User:Xenllium/Xenllium's circulating scales: Difference between revisions
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fixed typo (cube-root rather than square-root in ratio) |
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1010.42752466704 | 1010.42752466704 | ||
1102.60624112803 | 1102.60624112803 | ||
1200. | 1200.00000000000 | ||
</pre> | </pre> | ||
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|- | |- | ||
| D–A <br> G–D <br> A–E | | D–A <br> G–D <br> A–E | ||
| <math>\sqrt{10/3}</math> | | <math>\sqrt[3]{10/3}</math> | ||
| 694.78624 | | 694.78624 | ||
| −7.16876 | | −7.16876 | ||
| D–G <br> E–A <br> A–D | | D–G <br> E–A <br> A–D | ||
| <math>\sqrt{12/5}</math> | | <math>\sqrt[3]{12/5}</math> | ||
| 505.21376 | | 505.21376 | ||
| +7.16876 | | +7.16876 | ||
| Line 90: | Line 90: | ||
|- | |- | ||
| D–F♯ <br> F–A | | D–F♯ <br> F–A | ||
| <math>\sqrt{(45/32)^{2}}</math> | | <math>\sqrt[3]{(45/32)^{2}}</math> | ||
| 393.48248 | | 393.48248 | ||
| +7.16876 | | +7.16876 | ||
|- | |- | ||
| A–C♯ <br> B♭–D | | A–C♯ <br> B♭–D | ||
| <math>\sqrt{32805/16384}</math> | | <math>\sqrt[3]{32805/16384}</math> | ||
| 400.65124 | | 400.65124 | ||
| +14.33753 | | +14.33753 | ||
| Line 104: | Line 104: | ||
|- | |- | ||
| rowspan="3" | D♭–F <br> G♭–B♭ <br> A♭–C <br> B–D♯ | | rowspan="3" | D♭–F <br> G♭–B♭ <br> A♭–C <br> B–D♯ | ||
| rowspan="3" | <math>405 | | rowspan="3" | <math>512/405</math> | ||
| rowspan="3" | 405.86628 | | rowspan="3" | 405.86628 | ||
| rowspan="3" | +19.55257 | | rowspan="3" | +19.55257 | ||
|- | |- | ||
| D–F <br> F♯–A | | D–F <br> F♯–A | ||
| <math>\sqrt{2048/1215}</math> | | <math>\sqrt[3]{2048/1215}</math> | ||
| 301.30376 | | 301.30376 | ||
| −14.33753 | | −14.33753 | ||
|- | |- | ||
| rowspan="2" | A–C <br> B–D | | rowspan="2" | A–C <br> B–D | ||
| rowspan="2" | <math>\sqrt{128/75}</math> | | rowspan="2" | <math>\sqrt[3]{128/75}</math> | ||
| rowspan="2" | 308.47252 | | rowspan="2" | 308.47252 | ||
| rowspan="2" | −7.16876 | | rowspan="2" | −7.16876 | ||
Revision as of 11:56, 3 January 2024
Below are listed circulating scales introduced by Xenllium.
Xentwelve
Xentwelve is a 12-tone circulating scale based on 12 equal temperament. In summary, it is close to 1/3-comma meantone in the natural keys and Pythagorean tuning in the remote keys. The generator is a perfect fifth, which comes in three sizes, with eight pure fifths (at C–G, C♯–G♯, E♭–B♭, E–B, F–C, F♯–C♯, B♭–F and B–F♯, frequency ratio 3/2), three 1/3-comma meantone fifths (at D–A, G–D and A–E, frequency ratio (10/3)^(1/3)), and one narrow schismic fifth (at G♯–D♯ (A♭–E♭), frequency ratio 16384/10935). It derives two major thirds exact 5/4 (at C–E and G–B) and one minor third exact 6/5 (at E–G).
! xentwelve_a.scl ! Xentwelve, Xenllium's 12-tone circulating scale, Central A 12 ! 104.56252207087 196.74123853187 308.47252380165 400.65124026264 505.21376233352 602.60752120549 694.78623766648 806.51752293626 898.69623939726 1010.42752466704 1102.60624112803 1200.00000000000
| Fifth (7-step) | Fourth (5-step) | ||||||
|---|---|---|---|---|---|---|---|
| Occurrences | Ratio | Cents | Error from 3/2 |
Occurrences | Ratio | Cents | Error from 4/3 |
| D–A G–D A–E |
[math]\displaystyle{ \sqrt[3]{10/3} }[/math] | 694.78624 | −7.16876 | D–G E–A A–D |
[math]\displaystyle{ \sqrt[3]{12/5} }[/math] | 505.21376 | +7.16876 |
| G♯–D♯ (A♭–E♭) |
[math]\displaystyle{ 16384/10935 }[/math] | 700.00128 | −1.95372 | D♯–G♯ (E♭–A♭) |
[math]\displaystyle{ 10935/8192 }[/math] | 499.99872 | +1.95372 |
| C–G C♯–G♯ E♭–B♭ E–B F–C F♯–C♯ B♭–F B–F♯ |
[math]\displaystyle{ 3/2 }[/math] | 701.95500 | +0.00000 | C–F C♯–F♯ F–B♭ F♯–B G–C G♯–C♯ B♭–E♭ B–E |
[math]\displaystyle{ 4/3 }[/math] | 498.04500 | +0.00000 |
| Major third (4-step) | Minor third (3-step) | ||||||
|---|---|---|---|---|---|---|---|
| Occurrences | Ratio | Cents | Error from 5/4 |
Occurrences | Ratio | Cents | Error from 6/5 |
| C–E G–B |
[math]\displaystyle{ 5/4 }[/math] | 386.31371 | +0.00000 | C–E♭ C♯–E G–B♭ G♯–B |
[math]\displaystyle{ 32/27 }[/math] | 294.13500 | −21.50629 |
| D–F♯ F–A |
[math]\displaystyle{ \sqrt[3]{(45/32)^{2}} }[/math] | 393.48248 | +7.16876 | ||||
| A–C♯ B♭–D |
[math]\displaystyle{ \sqrt[3]{32805/16384} }[/math] | 400.65124 | +14.33753 | E♭–G♭ F–A♭ B♭–D♭ |
[math]\displaystyle{ 1215/1024 }[/math] | 296.08872 | −19.55257 |
| D♭–F G♭–B♭ A♭–C B–D♯ |
[math]\displaystyle{ 512/405 }[/math] | 405.86628 | +19.55257 | ||||
| D–F F♯–A |
[math]\displaystyle{ \sqrt[3]{2048/1215} }[/math] | 301.30376 | −14.33753 | ||||
| A–C B–D |
[math]\displaystyle{ \sqrt[3]{128/75} }[/math] | 308.47252 | −7.16876 | ||||
| E♭–G E–G♯ |
[math]\displaystyle{ 81/64 }[/math] | 407.82000 | +21.50629 | ||||
| E–G | [math]\displaystyle{ 6/5 }[/math] | 315.64129 | +0.00000 | ||||