View source for User:Xenllium/Xenllium's circulating scales

Below are listed circulating scales introduced by [[User:Xenllium|Xenllium]].

== 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), and one schisma-compressed 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 triad (at C–E–G) and a pure minor triad (at E–G–B).

<pre>
! 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
</pre>

=== Intervals ===
By the definition, there are no fifths larger than pure 3/2, no major thirds larger than Pythagorean 81/64, no minor thirds smaller than Pythagorean 32/27, and no whole tones larger than Pythagorean 9/8. Major thirds and minor thirds come in five sizes, whole tones and semitones come in four sizes respectively. 

<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"
! colspan="4" | Fifth (7-step)
! colspan="4" | Fourth (5-step)
|-
! Occurrences
! Ratio
! Cents
! Error <br>from 3/2
! Occurrences
! Ratio
! Cents
! Error <br>from 4/3
|-
| D–A <br> G–D <br> A–E
| <math>\sqrt[3]{10/3}</math>
| 694.78624
| &minus;7.16876
| D–G <br> E–A <br> A–D
| <math>\sqrt[3]{12/5}</math>
| 505.21376
| +7.16876
|-
| G♯–D♯ <br> (A♭–E♭)
| <math>16384/10935</math>
| 700.00128
| &minus;1.95372
| D♯–G♯ <br> (E♭–A♭) 
| <math>10935/8192</math>
| 499.99872
| +1.95372
|-
| C–G <br> C♯–G♯ <br> E♭–B♭ <br> E–B <br> F–C <br> F♯–C♯ <br> B♭–F <br> B–F♯
| <math>3/2</math>
| 701.95500
| +0.00000
| C–F <br> C♯–F♯ <br> F–B♭ <br> F♯–B <br> G–C <br> G♯–C♯ <br> B♭–E♭ <br> B–E
| <math>4/3</math>
| 498.04500
| +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"
! colspan="4" | Major third (4-step)
! colspan="4" | Minor third (3-step)
|-
! Occurrences
! Ratio
! Cents
! Error <br>from 5/4
! Occurrences
! Ratio
! Cents
! Error <br>from 6/5
|-
| C–E <br> G–B
| <math>5/4</math>
| 386.31371
| +0.00000
| rowspan="2" | C–E♭ <br> C♯–E <br> G–B♭ <br> G♯–B
| rowspan="2" | <math>32/27</math>
| rowspan="2" | 294.13500
| rowspan="2" | &minus;21.50629
|-
| D–F♯ <br> F–A
| <math>\sqrt[3]{(45/32)^{2}}</math>
| 393.48248
| +7.16876
|-
| A–C♯ <br> B♭–D
| <math>\sqrt[3]{32805/16384}</math>
| 400.65124
| +14.33753
| rowspan="2" | E♭–G♭ <br> F–A♭ <br> B♭–D♭
| rowspan="2" | <math>1215/1024</math>
| rowspan="2" | 296.08872
| rowspan="2" | &minus;19.55257
|-
| rowspan="3" | D♭–F <br> G♭–B♭ <br> A♭–C <br> B–D♯
| rowspan="3" | <math>512/405</math>
| rowspan="3" | 405.86628
| rowspan="3" | +19.55257
|-
| D–F <br> F♯–A
| <math>\sqrt[3]{2048/1215}</math>
| 301.30376
| &minus;14.33753
|-
| rowspan="2" | A–C <br> B–D
| rowspan="2" | <math>\sqrt[3]{128/75}</math>
| rowspan="2" | 308.47252
| rowspan="2" | &minus;7.16876
|-
| rowspan="2" | E♭–G <br> E–G♯
| rowspan="2" | <math>81/64</math>
| rowspan="2" | 407.82000
| rowspan="2" | +21.50629
|-
| E–G
| <math>6/5</math>
| 315.64129
| +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>

=== Music ===
* ''[https://youtube.com/watch?v=QOFBKfCYThI Xentwelve tuning]'' – demonstration of Xentwelve tuning
* [https://youtube.com/watch?v=bbJ0HjPAuaA <span lang="ja" style="font-family:Yu Gothic UI, Yu Gothic, Meiryo, MS PGothic, sans-serif">【オリジナル曲】かなしいこと</span> / The sadness]

== 31-tone circulating scales ==
'''Xenthirtyone''' is a 31-tone circulating scale based on [[31edo|31 equal temperament]]. There are two circulating scales, named ''Xenthirtyone I'' and ''Xenthirtyone II'', generated by a major third, which comes in three sizes, with twenty-two pure major thirds, eight 1/4-Würschmidt-comma-stretched major thirds, and one luna-comma-stretched major third.

=== Xenthirtyone I ===
<pre>
! xenthirtyone1.scl
!
Xenthirtyone I, Xenllium's 31-tone circulating scale
31
!
128/125
73.53374935096
16/15
2048/1875
262144/234375
9375/8192
75/64
6/5
348.11617794602
5/4
32/25
462.70878570247
4/3
512/375
65536/46875
46875/32768
375/256
3/2
737.29121429753
25/16
8/5
851.88382205398
5/3
128/75
16384/9375
234375/131072
1875/1024
15/8
1126.46625064904
125/64
2/1
</pre>

=== Xenthirtyone II ===
<pre>
! xenthirtyone2.scl
!
Xenthirtyone II, Xenllium's 31-tone circulating scale
31
!
128/125
79.25639432431
117.45393024313
155.65146616195
262144/234375
9375/8192
271.72110610838
309.91864202720
348.11617794602
5/4
32/25
465.57010818915
503.76764410797
541.96518002679
65536/46875
46875/32768
658.03481997321
696.23235589203
734.42989181085
25/16
8/5
851.88382205398
890.08135797280
928.27889389162
16384/9375
234375/131072
1044.34853383805
1082.54606975687
1120.74360567569
125/64
2/1
</pre>

[[Category:Pages with Scala files]]
[[Category:12-tone scales]]
[[Category:31-tone scales]]
[[Category:Tempered scales]]