User:Contribution/Successive superparticular complementary pair
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Todo: Finish the article and move it
When the article is finished and the table explained, move it to the main root |
For each pair of superparticular ratios [math]\displaystyle{ {s1}/{s2} }[/math] and [math]\displaystyle{ {s2}/{s3} }[/math], there exists a ratio [math]\displaystyle{ {a}/{b} }[/math] such that [math]\displaystyle{ {s1}/{s2} }[/math] and [math]\displaystyle{ {s2}/{s3} }[/math] are [math]\displaystyle{ {a}/{b} }[/math] complementary; it is observed that [math]\displaystyle{ a−b=1 }[/math] or [math]\displaystyle{ a−b=2 }[/math]. In other words, for each ratio [math]\displaystyle{ a/b }[/math] where [math]\displaystyle{ a−b=1 }[/math] or [math]\displaystyle{ a−b=2 }[/math], there exists a pair of superparticular ratios [math]\displaystyle{ {s1}/{s2} }[/math] and [math]\displaystyle{ {s2}/{s3} }[/math] that are [math]\displaystyle{ {a}/{b} }[/math] complementary.
Bellow is a table that show for equal divisions of [math]\displaystyle{ a/b }[/math] the cent error in the mapping of superparticular ratios [math]\displaystyle{ {s1}/{s2} }[/math] and [math]\displaystyle{ {s2}/{s3} }[/math] that are [math]\displaystyle{ a/b }[/math] complementary.
We can observe a converging sequence and pattern for low errors: 5, 7, 12; then 7, 9, 16; then 9, 11, 20; then 11, 13, 24; then 13, 15, 28; then 15, 17, 32; then 17, 19, 36; then 19, 21, 40; then 21, 23, 44; etc. --
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Todo: Tables
Explain the tables. |
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Todo: Pattern
Clarify the observed pattern and create a descriptive name for it, such as the "Alpha-Beta-Gamma pattern" or the "Alpha-Beta-Gamma class" when referring to the group of scales. Assign distinct names to each scale within this class. For instance, 5edo might be called "2/1 Alpha", 7edo could be "2/1 Beta", and 12edo could be "2/1 Gamma". Additionally, compute the Dave Benson optimization for each scale as an alternative tuning. Note: 23edo with stretched octave is "7/5 Alpha". Consider this second version for naming: "Alpha 2/1, Beta 2/1, Gamma 2/1, Alpha 7/5." Consistency and Clarity: The second version ("Alpha 2/1, Beta 2/1, Gamma 2/1, Alpha 7/5") places the descriptive name ("Alpha," "Beta," "Gamma") before the ratio. This makes it clear that "Alpha," "Beta," "Gamma," and so on are categories or types, while "2/1" and "7/5" are specific tunings or ratios within those categories. This ordering helps maintain a logical structure that is easier to follow. |
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Todo: Add data
Include columns for the following optimizations: Pure Equave, Dave Benson, TE, POTE, and CTE. To prevent the table from becoming overcrowded, choose between using cents or EDO. Cents are preferable since these are EDONOI. Retain the EDO column for pure equaves to provide a sense of scale, but place it outside the optimization subgroup column. |
| Tuning | Intervals | Mappings | ||||
|---|---|---|---|---|---|---|
| Name | Equal division | Steps per octave | Equave | SSC pair | Steps (Equave, SSC pair) | Errors (cent) |
| Alpha 3/1 | 3ed3/1 | 1.89278926071437 | 3/1 | 2/1, 3/2 | 3\3<3/1>, 2\3<3/1>, 1\3<3/1> | 0, 67.970, -67.970 |
| Beta 3/1 | 5ed3/1 | 3.15464876785729 | 5\5<3/1>, 3\5<3/1>, 2\5<3/1> | 0, -58.827, 58.827 | ||
| Gamma 3/1 | 8ed3/1 | 5.04743802857166 | 8\8<3/1>, 5\8<3/1>, 3\8<3/1> | 0, -11.278, 11.278 | ||
| Alpha 2/1 | 5ed2/1 | 5 | 2/1 | 3/2, 4/3 | 5\5<2/1>, 3\5<2/1>, 2\5<2/1> | 0, 18.045, -18.045 |
| Beta 2/1 | 7ed2/1 | 7 | 7\7<2/1>, 4\7<2/1>, 3\7<2/1> | 0, -16.241, 16.241 | ||
| Gamma 2/1 | 12ed2/1 | 12 | 12\12<2/1>, 7\12<2/1>, 5\12<2/1> | 0, -1.955, 1.955 | ||
| Alpha 5/3 | 7ed5/3 | 9.49840814199707 | 5/3 | 4/3, 5/4 | 7\7<5/3>, 4\7<5/3>, 3\7<5/3> | 0, 7.303, -7.303 |
| Beta 5/3 | 9ed5/3 | 12.2122390397105 | 9\9<5/3>, 5\9<5/3>, 4\9<5/3> | 0, -6.735, 6.735 | ||
| Gamma 5/3 | 16ed5/3 | 21.7106471817076 | 16\16<5/3>, 9\16<5/3>, 7\16<5/3> | 0, -0.593, 0.593 | ||
| Alpha 3/2 | 9ed3/2 | 15.3856016221631 | 3/2 | 5/4, 6/5 | 9\9<3/2>, 5\9<3/2>, 4\9<3/2> | 0, 3.661, -3.661 |
| Beta 3/2 | 11ed3/2 | 18.8046242048660 | 11\11<3/2>, 6\11<3/2>, 5\11<3/2> | 0, -3.429, 3.429 | ||
| Gamma 3/2 | 20ed3/2 | 34.1902258270291 | 20\20<3/2>, 11\20<3/2>, 9\20<3/2> | 0, -0.238, 0.238 | ||
| Alpha 7/5 | 11ed7/5 | 22.6604698881676 | 7/5 | 6/5, 7/6 | 11\11<7/5>, 6\11<7/5>, 5\11<7/5> | 0, 2.093, -2.093 |
| Beta 7/5 | 13ed7/5 | 26.7805553223799 | 13\13<7/5>, 7\13<7/5>, 6\13<7/5> | 0, -1.981, 1.981 | ||
| Gamma 7/5 | 24ed7/5 | 49.4410252105475 | 24\24<7/5>, 13\24<7/5>, 11\24<7/5> | 0, -0.114, 0.114 | ||
| Alpha 4/3 | 13ed4/3 | 31.3224709154917 | 4/3 | 7/6, 8/7 | 13\13<4/3>, 7\13<4/3>, 6\13<4/3> | 0, 1.307, -1.307 |
| Beta 4/3 | 15ed4/3 | 36.1413125947981 | 15\15<4/3>, 8\15<4/3>, 7\15<4/3> | 0, -1.247, 1.247 | ||
| Gamma 4/3 | 28ed4/3 | 67.4637835102899 | 28\28<4/3>, 15\28<4/3>, 13\28<4/3> | 0, -0.061, 0.061 | ||
| Alpha 9/7 | 15ed9/7 | 41.3713123417559 | 9/7 | 8/7, 9/8 | 15\15<9/7>, 8\15<9/7>, 7\15<9/7> | 0, 0.871, -0.871 |
| Beta 9/7 | 17ed9/7 | 46.8874873206567 | 17\17<9/7>, 9\17<9/7>, 8\17<9/7> | 0, -0.835, 0.835 | ||
| Gamma 9/7 | 32ed9/7 | 88.2587996624126 | 32\32<9/7>, 17\32<9/7>, 15\32<9/7> | 0, -0.036, 0.036 | ||
| Alpha 5/4 | 17ed5/4 | 52.8068232315916 | 5/4 | 9/8, 10/9 | 17\17<5/4>, 9\17<5/4>, 8\17<5/4> | 0, 0.609, -0.609 |
| Beta 5/4 | 19ed5/4 | 59.0193906706024 | 19\19<5/4>, 10\19<5/4>, 9\19<5/4> | 0, -0.587, 0.587 | ||
| Gamma 5/4 | 36ed5/4 | 111.826213902194 | 36\36<5/4>, 19\36<5/4>, 17\36<5/4> | 0, -0.022, 0.022 | ||
| Alpha 11/9 | 19ed11/9 | 65.6288971357202 | 11/9 | 10/9, 11/10 | 19\19<11/9>, 10\19<11/9>, 9\19<11/9> | 0, 0.443, -0.443 |
| Beta 11/9 | 21ed11/9 | 72.5372020973750 | 21\21<11/9>, 11\21<11/9>, 10\21<11/9> | 0, -0.428, 0.428 | ||
| Gamma 11/9 | 40ed11/9 | 138.166099233095 | 40\40<11/9>, 21\40<11/9>, 19\40<11/9> | 0, -0.015, 0.015 | ||
| Alpha 6/5 | 21ed6/5 | 79.8374643554025 | 6/5 | 11/10, 12/11 | 21\21<6/5>, 11\21<6/5>, 10\21<6/5> | 0, 0.332, -0.332 |
| Beta 6/5 | 23ed6/5 | 87.4410323892504 | 23\23<6/5>, 12\23<6/5>, 11\23<6/5> | 0, -0.322, 0.322 | ||
| Gamma 6/5 | 44ed6/5 | 167.278496744653 | 44\44<6/5>, 23\44<6/5>, 21\44<6/5> | 0, -0.010, 0.010 | ||
| Alpha 13/11 | 23ed13/11 | 95.4324773621886 | 13/11 | 12/11, 13/12 | 23\23<13/11>, 12\23<13/11>, 11\23<13/11> | 0, 0.255, -0.255 |
| Beta 13/11 | 25ed13/11 | 103.730953654553 | 25\25<13/11>, 13\25<13/11>, 12\25<13/11> | 0, -0.248, 0.248 | ||
| Gamma 13/11 | 48ed13/11 | 199.163431016741 | 48\48<13/11>, 25\48<13/11>, 23\48<13/11> | 0, -0.007, 0.007 | ||
| Alpha 7/6 | 25ed7/6 | 112.413902640048 | 7/6 | 13/12, 14/13 | 25\25<7/6>, 13\25<7/6>, 12\25<7/6> | 0, 0.200, -0.200 |
| Beta 7/6 | 27ed7/6 | 121.407014851252 | 27\27<7/6>, 14\27<7/6>, 13\27<7/6> | 0, -0.195, 0.195 | ||
| Gamma 7/6 | 52ed7/6 | 233.820917491300 | 52\52<7/6>, 27\52<7/6>, 25\52<7/6> | 0, -0.005, 0.005 | ||
| Alpha 15/13 | 27ed15/13 | 130.781715879411 | 15/13 | 14/13, 15/14 | 27\27<15/13>, 14\27<15/13>, 13\27<15/13> | 0, 0.160, -0.160 |
| Beta 15/13 | 29ed15/13 | 140.469250388997 | 29\29<15/13>, 15\29<15/13>, 14\29<15/13> | 0, -0.156, 0.156 | ||
| Gamma 15/13 | 56ed15/13 | 271.250966268408 | 56\56<15/13>, 29\56<15/13>, 27\56<15/13> | 0, -0.004, 0.004 | ||
| Alpha 8/7 | 29ed8/7 | 150.535899020849 | 8/7 | 15/14, 16/15 | 29\29<8/7>, 15\29<8/7>, 14\29<8/7> | 0, 0.130, -0.130 |
| Beta 8/7 | 31ed8/7 | 160.917685160217 | 31\31<8/7>, 16\31<8/7>, 15\31<8/7> | 0, -0.127, 0.127 | ||
| Gamma 8/7 | 60ed8/7 | 311.453584181066 | 60\60<8/7>, 31\60<8/7>, 29\60<8/7> | 0, -0.003, 0.003 | ||
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Todo: Temperaments
Compute the temperaments associated to each Alpha-Beta-Gamma scales. |
The fact
As a fact, all Alpha scales are (s1 + s2)ED(a / b), all Beta scales are (s2 + s3)ED(a / b), and all Gamma scales are (s1 + s2 + s2 + s3)ED(a / b).