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: Table
Explain the table. |
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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. |
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | ||
|---|---|---|---|---|---|---|---|---|---|---|
| 3/1 | 2/1, 3/2 | Alpha 3/1 | 3ed3/1 | 1.892789 | 633.985000 | 1.907395926960071 | 629.130000247253548 | 3/1 mapping: 3\3<3/1> | equal division error: 0.000 | optimization error: -14.565 |
| 2/1 mapping: 2\3<3/1> | equal division error: 67.970 | optimization error: 58.260 | ||||||||
| 3/2 mapping: 1\3<3/1> | equal division error: -67.970 | optimization error: -72.825 | ||||||||
| Beta 3/1 | 5ed3/1 | 3.154649 | 380.391000 | 3.141862316907629 | 381.939079106781893 | 3/1 mapping: 5\5<3/1> | equal division error: 0.000 | optimization error: 7.740 | ||
| 2/1 mapping: 3\5<3/1> | equal division error: -58.827 | optimization error: -54.183 | ||||||||
| 3/2 mapping: 2\5<3/1> | equal division error: 58.827 | optimization error: 61.923 | ||||||||
| Gamma 3/1 | 8ed3/1 | 5.047438 | 237.744375 | 5.042556213760587 | 237.974540913461853 | 3/1 mapping: 8\8<3/1> | equal division error: 0.000 | optimization error: 1.841 | ||
| 2/1 mapping: 5\8<3/1> | equal division error: -11.278 | optimization error: -10.127 | ||||||||
| 3/2 mapping: 3\8<3/1> | equal division error: 11.278 | optimization error: 11.969 | ||||||||
| 2/1 | 3/2, 4/3 | Alpha 2/1 | 5ed2/1 | 5.000000 | 240.000000 | 5.009912705090773 | 239.525131601720722 | 2/1 mapping: 5\5<2/1> | equal division error: 0.000 | optimization error: -2.374 |
| 3/2 mapping: 3\5<2/1> | equal division error: 18.045 | optimization error: 16.620 | ||||||||
| 4/3 mapping: 2\5<2/1> | equal division error: -18.045 | optimization error: -18.995 | ||||||||
| Beta 2/1 | 7ed2/1 | 7.000000 | 171.428571 | 6.991049802487100 | 171.648040552234965 | 2/1 mapping: 7\7<2/1> | equal division error: 0.000 | optimization error: 1.536 | ||
| 3/2 mapping: 4\7<2/1> | equal division error: -16.241 | optimization error: -15.363 | ||||||||
| 4/3 mapping: 3\7<2/1> | equal division error: 16.241 | optimization error: 16.899 | ||||||||
| Gamma 2/1 | 12ed2/1 | 12.000000 | 100.000000 | 11.997848091431052 | 100.017935787755848 | 2/1 mapping: 12\12<2/1> | equal division error: 0.000 | optimization error: 0.215 | ||
| 3/2 mapping: 7\12<2/1> | equal division error: -1.955 | optimization error: -1.829 | ||||||||
| 4/3 mapping: 5\12<2/1> | equal division error: 1.955 | optimization error: 2.045 | ||||||||
| Alpha 5/3 | 7ed5/3 | 9.498408 | 126.336959 | 9.505833538777849 | 126.238272015257927 | 5/3 mapping: 7\7<5/3> | equal division error: 0.000 | optimization error: -0.691 | ||
| 4/3 mapping: 4\7<5/3> | equal division error: 7.303 | optimization error: 6.908 | ||||||||
| 5/4 mapping: 3\7<5/3> | equal division error: -7.303 | optimization error: -7.599 | ||||||||
| Beta 5/3 | 9ed5/3 | 12.212239 | 98.262079 | 12.205382300878206 | 98.317280886290400 | 5/3 mapping: 9\9<5/3> | equal division error: 0.000 | optimization error: 0.497 | ||
| 4/3 mapping: 5\9<5/3> | equal division error: -6.735 | optimization error: -6.459 | ||||||||
| 5/4 mapping: 4\9<5/3> | equal division error: 6.735 | optimization error: 6.955 | ||||||||
| Gamma 5/3 | 16ed5/3 | 21.710647 | 55.272420 | 21.709439921550910 | 55.275493257141231 | 5/3 mapping: 16\16<5/3> | equal division error: 0.000 | optimization error: 0.049 | ||
| 4/3 mapping: 9\16<5/3> | equal division error: -0.593 | optimization error: -0.566 | ||||||||
| 5/4 mapping: 7\16<5/3> | equal division error: 0.593 | optimization error: 0.615 | ||||||||
| Alpha 3/2 | 9ed3/2 | 15.385602 | 77.995000 | 15.391523899692793 | 77.964989550121895 | 3/2 mapping: 9\9<3/2> | equal division error: 0.000 | optimization error: -0.270 | ||
| 5/4 mapping: 5\9<3/2> | equal division error: 3.661 | optimization error: 3.511 | ||||||||
| 6/5 mapping: 4\9<3/2> | equal division error: -3.661 | optimization error: -3.781 | ||||||||
| Beta 3/2 | 11ed3/2 | 18.804624 | 63.814091 | 18.799073639411081 | 63.832932569840843 | 3/2 mapping: 11\11<3/2> | equal division error: 0.000 | optimization error: 0.207 | ||
| 5/4 mapping: 6\11<3/2> | equal division error: -3.429 | optimization error: -3.316 | ||||||||
| 6/5 mapping: 5\11<3/2> | equal division error: 3.429 | optimization error: 3.523 | ||||||||
| Gamma 3/2 | 20ed3/2 | 34.190226 | 35.097750 | 34.189454092191388 | 35.098542280441702 | 3/2 mapping: 20\20<3/2> | equal division error: 0.000 | optimization error: 0.016 | ||
| 5/4 mapping: 11\20<3/2> | equal division error: -0.238 | optimization error: -0.230 | ||||||||
| 6/5 mapping: 9\20<3/2> | equal division error: 0.238 | optimization error: 0.246 | ||||||||
| Alpha 7/5 | 11ed7/5 | 22.660470 | 52.955654 | 22.665391113336561 | 52.944155871808760 | 7/5 mapping: 11\11<7/5> | equal division error: 0.000 | optimization error: -0.126 | ||
| 6/5 mapping: 6\11<7/5> | equal division error: 2.093 | optimization error: 2.024 | ||||||||
| 7/6 mapping: 5\11<7/5> | equal division error: -2.093 | optimization error: -2.150 | ||||||||
| Beta 7/5 | 13ed7/5 | 26.780555 | 44.808630 | 26.775895108856630 | 44.816428923157735 | 7/5 mapping: 13\13<7/5> | equal division error: 0.000 | optimization error: 0.101 | ||
| 6/5 mapping: 7\13<7/5> | equal division error: -1.981 | optimization error: -1.926 | ||||||||
| 7/6 mapping: 6\13<7/5> | equal division error: 1.981 | optimization error: 2.028 | ||||||||
| Gamma 7/5 | 24ed7/5 | 49.441025 | 24.271341 | 49.440489621601243 | 24.271604290013001 | 7/5 mapping: 24\24<7/5> | equal division error: 0.000 | optimization error: 0.006 | ||
| 6/5 mapping: 13\24<7/5> | equal division error: -0.114 | optimization error: -0.110 | ||||||||
| 7/6 mapping: 11\24<7/5> | equal division error: 0.114 | optimization error: 0.117 | ||||||||
| Alpha 4/3 | 13ed4/3 | 31.322471 | 38.311154 | 31.326679032092577 | 38.306007437643215 | 4/3 mapping: 13\13<4/3> | equal division error: 0.000 | optimization error: -0.067 | ||
| 7/6 mapping: 7\13<4/3> | equal division error: 1.307 | optimization error: 1.271 | ||||||||
| 8/7 mapping: 6\13<4/3> | equal division error: -1.307 | optimization error: -1.338 | ||||||||
| Beta 4/3 | 15ed4/3 | 36.141313 | 33.203000 | 36.137297503882719 | 33.206689013506551 | 4/3 mapping: 15\15<4/3> | equal division error: 0.000 | optimization error: 0.055 | ||
| 7/6 mapping: 8\15<4/3> | equal division error: -1.247 | optimization error: -1.217 | ||||||||
| 8/7 mapping: 7\15<4/3> | equal division error: 1.247 | optimization error: 1.273 | ||||||||
| Gamma 4/3 | 28ed4/3 | 67.463784 | 17.787321 | 67.463390164664623 | 17.787425106728855 | 4/3 mapping: 28\28<4/3> | equal division error: 0.000 | optimization error: 0.003 | ||
| 7/6 mapping: 15\28<4/3> | equal division error: -0.061 | optimization error: -0.060 | ||||||||
| 8/7 mapping: 13\28<4/3> | equal division error: 0.061 | optimization error: 0.062 | ||||||||
| Alpha 9/7 | 15ed9/7 | 41.371312 | 29.005606 | 41.374987163985893 | 29.003030145820039 | 9/7 mapping: 15\15<9/7> | equal division error: 0.000 | optimization error: -0.039 | ||
| 8/7 mapping: 8\15<9/7> | equal division error: 0.871 | optimization error: 0.850 | ||||||||
| 9/8 mapping: 7\15<9/7> | equal division error: -0.871 | optimization error: -0.889 | ||||||||
| Beta 9/7 | 17ed9/7 | 46.887487 | 25.593182 | 46.883960906871343 | 25.595107085419638 | 9/7 mapping: 17\17<9/7> | equal division error: 0.000 | optimization error: 0.033 | ||
| 8/7 mapping: 9\17<9/7> | equal division error: -0.835 | optimization error: -0.818 | ||||||||
| 9/8 mapping: 8\17<9/7> | equal division error: 0.835 | optimization error: 0.851 | ||||||||
| Gamma 9/7 | 32ed9/7 | 88.258800 | 13.596378 | 88.258498580415662 | 13.596424359141285 | 9/7 mapping: 32\32<9/7> | equal division error: 0.000 | optimization error: 0.001 | ||
| 8/7 mapping: 17\32<9/7> | equal division error: -0.036 | optimization error: -0.035 | ||||||||
| 9/8 mapping: 15\32<9/7> | equal division error: 0.036 | optimization error: 0.036 | ||||||||
| Alpha 5/4 | 17ed5/4 | 52.806823 | 22.724336 | 52.810084374305705 | 22.722932830303330 | 5/4 mapping: 17\17<5/4> | equal division error: 0.000 | optimization error: -0.024 | ||
| 9/8 mapping: 9\17<5/4> | equal division error: 0.609 | optimization error: 0.596 | ||||||||
| 10/9 mapping: 8\17<5/4> | equal division error: -0.609 | optimization error: -0.620 | ||||||||
| Beta 5/4 | 19ed5/4 | 59.019391 | 20.332301 | 59.016247125030467 | 20.333383745288099 | 5/4 mapping: 19\19<5/4> | equal division error: 0.000 | optimization error: 0.021 | ||
| 9/8 mapping: 10\19<5/4> | equal division error: -0.587 | optimization error: -0.576 | ||||||||
| 10/9 mapping: 9\19<5/4> | equal division error: 0.587 | optimization error: 0.597 | ||||||||
| Gamma 5/4 | 36ed5/4 | 111.826214 | 10.730936 | 111.825976049765954 | 10.730959320810789 | 5/4 mapping: 36\36<5/4> | equal division error: 0.000 | optimization error: 0.001 | ||
| 9/8 mapping: 19\36<5/4> | equal division error: -0.022 | optimization error: -0.022 | ||||||||
| 10/9 mapping: 17\36<5/4> | equal division error: 0.022 | optimization error: 0.023 | ||||||||
| Alpha 11/9 | 19ed11/9 | 65.628897 | 18.284628 | 65.631828119476568 | 18.283811900157846 | 11/9 mapping: 19\19<11/9> | equal division error: 0.000 | optimization error: -0.016 | ||
| 10/9 mapping: 10\19<11/9> | equal division error: 0.443 | optimization error: 0.434 | ||||||||
| 11/10 mapping: 9\19<11/9> | equal division error: -0.443 | optimization error: -0.450 | ||||||||
| Beta 11/9 | 21ed11/9 | 72.537202 | 16.543235 | 72.534366561494206 | 16.543881981552112 | 11/9 mapping: 21\21<11/9> | equal division error: 0.000 | optimization error: 0.014 | ||
| 10/9 mapping: 11\21<11/9> | equal division error: -0.428 | optimization error: -0.421 | ||||||||
| 11/10 mapping: 10\21<11/9> | equal division error: 0.428 | optimization error: 0.435 | ||||||||
| Gamma 11/9 | 40ed11/9 | 138.166099 | 8.685199 | 138.165906595462172 | 8.685210625176124 | 11/9 mapping: 40\40<11/9> | equal division error: 0.000 | optimization error: 0.000 | ||
| 10/9 mapping: 21\40<11/9> | equal division error: -0.015 | optimization error: -0.014 | ||||||||
| 11/10 mapping: 19\40<11/9> | equal division error: 0.015 | optimization error: 0.015 | ||||||||
| Alpha 6/5 | 21ed6/5 | 79.837464 | 15.030537 | 79.840125772190183 | 15.030036443379233 | 6/5 mapping: 21\21<6/5> | equal division error: 0.000 | optimization error: -0.011 | ||
| 11/10 mapping: 11\21<6/5> | equal division error: 0.332 | optimization error: 0.326 | ||||||||
| 12/11 mapping: 10\21<6/5> | equal division error: -0.332 | optimization error: -0.337 | ||||||||
| Beta 6/5 | 23ed6/5 | 87.441032 | 13.723534 | 87.438449973495273 | 13.723939529620542 | 6/5 mapping: 23\23<6/5> | equal division error: 0.000 | optimization error: 0.009 | ||
| 11/10 mapping: 12\23<6/5> | equal division error: -0.322 | optimization error: -0.317 | ||||||||
| 12/11 mapping: 11\23<6/5> | equal division error: 0.322 | optimization error: 0.326 | ||||||||
| Gamma 6/5 | 44ed6/5 | 167.278497 | 7.173666 | 167.278337553931523 | 7.173672440480304 | 6/5 mapping: 44\44<6/5> | equal division error: 0.000 | optimization error: 0.000 | ||
| 11/10 mapping: 23\44<6/5> | equal division error: -0.010 | optimization error: -0.010 | ||||||||
| 12/11 mapping: 21\44<6/5> | equal division error: 0.010 | optimization error: 0.010 | ||||||||
| Alpha 13/11 | 23ed13/11 | 95.432477 | 12.574336 | 95.434914550823771 | 12.574014506618971 | 13/11 mapping: 23\23<13/11> | equal division error: 0.000 | optimization error: -0.007 | ||
| 12/11 mapping: 12\23<13/11> | equal division error: 0.255 | optimization error: 0.251 | ||||||||
| 13/12 mapping: 11\23<13/11> | equal division error: -0.255 | optimization error: -0.259 | ||||||||
| Beta 13/11 | 25ed13/11 | 103.730954 | 11.568389 | 103.728582924336770 | 11.568653173208022 | 13/11 mapping: 25\25<13/11> | equal division error: 0.000 | optimization error: 0.007 | ||
| 12/11 mapping: 13\25<13/11> | equal division error: -0.248 | optimization error: -0.245 | ||||||||
| 13/12 mapping: 12\25<13/11> | equal division error: 0.248 | optimization error: 0.251 | ||||||||
| Gamma 13/11 | 48ed13/11 | 199.163431 | 6.025202 | 199.163297261207502 | 6.025206534044126 | 13/11 mapping: 48\48<13/11> | equal division error: 0.000 | optimization error: 0.000 | ||
| 12/11 mapping: 25\48<13/11> | equal division error: -0.007 | optimization error: -0.007 | ||||||||
| 13/12 mapping: 23\48<13/11> | equal division error: 0.007 | optimization error: 0.007 | ||||||||
| Alpha 7/6 | 25ed7/6 | 112.413903 | 10.674836 | 112.416150402630623 | 10.674622780642016 | 7/6 mapping: 25\25<7/6> | equal division error: 0.000 | optimization error: -0.005 | ||
| 13/12 mapping: 13\25<7/6> | equal division error: 0.200 | optimization error: 0.197 | ||||||||
| 14/13 mapping: 12\25<7/6> | equal division error: -0.200 | optimization error: -0.203 | ||||||||
| Beta 7/6 | 27ed7/6 | 121.407015 | 9.884108 | 121.404823766036118 | 9.884286000962910 | 7/6 mapping: 27\27<7/6> | equal division error: 0.000 | optimization error: 0.005 | ||
| 13/12 mapping: 14\27<7/6> | equal division error: -0.195 | optimization error: -0.193 | ||||||||
| 14/13 mapping: 13\27<7/6> | equal division error: 0.195 | optimization error: 0.197 | ||||||||
| Gamma 7/6 | 52ed7/6 | 233.820917 | 5.132133 | 233.820803527976982 | 5.132135301452842 | 7/6 mapping: 52\52<7/6> | equal division error: 0.000 | optimization error: 0.000 | ||
| 13/12 mapping: 27\52<7/6> | equal division error: -0.005 | optimization error: -0.005 | ||||||||
| 14/13 mapping: 25\52<7/6> | equal division error: 0.005 | optimization error: 0.005 | ||||||||
| Alpha 15/13 | 27ed15/13 | 130.781716 | 9.175595 | 130.783801507844919 | 9.175448229557843 | 15/13 mapping: 27\27<15/13> | equal division error: 0.000 | optimization error: -0.004 | ||
| 14/13 mapping: 14\27<15/13> | equal division error: 0.160 | optimization error: 0.158 | ||||||||
| 15/14 mapping: 13\27<15/13> | equal division error: -0.160 | optimization error: -0.162 | ||||||||
| Beta 15/13 | 29ed15/13 | 140.469250 | 8.542795 | 140.467213664559518 | 8.542918797162452 | 15/13 mapping: 29\29<15/13> | equal division error: 0.000 | optimization error: 0.004 | ||
| 14/13 mapping: 15\29<15/13> | equal division error: -0.156 | optimization error: -0.154 | ||||||||
| 15/14 mapping: 14\29<15/13> | equal division error: 0.156 | optimization error: 0.158 | ||||||||
| Gamma 15/13 | 56ed15/13 | 271.250966 | 4.423947 | 271.250868008139347 | 4.423948976871078 | 15/13 mapping: 56\56<15/13> | equal division error: 0.000 | optimization error: 0.000 | ||
| 14/13 mapping: 29\56<15/13> | equal division error: -0.004 | optimization error: -0.004 | ||||||||
| 15/14 mapping: 27\56<15/13> | equal division error: 0.004 | optimization error: 0.004 | ||||||||
| Alpha 8/7 | 29ed8/7 | 150.535899 | 7.971520 | 150.537844310638475 | 7.971417456488689 | 8/7 mapping: 29\29<8/7> | equal division error: 0.000 | optimization error: -0.003 | ||
| 15/14 mapping: 15\29<8/7> | equal division error: 0.130 | optimization error: 0.128 | ||||||||
| 16/15 mapping: 14\29<8/7> | equal division error: -0.130 | optimization error: -0.131 | ||||||||
| Beta 8/7 | 31ed8/7 | 160.917685 | 7.457229 | 160.915782495277457 | 7.457316997698579 | 8/7 mapping: 31\31<8/7> | equal division error: 0.000 | optimization error: 0.003 | ||
| 15/14 mapping: 16\31<8/7> | equal division error: -0.127 | optimization error: -0.126 | ||||||||
| 16/15 mapping: 15\31<8/7> | equal division error: 0.127 | optimization error: 0.128 | ||||||||
| Gamma 8/7 | 60ed8/7 | 311.453584 | 3.852902 | 311.453498588281532 | 3.852902617691610 | 8/7 mapping: 60\60<8/7> | equal division error: 0.000 | optimization error: 0.000 | ||
| 15/14 mapping: 31\60<8/7> | equal division error: -0.003 | optimization error: -0.003 | ||||||||
| 16/15 mapping: 29\60<8/7> | equal division error: 0.003 | optimization error: 0.003 |
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Todo: Temperaments
Compute the temperaments associated to each Alpha-Beta-Gamma scales. |
Coincidence?
As a coincidence (?), 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).