User:Xenllium/Xenllium's circulating scales: Difference between revisions
fixed typo (cube-root rather than square-root in ratio) |
No edit summary |
||
| Line 2: | Line 2: | ||
== Xentwelve == | == Xentwelve == | ||
'''Xentwelve''' is a 12-tone circulating scale based on [[12edo|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 | '''Xentwelve''' is a 12-tone circulating scale based on [[12edo|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), 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), with a pure major chord (at C–E–G) and a pure minor chord (at E–G–B). | ||
<pre> | <pre> | ||
| Line 24: | Line 24: | ||
</pre> | </pre> | ||
<div class="toccolours mw-collapsible mw-collapsed" style="width:500px; overflow:auto;"> | |||
<div style="line-height:1.6;">'''Sizes and occurrences of fifth and fourth'''</div> | |||
<div class="mw-collapsible-content"> | |||
{| class="wikitable center-all" | {| class="wikitable center-all" | ||
! colspan="4" | Fifth (7-step) | ! colspan="4" | Fifth (7-step) | ||
! colspan="4" | Fourth (5-step) | ! colspan="4" | Fourth (5-step) | ||
| Line 65: | Line 67: | ||
| +0.00000 | | +0.00000 | ||
|} | |} | ||
</div></div> | |||
<div class="toccolours mw-collapsible mw-collapsed" style="width:500px; overflow:auto;"> | |||
<div style="line-height:1.6;">'''Sizes and occurrences of major third and minor third'''</div> | |||
<div class="mw-collapsible-content"> | |||
{| class="wikitable center-all left-4 left-8" | {| class="wikitable center-all left-4 left-8" | ||
! colspan="4" | Major third (4-step) | ! colspan="4" | Major third (4-step) | ||
! colspan="4" | Minor third (3-step) | ! colspan="4" | Minor third (3-step) | ||
| Line 128: | Line 134: | ||
| +0.00000 | | +0.00000 | ||
|} | |} | ||
</div></div> | |||
<div class="toccolours mw-collapsible mw-collapsed" style="width:500px; overflow:auto;"> | |||
<div style="line-height:1.6;">'''Sizes and occurrences of whole tone and semitone'''</div> | |||
<div class="mw-collapsible-content"> | |||
{| class="wikitable center-all left-3 left-6" | |||
! colspan="3" | Whole tone | |||
! colspan="3" | Semitone | |||
|- | |||
! Occurrences | |||
! Ratio | |||
! Cents | |||
! Occurrences | |||
! Ratio | |||
! Cents | |||
|- | |||
| D–E <br> G–A | |||
| <math>\sqrt[3]{25/18}</math> | |||
| 189.57248 | |||
| rowspan="3" | C–D♭ <br> D♯–E <br> F–G♭ <br> G–A♭ <br> A♯–B | |||
| rowspan="3" | <math>135/128</math> | |||
| rowspan="3" | 92.17872 | |||
|- | |||
| C–D <br> A–B | |||
| <math>\sqrt[3]{45/32}</math> | |||
| 196.74124 | |||
|- | |||
| rowspan="2" | D♭–E♭ <br> A♭–B♭ | |||
| rowspan="2" | <math>4096/3645</math> | |||
| rowspan="2" | 201.95628 | |||
|- | |||
| rowspan="2" | D–E♭<br>G♯–A | |||
| rowspan="2" | <math>\sqrt[3]{1048576/885735}</math> | |||
| rowspan="2" | 97.39376 | |||
|- | |||
| rowspan="3" | E♭–F <br> E–F♯ <br> F–G <br> F♯–G♯ <br> B♭–C <br>B–C♯ | |||
| rowspan="3" | <math>9/8</math> | |||
| rowspan="3" | 203.91000 | |||
|- | |||
| C♯–D <br> A–B♭ | |||
| <math>\sqrt[3]{65536/54675}</math> | |||
| 104.56252 | |||
|- | |||
| E–F <br> F♯–G <br> B–C | |||
| <math>16/15</math> | |||
| 111.73129 | |||
|} | |||
</div></div> | |||
[[Category:12-tone scales]] | [[Category:12-tone scales]] | ||
[[Category:Tempered scales]] | [[Category:Tempered scales]] | ||