User:Xenllium/Xenllium's circulating scales - Revision history

Xenllium at 14:02, 22 September 2024

2024-09-22T14:02:16Z

← Older revision		Revision as of 14:02, 22 September 2024
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	=== Music ===		=== Music ===
	* ''[https://youtube.com/watch?v=QOFBKfCYThI Xentwelve tuning]'' – demonstration of Xentwelve tuning		* ''[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 ==		== 31-tone circulating scales ==

Xenllium at 12:27, 20 June 2024

2024-06-20T12:27:47Z

← Older revision		Revision as of 12:27, 20 June 2024
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	\|}		\|}
	</div></div>		</div></div>

			=== Music ===
			* ''[https://youtube.com/watch?v=QOFBKfCYThI Xentwelve tuning]'' – demonstration of Xentwelve tuning

	== 31-tone circulating scales ==		== 31-tone circulating scales ==

Xenllium at 00:11, 30 March 2024

2024-03-30T00:11:22Z

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

Xenllium at 22:18, 11 January 2024

2024-01-11T22:18:12Z

← Older revision		Revision as of 22:18, 11 January 2024
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	== 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), 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 triad (at C–E–G) and a pure minor triad (at E–G–B).		'''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>		<pre>
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	== 31-tone circulating scales ==		== 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''' 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 ===		=== Xenthirtyone I ===

Xenllium: /* Xentwelve */

2024-01-03T15:28:03Z

Xentwelve

@@ 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), 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).
+'''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 triad (at C–E–G) and a pure minor triad (at E–G–B).
 <pre>
@@ Line 184: / Line 184: @@
 |}
 </div></div>
 [[Category:Pages with Scala files]]
 [[Category:12-tone scales]]
 [[Category:Tempered scales]]

Xenllium at 13:26, 3 January 2024

2024-01-03T13:26:56Z

← Older revision		Revision as of 13:26, 3 January 2024
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	!		!
	Xentwelve, Xenllium's 12-tone circulating scale, Central A		Xentwelve, Xenllium's 12-tone circulating scale, Central A
	12		12
	!		!
	104.56252207087		104.56252207087
	196.74123853187		196.74123853187
	308.47252380165		308.47252380165
	400.65124026264		400.65124026264
	505.21376233352		505.21376233352
	602.60752120549		602.60752120549
	694.78623766648		694.78623766648
	806.51752293626		806.51752293626
	898.69623939726		898.69623939726
	1010.42752466704		1010.42752466704
	1102.60624112803		1102.60624112803
	1200.00000000000		1200.00000000000
	</pre>		</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 class="toccolours mw-collapsible mw-collapsed" style="width:500px; overflow:auto;">
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	\|}		\|}
	</div></div>		</div></div>


	<div class="toccolours mw-collapsible mw-collapsed" style="width:500px; overflow:auto;">		<div class="toccolours mw-collapsible mw-collapsed" style="width:500px; overflow:auto;">
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	\|}		\|}
	</div></div>		</div></div>


	<div class="toccolours mw-collapsible mw-collapsed" style="width:500px; overflow:auto;">		<div class="toccolours mw-collapsible mw-collapsed" style="width:500px; overflow:auto;">
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	</div></div>		</div></div>

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

Xenllium at 12:52, 3 January 2024

2024-01-03T12:52:16Z

@@ 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♯–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''' 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>
@@ Line 24: / Line 24: @@
 </pre>
 {| class="wikitable center-all"
 ! colspan="4" | Fifth (7-step)
 ! colspan="4" | Fourth (5-step)
@@ Line 65: / Line 67: @@
 | +0.00000
 |}
 {| class="wikitable center-all left-4 left-8"
 ! colspan="4" | Major third (4-step)
 ! colspan="4" | Minor third (3-step)
@@ Line 128: / Line 134: @@
 | +0.00000
 |}
 [[Category:12-tone scales]]
 [[Category:Tempered scales]]

Xenllium: fixed typo (cube-root rather than square-root in ratio)

2024-01-03T11:56:53Z

fixed typo (cube-root rather than square-root in ratio)

← Older revision		Revision as of 11:56, 3 January 2024
Line 21:		Line 21:
	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
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	\|-		\|-
	\| 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
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	\|-		\|-
	\| 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~~/256~~</math>		\| 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

Xenllium at 11:48, 3 January 2024

2024-01-03T11:48:30Z

@@ Line 24: / Line 24: @@
 </pre>
-{| class="wikitable center-all left-all"
+{| class="wikitable center-all"
 |+ Sizes and occurrences of fifth and fourth
 ! colspan="4" | Fifth (7-step)
@@ Line 63: / Line 63: @@
 | <math>4/3</math>
 | 498.04500
 | +0.00000
 |}

Xenllium at 10:25, 3 January 2024

2024-01-03T10:25:11Z

@@ 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).
 <pre>
@@ Line 23: / Line 23: @@
 .
 </pre>
 [[Category:12-tone scales]]
 [[Category:Tempered scales]]