Lériendil: Text replacement - "Eigenmonzo
(unchanged-interval)" to "Unchanged interval
(eigenmonzo)"

2025-06-21T06:56:19Z

Text replacement - "Eigenmonzo<br>(unchanged-interval)" to "Unchanged interval<br>(eigenmonzo)"

← Older revision		Revision as of 06:56, 21 June 2025
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	{\| class="wikitable center-all"		{\| class="wikitable center-all"
	\|-		\|-
	! [[Eigenmonzo\|~~Eigenmonzo~~<br>(~~unchanged-interval~~)]]		! [[Eigenmonzo\|Unchanged interval<br>(eigenmonzo)]]
	! Generator<br>(¢)		! Generator<br>(¢)
	! Comments		! Comments

FloraC: - parent category

2025-04-28T14:36:13Z

- parent category

← Older revision		Revision as of 14:36, 28 April 2025
Line 617:		Line 617:
	* [[Quintupole97]]		* [[Quintupole97]]

	[[Category:~~Temperaments~~]]		[[Category:Quintupole\| ]] <!-- main article -->
	[[Category:Rank-2 temperaments]]		[[Category:Rank-2 temperaments]]
	[[Category:Quintaleap family]]		[[Category:Quintaleap family]]
	[[Category:Octagar temperaments]]		[[Category:Octagar temperaments]]
	[[Category:Mistismic temperaments]]		[[Category:Mistismic temperaments]]
	~~[[Category:Quintupole]]~~

Lériendil at 23:40, 19 April 2025

2025-04-19T23:40:26Z

← Older revision		Revision as of 23:40, 19 April 2025
Line 618:		Line 618:

	[[Category:Temperaments]]		[[Category:Temperaments]]
			[[Category:Rank-2 temperaments]]
	[[Category:Quintaleap family]]		[[Category:Quintaleap family]]
	[[Category:Octagar temperaments]]		[[Category:Octagar temperaments]]
	[[Category:Mistismic temperaments]]		[[Category:Mistismic temperaments]]
	[[Category:Quintupole]]		[[Category:Quintupole]]

Xenllium at 11:03, 18 June 2023

2023-06-18T11:03:41Z

← Older revision		Revision as of 11:03, 18 June 2023
Line 1:		Line 1:
	'''Quintupole''' is a temperament for the 7, 11, 13, 17, and 19 [[Harmonic limit\|prime limits]]. It is a member of [[quintaleap family]], [[octagar temperaments]], and [[mistismic temperaments]]. It has a semitone inverval which represents [[18/17]], [[35/33]], [[52/49]], [[55/52]] and [[135/128]] as a generator. [[121edo\|121EDO]] is an excellent tuning for quintupole, with a semitone generator of 10\121, and MOS of 13, 25, 37, 49, 61, 73, 85, 97, or 109 notes are available.		'''Quintupole''' is a temperament for the 7, 11, 13, 17, and 19 [[Harmonic limit\|prime limits]]. It is a member of [[quintaleap family]], [[octagar temperaments]], and [[mistismic temperaments]]. It has a semitone inverval which represents [[18/17]], [[35/33]], [[52/49]], [[55/52]] and [[135/128]] as a generator, five of them give a fourth (~2.2 cents flat of [[4/3]]), sixteen give [[5/2]], and thirty-four give [[7/1\|seventh harmonic]]. [[121edo\|121EDO]] is an excellent tuning for quintupole, with a semitone generator of 10\121, and MOS of 13, 25, 37, 49, 61, 73, 85, 97, or 109 notes are available.

	== Temperament data ==		== Temperament data ==
Line 78:		Line 78:

	== Interval chain ==		== Interval chain ==
			Quintupole is considered as a [[cluster temperament]] with twelve clusters of notes in an octave. The chroma interval between adjacent notes in each cluster represents 100/99～120/119～171/170～196/195～209/208～210/209～221/220～225/224～289/288～324/323～441/440～513/512 all tempered together.

	{\| class="wikitable center-1 right-2"		{\| class="wikitable center-1 right-2"
	! Number of <br>generator		! Number of <br>generator
Line 85:		Line 87:
	\| 0		\| 0
	\| 0.000		\| 0.000
	\| 1/1		\| [[1/1]]
	\|-		\|-
	\| 1		\| 1
Line 604:		Line 606:
	\|		\|
	\|}		\|}

			== Scales ==
			* [[Quintupole13]]
			* [[Quintupole25]]
			* [[Quintupole37]]
			* [[Quintupole49]]
			* [[Quintupole61]]
			* [[Quintupole73]]
			* [[Quintupole85]]
			* [[Quintupole97]]

	[[Category:Temperaments]]		[[Category:Temperaments]]

Xenllium: Created page with "'''Quintupole''' is a temperament for the 7, 11, 13, 17, and 19 prime limits. It is a member of quintaleap family, octagar temperaments, and misti..."

2023-06-18T09:14:10Z

Created page with "'''Quintupole''' is a temperament for the 7, 11, 13, 17, and 19 prime limits. It is a member of quintaleap family, octagar temperaments, and misti..."

New page

'''Quintupole''' is a temperament for the 7, 11, 13, 17, and 19 [[Harmonic limit|prime limits]]. It is a member of [[quintaleap family]], [[octagar temperaments]], and [[mistismic temperaments]]. It has a semitone inverval which represents [[18/17]], [[35/33]], [[52/49]], [[55/52]] and [[135/128]] as a generator. [[121edo|121EDO]] is an excellent tuning for quintupole, with a semitone generator of 10\121, and MOS of 13, 25, 37, 49, 61, 73, 85, 97, or 109 notes are available.

== Temperament data ==
<div class="toccolours mw-collapsible mw-collapsed" style="width:600px; overflow:auto;">
<div style="line-height:1.6;">'''Quintupole temperament (109&121)'''</div>
<div class="mw-collapsible-content">
Subgroup: 2.3.5.7.11.13.17.19

[[Mapping]]: [{{val| 1 2 1 0 -1 -2 5 4 }}, {{val| 0 -5 16 34 54 69 -11 3 }}]

[[Comma list]]:
* 7-limit: 4000/3969, 458752/455625
* 11-limit: 896/891, 1375/1372, 4375/4356
* 13-limit: 352/351, 364/363, 625/624, 2704/2695
* 17-limit: 256/255, 352/351, 364/363, 375/374, 442/441
* 19-limit: 190/189, 256/255, 352/351, 361/360, 364/363, 375/374

[[Constrained tuning|Constrained TE generator]]:
* 7-limit: ~135/128 = 99.12306
* 11-limit: ~35/33 = 99.10969
* 13-limit: ~35/33 = 99.12210
* 17-limit: ~18/17 = 99.12595
* 19-limit: ~18/17 = 99.12598

[[POTE generator]]:
* 7-limit: ~135/128 = 99.17546
* 11-limit: ~35/33 = 99.15582
* 13-limit: ~35/33 = 99.16472
* 17-limit: ~18/17 = 99.17178
* 19-limit: ~18/17 = 99.16377

[[TOP tuning|TOP generator]]s:
* 7-limit: ~2 = 1199.32542, ~135/128 = 99.11971
* 11-limit: ~2 = 1199.34360, ~35/33 = 99.10159
* 13-limit: ~2 = 1199.32361, ~35/33 = 99.10883
* 17-limit: ~2 = 1199.22471, ~18/17 = 99.10770
* 19-limit: ~2 = 1199.36954, ~18/17 = 99.11167

[[Diamond monotone]] ranges:
* 7 and 9-odd-limit: ~135/128 = [98.63014, 100.00000] (6\73 to 1\12)
* 11-odd-limit: ~35/33 = [98.96907, 100.00000] (8\97 to 1\12)
* 13 and 15-odd-limit: ~35/33 = [99.08257, 99.24812] (9\109 to 11\133)
* 17, 19, and 21-odd-limit: ~18/17 = [99.08257, 99.24812] (9\109 to 11\133)

[[Diamond tradeoff]] ranges:
* 7 and 9-odd-limit: ~135/128 = [99.02846, 99.60900]
* 11 and 13-odd-limit: ~35/33 = [99.02846, 99.60900]
* 15-odd-limit: ~35/33 = [98.93352, 99.60900]
* 17-odd-limit: ~18/17 = [98.93352, 99.60900]
* 19-odd-limit: ~18/17 = [98.84446, 99.60900]
* 21-odd-limit: ~18/17 = [98.80517, 99.60900]

Diamond monotone and tradeoff ranges:
* 7 and 9-odd-limit: ~135/128 = [99.02846, 99.60900]
* 11-odd-limit: ~35/33 = [99.02846, 99.60900]
* 13 and 15-odd-limit: ~35/33 = [99.08257, 99.24812]
* 17, 19, and 21-odd-limit: ~18/17 = [99.08257, 99.24812]

[[Optimal ET sequence]]s (Constrained TE):
* 7-limit: {{Optimal ET sequence| 12, 61cdd, 73cd, 85d, 97, 109, 230, 569bbd }}
* 11-limit: {{Optimal ET sequence| 12, …, 85dee, 97e, 109, 230, 339b, 448b }}
* 13-limit: {{Optimal ET sequence| 12f, …, 97ef, 109, 230, 339b, 569bbdef }}
* 17 and 19-limit: {{Optimal ET sequence| 12f, …, 97efg, 109, 230g }}

[[Optimal ET sequence]]s (POTE):
* 7-limit: {{Optimal ET sequence| 12, 61cdd, 73cd, 85d, 97, 109, 121 }}
* 11-limit: {{Optimal ET sequence| 12, …, 85dee, 97e, 109, 121, 351bde, 472bdee }}
* 13-limit: {{Optimal ET sequence| 12f, …, 97ef, 109, 121 }}
* 17 and 19-limit: {{Optimal ET sequence| 12f, …, 97efg, 109, 121 }}

[[Badness]]:
* 7-limit: 0.111620
* 11-limit: 0.056501
* 13-limit: 0.038431
* 17-limit: 0.028721
* 19-limit: 0.023818
</div></div>

== Interval chain ==
{| class="wikitable center-1 right-2"
! Number of <br>generator
! Cents <br>value*
! Approximate Ratios
|-
| 0
| 0.000
| 1/1
|-
| 1
| 99.164
| [[18/17]], [[35/33]]
|-
| 2
| 198.328
| [[28/25]]
|-
| 3
| 297.491
| [[19/16]], [[25/21]]
|-
| 4
| 396.655
| [[34/27]]
|-
| 5
| 495.819
| [[4/3]]
|-
| 6
| 594.983
| [[24/17]]
|-
| 7
| 694.146
|
|-
| 8
| 793.310
| [[19/12]], [[30/19]]
|-
| 9
| 892.474
|
|-
| 10
| 991.638
| [[16/9]]
|-
| 11
| 1090.801
| [[15/8]], [[32/17]]
|-
| 12
| 1189.965
|
|-
| 13
| 89.129
| [[19/18]], [[20/19]], [[21/20]]
|-
| 14
| 188.293
| [[19/17]]
|-
| 15
| 287.457
| [[13/11]]
|-
| 16
| 386.620
| [[5/4]]
|-
| 17
| 485.784
|
|-
| 18
| 584.948
| [[7/5]]
|-
| 19
| 684.112
|
|-
| 20
| 783.275
| [[11/7]]
|-
| 21
| 882.439
| [[5/3]]
|-
| 22
| 981.603
| [[30/17]]
|-
| 23
| 1080.767
| [[28/15]]
|-
| 24
| 1179.930
|
|-
| 25
| 79.094
| [[22/21]]
|-
| 26
| 178.258
| [[10/9]], [[21/19]]
|-
| 27
| 277.422
| [[20/17]]
|-
| 28
| 376.586
|
|-
| 29
| 475.749
| [[21/16]]
|-
| 30
| 574.913
|
|-
| 31
| 674.077
| [[28/19]]
|-
| 32
| 773.241
|
|-
| 33
| 872.404
|
|-
| 34
| 971.568
| [[7/4]]
|-
| 35
| 1070.732
| [[13/7]]
|-
| 36
| 1169.896
|
|-
| 37
| 69.060
| [[25/24]], [[26/25]]
|-
| 38
| 168.223
| [[11/10]]
|-
| 39
| 267.387
| [[7/6]]
|-
| 40
| 366.551
| [[21/17]], [[26/21]]
|-
| 41
| 465.715
|
|-
| 42
| 564.878
|
|-
| 43
| 664.042
| [[22/15]]
|-
| 44
| 763.206
| [[14/9]]
|-
| 45
| 862.370
| [[28/17]]
|-
| 46
| 961.533
|
|-
| 47
| 1060.697
|
|-
| 48
| 1159.861
| [[39/20]]
|-
| 49
| 59.025
| [[33/32]]
|-
| 50
| 158.189
|
|-
| 51
| 257.352
| [[22/19]]
|-
| 52
| 356.516
|
|-
| 53
| 455.680
| [[13/10]]
|-
| 54
| 554.844
| [[11/8]]
|-
| 55
| 654.007
|
|-
| 56
| 753.171
|
|-
| 57
| 852.335
|
|-
| 58
| 951.499
| [[26/15]]
|-
| 59
| 1050.662
| [[11/6]]
|-
| 60
| 1149.826
| [[35/18]]
|-
| 61
| 48.990
|
|-
| 62
| 148.154
|
|-
| 63
| 247.318
|
|-
| 64
| 346.481
| [[11/9]]
|-
| 65
| 445.645
| [[22/17]]
|-
| 66
| 544.809
| [[26/19]]
|-
| 67
| 643.973
|
|-
| 68
| 743.136
|
|-
| 69
| 842.300
| [[13/8]]
|-
| 70
| 941.464
|
|-
| 71
| 1040.628
|
|-
| 72
| 1139.791
|
|-
| 73
| 38.955
|
|-
| 74
| 138.119
| [[13/12]]
|-
| 75
| 237.283
|
|-
| 76
| 336.447
| (close to [[17/14]])
|-
| 77
| 435.610
| (close to [[9/7]])
|-
| 78
| 534.774
|
|-
| 79
| 633.938
| [[13/9]]
|-
| 80
| 733.102
| [[26/17]]
|}
<nowiki>*</nowiki> in 19-limit POTE tuning

== Tuning spectrum ==
Gencom: [2 18/17; 190/189 256/255 352/351 361/360 364/363 375/374]

Gencom mapping: [{{val| 1 2 1 0 -1 -2 5 4 }}, {{val| 0 -5 16 34 54 69 -11 3 }}]

{| class="wikitable center-all"
|-
! [[Eigenmonzo|Eigenmonzo<br>(unchanged-interval)]]
! Generator<br>(¢)
! Comments
|-
| 21/20
| 98.8052
|
|-
| 19/15
| 98.8445
|
|-
| 16/15
| 98.9335
|
|-
| 18/17
| 98.9546
|
|-
| 21/19
| 98.9718
|
|-
| 21/16
| 98.9924
|
|-
| 7/5
| 99.0285
|
|-
| 19/14
| 99.0746
|
|-
| 11/10
| 99.0791
|
|-
| 8/7
| 99.0831
|
|-
| 22/19
| 99.0942
|
|-
| 11/8
| 99.0985
|
|-
| 14/11
| 99.1246
|
|-
| 13/10
| 99.1361
|
|-
| 26/19
| 99.1366
|
|-
| 16/13
| 99.1381
|
|-
| 20/19
| 99.1385
|
|-
| 15/11
| 99.1406
|
|-
| 12/11
| 99.1417
|
|-
| 5/4
| 99.1446
|
|-
| 21/17
| 99.1456
|
|-
| 7/6
| 99.1505
| 7 and 21-odd-limit minimax
|-
| 15/14
| 99.1547
|
|-
| 13/12
| 99.1699
|
|-
| 19/16
| 99.1710
|
|-
| 22/17
| 99.1748
|
|-
| 15/13
| 99.1769
|
|-
| 11/9
| 99.1782
| 11, 13, 15, 17 and 19-odd-limit minimax
|-
| 14/13
| 99.1915
|
|-
| 17/13
| 99.1947
|
|-
| 17/14
| 99.1971
|
|-
| 18/13
| 99.1977
|
|-
| 9/7
| 99.2026
| 9-odd-limit minimax
|-
| 22/21
| 99.2215
|
|-
| 17/15
| 99.2415
|
|-
| 26/21
| 99.2437
|
|-
| 6/5
| 99.2552
| 5-odd-limit minimax
|-
| 13/11
| 99.2807
|
|-
| 20/17
| 99.3096
|
|-
| 10/9
| 99.3232
|
|-
| 24/19
| 99.4448
|
|-
| 19/17
| 99.4684
|
|-
| 24/17
| 99.4999
|
|-
| 19/18
| 99.5079
|
|-
| 17/16
| 99.5495
|
|-
| 4/3
| 99.6090
|
|}

[[Category:Temperaments]]
[[Category:Quintaleap family]]
[[Category:Octagar temperaments]]
[[Category:Mistismic temperaments]]
[[Category:Quintupole]]

Quintupole - Revision history

Lériendil: Text replacement - "Eigenmonzo(unchanged-interval)" to "Unchanged interval(eigenmonzo)"

FloraC: - parent category

Lériendil at 23:40, 19 April 2025

Xenllium at 11:03, 18 June 2023

Xenllium: Created page with "'''Quintupole''' is a temperament for the 7, 11, 13, 17, and 19 prime limits. It is a member of quintaleap family, octagar temperaments, and misti..."

Lériendil: Text replacement - "Eigenmonzo
(unchanged-interval)" to "Unchanged interval
(eigenmonzo)"