Pajara - Revision history

Overthink: /* Interval chains */ oops

2026-06-05T18:02:27Z

Interval chains: oops

← Older revision		Revision as of 18:02, 5 June 2026
Line 141:		Line 141:
	\| 22/21, 33/32, 81/80		\| 22/21, 33/32, 81/80
	\|}		\|}
	<nowiki/>* In 2.3.5.7.17-subgroup CWE tuning, octave-reduced		<nowiki/>* In 2.3.5.7.11.17-subgroup CWE tuning, octave-reduced

	== Chords and harmony ==		== Chords and harmony ==

Overthink: /* Interval chains */ a few missing ones

2026-06-05T18:02:10Z

Interval chains: a few missing ones

← Older revision		Revision as of 18:02, 5 June 2026
Line 103:		Line 103:
	\| '''1/1'''		\| '''1/1'''
	\| 600.0		\| 600.0
	\| 7/5, 10/7		\| 7/5, 10/7, 17/12, 24/17
	\|-		\|-
	\| 1		\| 1
Line 125:		Line 125:
	\| 4		\| 4
	\| 438.2		\| 438.2
	\| 9/7		\| 9/7, 22/17
	\| 1038.2		\| 1038.2
	\| 9/5, 11/6		\| 9/5, 11/6

Overthink: /* Interval chains */ add intervals of 17

2026-06-05T17:59:36Z

Interval chains: add intervals of 17

← Older revision		Revision as of 17:59, 5 June 2026
Line 30:		Line 30:
	There are two different mappings of the 11-limit. One is just called ''pajara'' and is slightly more complex but suffers almost no loss of accuracy compared to the 7-limit. It is best tuned flat of 22edo, with the optimum at around 707-708 cents. The other, called ''pajarous'' to avoid confusion, maps the 11th harmonic slightly simpler, but it equates [[12/11]] with [[10/9]], and the only tuning equating [[11/10]] with both is 22edo.		There are two different mappings of the 11-limit. One is just called ''pajara'' and is slightly more complex but suffers almost no loss of accuracy compared to the 7-limit. It is best tuned flat of 22edo, with the optimum at around 707-708 cents. The other, called ''pajarous'' to avoid confusion, maps the 11th harmonic slightly simpler, but it equates [[12/11]] with [[10/9]], and the only tuning equating [[11/10]] with both is 22edo.

	In the following tables, odd harmonics ~~1–11~~ and their inverses are in '''bold'''.		In the following tables, odd harmonics 1–21 and their inverses are in '''bold'''.

	{\| class="wikitable center-1 right-2 right-4"		{\| class="wikitable center-1 right-2 right-4"
Line 48:		Line 48:
	\| '''1/1'''		\| '''1/1'''
	\| 600.0		\| 600.0
	\| 7/5, 10/7		\| 7/5, 10/7, 17/12, 24/17
	\|-		\|-
	\| 1		\| 1
	\| 707.2		\| 707.4
	\| '''3/2'''		\| '''3/2''', '''32/21'''
	\| 107.2		\| 107.4
	\| 15/14, 16/15, 21/20		\| 15/14, '''16/15''', '''17/16''',<br>18/17, 21/20
	\|-		\|-
	\| 2		\| 2
	\| 214.4		\| 214.7
	\| '''8/7''', '''9/8'''		\| '''8/7''', '''9/8''', 17/15
	\| 814.4		\| 814.7
	\| '''8/5'''		\| '''8/5''', 34/21
	\|-		\|-
	\| 3		\| 3
	\| ~~921~~.5		\| 922.1
	\| 12/7		\| 12/7, 17/10
	\| ~~321~~.5		\| 322.1
	\| 6/5		\| 6/5, 17/14
	\|-		\|-
	\| 4		\| 4
	\| ~~428~~.7		\| 429.5
	\| 9/7, 14/11		\| 9/7, 14/11
	\| ~~1028~~.7		\| 1029.5
	\| 9/5, 20/11		\| 9/5, 20/11
	\|-		\|-
	\| 5		\| 5
	\| ~~1135~~.9		\| 1136.9
	\| 21/11, 27/14, 48/25, <br>64/33, 96/49		\| 21/11, 27/14, 48/25, <br>64/33, 96/49
	\| ~~535~~.9		\| 536.9
	\| 15/11, 27/20		\| 15/11, 27/20
	\|-		\|-
	\| 6		\| 6
	\| ~~643~~.1		\| 644.2
	\| '''16/11'''		\| '''16/11''', 36/25, 72/49
	\| 43.1		\| 44.2
	\| 45/44, 56/55, 81/80		\| 45/44, 56/55, 81/80
	\|}		\|}
Line 106:		Line 106:
	\|-		\|-
	\| 1		\| 1
	\| 709.6		\| 709.5
	\| '''3/2'''		\| '''3/2''', '''32/21'''
	\| 109.6		\| 109.5
	\| 15/14, 16/15, 21/20		\| 15/14, '''16/15''', '''17/16''',<br>18/17, 21/20
	\|-		\|-
	\| 2		\| 2
	\| 219.1		\| 219.1
	\| '''8/7''', '''9/8'''		\| '''8/7''', '''9/8''', 17/15
	\| 819.1		\| 819.1
	\| '''8/5'''		\| '''8/5''', 34/21
	\|-		\|-
	\| 3		\| 3
	\| 928.7		\| 928.6
	\| 12/7		\| 12/7, 17/10
	\| 328.7		\| 328.6
	\| 6/5, 11/9		\| 6/5, 11/9, 17/14
	\|-		\|-
	\| 4		\| 4
Line 130:		Line 130:
	\|-		\|-
	\| 5		\| 5
	\| 1147.8		\| 1147.7
	\| 27/14, 48/25, 55/28, <br>88/45, 96/49		\| 27/14, 48/25, 55/28, <br>88/45, 96/49
	\| 547.8		\| 547.7
	\| '''11/8''', 27/20		\| '''11/8''', 27/20
	\|-		\|-
Line 141:		Line 141:
	\| 22/21, 33/32, 81/80		\| 22/21, 33/32, 81/80
	\|}		\|}
	<nowiki/>* In 11-~~limit~~ CWE tuning, octave-reduced		<nowiki/>* In 2.3.5.7.17-subgroup CWE tuning, octave-reduced

	== Chords and harmony ==		== Chords and harmony ==

Roeesi: /* Tunings */ Added the minimax tunings according to the power limit method proposed by Dave & Doug and independently by myself.

2026-04-14T17:44:35Z

Tunings: Added the minimax tunings according to the power limit method proposed by Dave & Doug and independently by myself.

← Older revision		Revision as of 17:44, 14 April 2026
Line 314:		Line 314:
	\| 11/8		\| 11/8
	\| 708.114		\| 708.114
	\|		\| 11- and 15-odd-limit minimax
			\|-
			\|
			\|36/35
			\|708.128
			\|9-odd-limit minimax
	\|-		\|-
	\|		\|
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	\| 709.091		\| 709.091
	\| Upper bound of 11-odd-limit diamond monotone		\| Upper bound of 11-odd-limit diamond monotone
			\|-
			\|
			\|48/35
			\|709.363
			\|7-odd-limit minimax
	\|-		\|-
	\|		\|

Overthink: /* Scala files */ add file

2026-04-07T18:15:35Z

Scala files: add file

← Older revision		Revision as of 18:15, 7 April 2026
Line 162:		Line 162:

	=== Scala files ===		=== Scala files ===
			* [[Pajara12]]
	* [[12-22h]]		* [[12-22h]]

Overthink: /* Interval chains */ explain why

2026-04-07T18:11:32Z

Interval chains: explain why

← Older revision		Revision as of 18:11, 7 April 2026
Line 28:		Line 28:

	== Interval chains ==		== Interval chains ==
	There are two different mappings of the 11-limit. One is just called ''pajara'' and is slightly more complex but suffers almost no loss of accuracy compared to the 7-limit. It is best tuned flat of 22edo. The other, called ''pajarous'' to avoid confusion, maps the 11th harmonic slightly simpler, but ~~22edo is the only~~ [[11~~-odd-limit~~]] [[~~diamond monotone~~]] ~~tuning~~, ~~where primes [[3/1\|3]]~~ and [[5/~~1\|5~~]] ~~are less accurate than in optimal tunings of canonical 11-limit pajara~~.		There are two different mappings of the 11-limit. One is just called ''pajara'' and is slightly more complex but suffers almost no loss of accuracy compared to the 7-limit. It is best tuned flat of 22edo, with the optimum at around 707-708 cents. The other, called ''pajarous'' to avoid confusion, maps the 11th harmonic slightly simpler, but it equates [[12/11]] with [[10/9]], and the only tuning equating [[11/10]] with both is 22edo.

	In the following tables, odd harmonics 1–11 and their inverses are in '''bold'''.		In the following tables, odd harmonics 1–11 and their inverses are in '''bold'''.

FloraC: /* Tunings */ review minimax tuning

2026-03-19T17:51:11Z

Tunings: review minimax tuning

@@ Line 183: / Line 183: @@
 | POTE: ~3/2 = 707.0477{{c}}
 |}
 {| class="wikitable mw-collapsible mw-collapsed"
 |+ style="font-size: 105%; white-space: nowrap;" | 11-limit norm-based tunings
@@ Line 198: / Line 197: @@
 | CWE: ~3/2 = 707.1826{{c}}
 | POTE: ~3/2 = 706.8851{{c}}
 |}
@@ Line 267: / Line 289: @@
 | 706.843
 | 7- and 11-limit POTT
 |-
 | 33\56
@@ Line 277: / Line 294: @@
 | 707.143
 | 56d val
 |-
+|
@@ Line 287: / Line 299: @@
 | 707.234
+|
 |-
+|
@@ Line 317: / Line 324: @@
 | 708.771
+|
 |-
 | 13\22

Overthink: /* Chords and harmony */ add some text

2026-03-16T19:34:34Z

Chords and harmony: add some text

← Older revision		Revision as of 19:34, 16 March 2026
Line 145:		Line 145:
	== Chords and harmony ==		== Chords and harmony ==
	{{See also\| Chords of pajara }}		{{See also\| Chords of pajara }}

			In pajara, a decatonic system of interval classification based on the [[2L 8s]] (jaric) [[mos scale]] is preferred over the [[diatonic]] interval classification system traditionally used in western music, which is used in [[meantone]]. If we count scale degrees similarly to diatonic, then [[2/1]] is a "hendecave" (11ve), as there are 10 scale degrees, and we repeat at 2/1 at the 11th. In this system, [[3/2]] is a perfect 7th, and [[4/3]] is a perfect 5th. The intervals [[5/4]] and [[6/5]] are major and minor decatonic 4ths respectively, rather than being major and minor 3rds by diatonic interval classification in meantone. Importantly, [[7/4]] is now a major decatonic 9th, with [[12/7]] being its minor counterpart. This is in contrast to diatonic, where 7/4 is considered a subminor 7th, and 12/7 a supermajor 6th.

			By decatonic interval classification, the [[4:5:6:7]] tetrad is written as P1–M4–P7–M9. It can be considered the ''major tetrad'', since the non-perfect intervals, those being the decatonic 4th and 9th, are both major intervals. If we instead use a minor interval for the 4th and 9th; that is, a P1–m4–P7–m9 chord, then we get a tetrad approximating [[70:84:105:120\|1/(12:10:8:7)]], which can be considered the ''minor tetrad''.

			{{Todo\|complete section}}

	== Scales ==		== Scales ==

FloraC: + link

2026-03-03T07:48:55Z

+ link

← Older revision		Revision as of 07:48, 3 March 2026
Line 142:		Line 142:
	\|}		\|}
	<nowiki/>* In 11-limit CWE tuning, octave-reduced		<nowiki/>* In 11-limit CWE tuning, octave-reduced

			== Chords and harmony ==
			{{See also\| Chords of pajara }}

	== Scales ==		== Scales ==

Roeesi: reduction by 2

2026-03-01T09:17:41Z

reduction by 2

← Older revision		Revision as of 09:17, 1 March 2026
Line 260:		Line 260:
	\|-		\|-
	\|		\|
	\|{{Monzo\|~~94 100~~ -10 -10 -58}}		\|{{Monzo\|37 50 -5 -5 -29}}
	\|707.106		\|707.106
	\|11-odd-limit minimax		\|11-odd-limit minimax
Line 270:		Line 270:
	\|-		\|-
	\|		\|
	\|{{Monzo\|~~95 110~~ -12 -68}}		\|{{Monzo\|48 55 0 -6 -34}}
	\|707.195		\|707.195
	\|15-odd-limit minimax		\|15-odd-limit minimax