User:Xenllium/Xenllium's circulating scales: Difference between revisions

Revision as of 10:25, 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.

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{10/3} }[/math]	694.78624	−7.16876	D–G E–A A–D	[math]\displaystyle{ \sqrt{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

@@ Line 2: / Line 2: @@
 == 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, frequency ratio (10/3)^(1/3)), and one narrow schismic fifth (at G♯(A♭)-D♯(E♭), frequency ratio [[16384/10935]]).
+'''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, 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).
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 .
 </pre>
+{| class="wikitable center-all left-all"
+|+ Sizes and occurrences of fifth and fourth
+! 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{10/3}</math>
+| 694.78624
+| &minus;7.16876
+| D–G <br> E–A <br> A–D
+| <math>\sqrt{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
+|}
 [[Category:12-tone scales]]
 [[Category:Tempered scales]]

User:Xenllium/Xenllium's circulating scales: Difference between revisions

Revision as of 10:25, 3 January 2024

Xentwelve

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