User:Xenllium/Xenllium's circulating scales

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), 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).

! 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

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.

Sizes and occurrences of fifth and fourth

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

Sizes and occurrences of major third and minor third

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

Sizes and occurrences of whole tone and semitone

Whole tone			Semitone
Occurrences	Ratio	Cents	Occurrences	Ratio	Cents
D–E G–A	[math]\displaystyle{ \sqrt[3]{25/18} }[/math]	189.57248	C–D♭ D♯–E F–G♭ G–A♭ A♯–B	[math]\displaystyle{ 135/128 }[/math]	92.17872
C–D A–B	[math]\displaystyle{ \sqrt[3]{45/32} }[/math]	196.74124
D♭–E♭ A♭–B♭	[math]\displaystyle{ 4096/3645 }[/math]	201.95628
			D–E♭ G♯–A	[math]\displaystyle{ \sqrt[3]{1048576/885735} }[/math]	97.39376
E♭–F E–F♯ F–G F♯–G♯ B♭–C B–C♯	[math]\displaystyle{ 9/8 }[/math]	203.91000
			C♯–D A–B♭	[math]\displaystyle{ \sqrt[3]{65536/54675} }[/math]	104.56252
			E–F F♯–G B–C	[math]\displaystyle{ 16/15 }[/math]	111.73129

Music

• Xentwelve tuning – demonstration of Xentwelve tuning
• 【オリジナル曲】かなしいこと / The sadness

31-tone circulating scales

Xenthirtyone is a 31-tone circulating scale based on 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

! 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

Xenthirtyone II

! 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