 10976/10935
Minimax tuning:
7+9 limit: [|1 0 0 0>, |15/8 0 -1/8 0>,
|0 0 1 0>, |65/24 0 1/24 0>]
Eigenmonzos: 2, 5/4
POTE generator: ~8/7 = 233.930
Mapping generator: ~8/7
Map: [<1 1 7 3|, <0 3 -24 -1|]
Wedgie: <<3 -24 -1 -45 -10 65||
EDOs: 36, 41, 77, 118, 277d
Badness: 0.0475
11-limit
Commas: 385/384, 441/440, 10976/10935
Minimax tuning:
[|1 0 0 0 0>, |15/8 0 -1/8 0 0>,
|0 0 1 0 0>, |65/24 0 1/24 0 0>,
|37/6 0 -7/6 0 0>]
Eigenmonzos: 2, 5/4
POTE generator: ~8/7 = 233.931
Mapping generator: ~8/7
Map: [<1 1 7 3 -2|, <0 3 -24 -1 28|]
Edos: 41, 77, 118, 159, 200, 277d
Badness: 0.0266
13-limit
Commas: 196/195, 352/351, 385/384, 729/728
POTE generator: ~8/7 = 233.890
Mapping generator: ~8/7
Map: [<1 1 7 3 -2 0|, <0 3 -24 -1 28 19|]
EDOs: 41, 77, 118
Badness: 0.0284
Pogo
Commas: 32805/32768, 118098/117649
POTE generator: ~9/7 = 433.901
Mapping generator: ~9/7
Map: [<2 1 22 2|, <0 3 -24 5|]
EDOs: 36, 94, 130, 171, 183, 224, 354
Badness: 0.0796
11-limit
Commas: 540/539, 4000/3993, 32805/32768
POTE generator: ~9/7 = 433.911
Mapping generator: ~9/7
Map: [<2 1 22 2 25|, <0 3 -24 5 -25|]
EDOs: 36, 94, 130, 224, 354, 578
Badness: 0.0319
13-limit
Commas: 540/539, 729/728, 4000/3993, 4225/4224
POTE generator: ~9/7 = 433.911
Map: [<2 1 22 2 25 -2|, <0 3 -24 5 -25 13|]
EDOs: 36, 94, 130, 224, 354, 578
Badness: 0.0175
Sanjaab
Commas: 225/224, 3125/3087, 1331/1323
POTE generator: ~11/10 = 165.974
Map: [<1 2 -1 -3 0|, <0 -3 24 42 25|]
EDOs: 29, 65d, 94, 441cde, 535cde, 629cde
Badness: 0.0580
13-limit
Commas: 225/224, 275/273, 847/845, 1331/1323
POTE generator: ~11/10 = 165.963
Map: [<1 2 -1 -3 0 -1|, <0 -3 24 42 25 34|]
EDOs: 29, 65d, 94
Badness: 0.0338
Squirrel
Commmas: 686/675, 32805/32768
POTE generator: ~160/147 = 166.140
Map: [<1 2 -1 1|, <0 -3 24 13|]
EDOs: 29, 36, 65
Badness: 0.1747
11-limit
Commas: 245/242, 686/675, 896/891
POTE generator: ~11/10 = 166.097
Map: [<1 2 -1 1 0|, <0 -3 24 13 25|]
EDOs: 29, 36, 65
Badness: 0.0683
13-limit
Commas: 91/90, 169/168, 245/242, 896/891
POTE generator: ~11/10 = 166.054
Map: [<1 2 -1 1 0 3|, <0 -3 24 13 25 5|]
EDOs: 29, 36, 65f, 94df, 159df
Badness: 0.0437
Schism
Commas: 64/63, 360/343
POTE generator: ~3/2 = 701.556
Mapping generator: ~3
Map: [<1 0 15 6|, <0 1 -8 -2|]
Wedgie: <<1 -8 -2 -15 -6 18||
EDOs: 12, 41d, 53d
Badness: 0.0566
11-limit
Commas: 45/44, 64/63, 99/98
POTE generator ~3/2 = 702.136
Mapping generator: ~3
Map: [<1 0 15 6 13|, <0 1 -8 -2 -6|]
EDOs: 12, 29de, 41de
Badness: 0.0375
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
POTE generator: 701.757
Mapping generator: ~3
Map: [<1 0 15 -59|, <0 1 -8 39|]
Wedgie: <<1 -8 39 -15 59 113||
Edos: 53, 118, 171, 1079, 1250, 1421
Badness: 0.0141
Grackle
Commas: 126/125, 32805/32768
7-limit minimax
Eigenmonzos: 2, 7/6
9-limit minimax
Eigenmonzos: 2, 9/7
POTE generator: 701.239
Mapping generator: ~3
Map: [<1 0 15 -44|, <0 1 -8 -26|]
Wedgie: <<1 -8 -26 -15 -44 -38||
Edos: 77, 89, 101, 166c, 243c
Badness: 0.070
Bischismic
Commas: 3136/3125, 32805/32768
7-limit minimax
Eigenmonzos: 2, 7/6
9-limit minimax
Eigenmonzos: 2, 9/7
POTE generator: 701.592
Mapping generator: ~3
Map: [<2 0 30 69|, <0 1 -8 -20|]
Wedgie: <<2 -16 -40 -30 -69 -48||
Edos: 118, 130, 248, 378, 508
Badness: 0.0547
11-limit
Commas: 441/440, 3136/3125, 8019/8000
POTE generator: ~3/2 = 701.612
Mapping generator: ~3
Map: [<2 0 30 69 102|, <0 1 -8 -20 -30|]
EDOs: 12, 118, 130, 248
Badness: 0.0282
13-limit
Commas: 441/440, 729/728, 1001/1000, 3136/3125
POTE generator: ~3/2 = 701.590
Mapping generator: ~3
Map: [<2 0 30 69 102 -75|, <0 1 -8 -20 -30 26|]
EDOs: 12, 118, 130, 248, 378
Badness: 0.0287
Kleischismic
Commas: 32805/32768, 1500625/1492992
POTE generator: ~36/35 = 50.920
Mapping generator: ~35/24
Map: [<2 1 22 -15|, <0 2 -16 19|]
Wedgie: <<4 -32 38 -60 49 178||
EDOs: 24, 94, 118, 212, 330, 542d, 872cd
Badness: 0.1106
11-limit
Commas: 385/384, 9801/9800, 14641/14580
POTE generator: ~36/35 = 50.918
Mapping generator: ~16/11
Map: [<2 1 22 -15 8|, <0 2 -16 19 -1|]
EDOs: 24, 94, 118, 212, 330e, 542de
Badness: 0.0367
13-limit
Commas: 352/351, 385/384, 729/728, 1575/1573
POTE generator: ~36/35 = 50.938
Mapping generator: ~16/11
Map: [<2 1 22 -15 8 15|, <0 2 -16 19 -1 -7|]
EDOs: 24, 94, 118, 212f
Badness: 0.0376
Term
Commas: 32805/32768, 250047/250000
7-limit minimax
Eigenmonzos: 2, 6/5
9-limit minimax
Eigenmonzos: 2, 9/7
POTE generator: ~3/2 = 701.742
Mapping generator: ~3
Map: [<3 0 45 94|, <0 1 -8 -18|]
Wedgie: <<3 -24 -54 -45 -94 -58||
Edos: 171, 1038, 1209
Badness: 0.0200
Sesquiquartififths
Commas: 2401/2400, 32805/32768
7-limit minimax
Eigenmonzos: 2, 7/6
9-limit minimax
Eigenmonzos: 2, 9/7
POTE generator: ~448/405 = 175.434
Mapping generator: ~448/405
Map: [<1 1 7 5|, <0 4 -32 -15|]
Wedgie: <<4 -32 -15 -60 -35 55||
Edos: 171, 643, 2100edo
Badness: 0.0112
Sesquart
Commas: 243/242, 441/440, 16384/16335
POTE generator: ~256/231 = 175.406
Mapping generator: ~256/231
Map: [<1 1 7 5 2|, <0 4 -32 -15 10|]
EDOs: 41, 89, 130
Badness: 0.0293
13-limit
Commas: 243/242, 364/363, 441/440, 105644/105625
POTE generator: ~256/231 = 175.409
Map: [<1 1 7 5 2 -2|, <0 4 -32 -15 10 39|]
EDOs: 41, 89, 130, 301e, 431e
Badness: 0.0224
Bisesqui
Commas: 2401/2400, 9801/9800, 32805/32768
POTE generator: ~448/405 = 175.435
Map: [<2 2 14 10 23|, <0 4 -32 -15 -55|]
EDOs: 130, 212, 342, 1156, 1498
Badness: 0.0170
Sextilififths
Commas: 32768/32805, 235298/234375
POTE generator: ~21/20 = 83.053
Mapping generator: ~21/20
Map: [<1 2 -1 -1|, <0 -6 48 55|]
EDOs: 29, 101, 130, 289, 419
Badness: 0.1088
11-limit
Commas: 441/440, 4000/3993, 235298/234375
POTE generator: ~21/20 = 83.049
Mapping generator: ~21/20
Map: [<1 2 -1 -1 0|, <0 -6 48 55 50|]
EDOs: 29, 130, 289
Badness: 0.0455
13-limit
Commas: 363/363, 441/440, 676/675, 10985/10976
POTE generator: ~21/20 = 83.049
Mapping generator: ~21/20
Map: [<1 2 -1 -1 0 1|, <0 -6 48 55 50 39|]
EDOs: 29, 130, 289
Badness: 0.0253
Tsaharuk
Commas: 32805/32768, 420175/419904
POTE generator: ~243/224 = 140.350
Mapping generator: ~243/224
Map: [<1 1 7 0|, <0 5 -40 24|]
Wedgie: <<5 -40 24 -75 24 168||
EDOs: 17, 26, 43, 60, 77, 94, 171
Badness: 0.0307
Quanharuk
Commas: 16875/16807, 32805/32768
POTE generator: ~56/45 = 380.355
Mapping generator: ~56/45
Map: [<1 0 15 12|, <0 5 -40 -29|]
Wedgie: <<5 -40 -29 -75 -60 45||
EDOs: 41, 142, 183, 224, 1303d, 1527cd, 1751cd, 1975cd
Badness: 0.0720
11-limit
Commas: 540/539, 1375/1372, 32805/32768
POTE generator: ~56/45 = 380.352
Mapping generator: ~56/45
Map: [<1 0 15 12 -7|, <0 5 -40 -29 33|]
EDOs: 41, 142, 183, 224, 631d, 855d, 1079d
Badness: 0.0315
13-limit
Commas: 540/539, 1375/1372, 4096/4095, 6656/6655
POTE generator: ~56/45 = 380.351
Map: [<1 0 15 12 -7 -15|, <0 5 -40 -29 33 59|]
EDOs: 41, 142, 183, 224, 631d, 855d
Badness: 0.0214
Octant
Commas: 32805/32768, 2259436291848/2251875390625
POTE generator: ~3/2 = 701.713
Mapping generator: ~3
Map: [<8 0 120 -117|, <0 1 -8 11|]
Wedgie: <<8 -64 88 -120 117 384||
EDOs: 24, 224, 472, 696, 1168
Badness: 0.1572
11-limit
Commas: 9801/9800, 32805/32768, 46656/46585
POTE generator: ~3/2 = 701.713
Mapping generator: ~3
Map: [<8 0 120 -117 15|, <0 1 -8 11 1|]
EDOs: 24, 224, 472, 696, 1168
Badness: 0.0448
13-limit
Commas: 729/728, 1575/1573, 2200/2197, 6656/6655
POTE generator: ~3/2 = 701.725
Mapping generator: ~3
Map: [<8 0 120 -117 15 93|, <0 1 -8 11 1 -5|]
EDOs: 24, 224, 472, 696
Badness: 0.0304