Pajara - Revision history

Overthink: 7-limit simplicity

2026-06-02T16:28:42Z

7-limit simplicity

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	'''Pajara''', 10 & 12, is a regular temperament wherein the octave is split into two tritone periods, and the generator is a fifth (3/2). A fifth minus a tritone is 16/15 ([[Diaschismic]] tempering), and therefore the 5/4 major third is found two generators below the tritone. Pajara makes the further equivalence of 5/4 plus a period to 7/4 ([[Jubilismic]] tempering) and therefore twice 4/3 is 7/4 ([[Archy]] tempering). The result is a 10-form system generated by a fifth tuned somewhere around 710 cents.		'''Pajara''', 10 & 12, is a regular temperament wherein the octave is split into two tritone periods, and the generator is a fifth (3/2). A fifth minus a tritone is 16/15 ([[Diaschismic]] tempering), and therefore the 5/4 major third is found two generators below the tritone. Pajara makes the further equivalence of 5/4 plus a period to 7/4 ([[Jubilismic]] tempering) and therefore twice 4/3 is 7/4 ([[Archy]] tempering). The result is a 10-form system generated by a fifth tuned somewhere around 710 cents.

	There are five [[patent]] tunings of Pajara: [[12edo\|12]], [[22edo\|22]], [[54edo\|54]], [[32edo\|32]], and [[10edo\|10]] (which is also the [[20edo]] [[val]] for the [[7-limit]]); of these 22, 54, and 32 are considered non-trivial, and 22edo is the generally assumed "canonical" tuning. As regards non-patent vals, [[34edo\|34d]] (using [[17edo]]'s mapping of 7) is commonly considered as well.		There are five [[patent]] tunings of Pajara: [[12edo\|12]], [[22edo\|22]], [[54edo\|54]], [[32edo\|32]], and [[10edo\|10]] (which is also the [[20edo]] [[val]] for the [[7-limit]]); of these 22, 54, and 32 are considered non-trivial, and 22edo is the generally assumed "canonical" tuning. As regards non-patent vals, [[34edo\|34d]] (using [[17edo]]'s mapping of 7) and [[56edo\|56d]] are commonly considered as well.

			Pajara is arguably the most accurate temperament that contains full 7-limit tetrads in small (5—10 note) [[MOS]] scales.

	== Extensions ==		== Extensions ==

Lériendil: tried to make coverage a little more balanced

2026-05-19T03:15:18Z

tried to make coverage a little more balanced

← Older revision		Revision as of 03:15, 19 May 2026
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	'''Pajara''', 22 & 32, is a regular temperament wherein the octave is split into two tritone periods, and the generator is a fifth (3/2). A fifth minus a tritone is 16/15 ([[Diaschismic]] tempering), and therefore the 5/4 major third is found two generators below the tritone. Pajara makes the further equivalence of 5/4 plus a period to 7/4 ([[Jubilismic]] tempering) and therefore twice 4/3 is 7/4 ([[Archy]] tempering). The result is a 10-form system generated by a fifth tuned somewhere around 710 cents. There are five patent tunings of Pajara: 12, 22, 54, 32, and 10 (which is also the 20edo val for the 7-limit); of these 22, 54, and 32 are considered ~~"reasonable"~~ and 22edo is the generally assumed tuning.		'''Pajara''', 10 & 12, is a regular temperament wherein the octave is split into two tritone periods, and the generator is a fifth (3/2). A fifth minus a tritone is 16/15 ([[Diaschismic]] tempering), and therefore the 5/4 major third is found two generators below the tritone. Pajara makes the further equivalence of 5/4 plus a period to 7/4 ([[Jubilismic]] tempering) and therefore twice 4/3 is 7/4 ([[Archy]] tempering). The result is a 10-form system generated by a fifth tuned somewhere around 710 cents.

			There are five [[patent]] tunings of Pajara: [[12edo\|12]], [[22edo\|22]], [[54edo\|54]], [[32edo\|32]], and [[10edo\|10]] (which is also the [[20edo]] [[val]] for the [[7-limit]]); of these 22, 54, and 32 are considered non-trivial, and 22edo is the generally assumed "canonical" tuning. As regards non-patent vals, [[34edo\|34d]] (using [[17edo]]'s mapping of 7) is commonly considered as well.

	== Extensions ==		== Extensions ==
	There are two main extensions of Pajara to the 11-limit: Pajarous (10 & 22) and ~~a temperament called "~~Undecimal Pajara" (12 & 22, which is supported by only those two patent vals) ~~which is pending a rename~~. Undecimal Pajara is best flat of 22edo; Pajarous is best sharp of 22edo.		There are two main extensions of Pajara to the [[11-limit]]: Pajarous (10 & 22) and Undecimal Pajara (12 & 22, which is supported by only those two patent vals). Undecimal Pajara is best flat of 22edo; Pajarous is best sharp of 22edo. Note that although Pajarous is supported patently by 32 and 54edo, no extension of Pajara can be [[monotonic]] in the [[11-odd-limit]] sharp of 22edo, which calls into question the utility of involving prime 11 in Pajara at all.

			As with all Diaschismic temperaments, the generating semitone can be identified as 17/16~18/17 and prime 17 hence comes for free.

	== Tuning considerations ==		== Tuning considerations ==
	Most optimization methods place the optimal tuning of Pajara's perfect fifth at around 707 cents (hence the focus on 12 & 22 as an extension). However, Pajara ~~is usually not the best interpretation of~~ those structures. EDOs with tunings of fifths flat of that of 22edo do not support Pajara in the patent val (except for 12edo), and 12edo ~~alongside rank-2 Pajara structures with that tuning are~~ generally extremely inaccurate in the 7-limit due to the fact that Archy temperament forces 9/8 and 8/7 together.		Most optimization methods place the optimal tuning of Pajara's perfect fifth at around 707 cents (hence the focus on 12 & 22 as an extension). However, interpretations more accurate than Pajara exist for those structures (although they may well be more complex). EDOs with tunings of fifths flat of that of 22edo do not support Pajara in the patent val (except for 12edo), and 12edo is generally extremely inaccurate in the 7-limit due to the fact that Archy temperament forces 9/8 and 8/7 together.

	Conversely, systems with fifths sharp of 32edo narrow the distinction between 5/4 and 6/5 considerably, detuning 6/5 to a neutral sound. Therefore, the tuning range of Pajara can be considered to lie between about 709 to 712.5 cents.		Conversely, systems with fifths sharp of 32edo narrow the distinction between 5/4 and 6/5 considerably, detuning 6/5 to a neutral sound. Therefore, the tuning range of Pajara can be considered to lie between about 709 to 712.5 cents.

Vector: ts is not pajara :sob:

2026-05-18T08:04:30Z

ts is not pajara :sob:

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	Additionally, there is a MODMOS sssLsssssL, which is considered the "pentachordal" pajara MOS due to being constructed from an sssL [[Tetrachord#Pentachord\|pentachord]].		Additionally, there is a MODMOS sssLsssssL, which is considered the "pentachordal" pajara MOS due to being constructed from an sssL [[Tetrachord#Pentachord\|pentachord]].

	To extend to the 11-limit, Pajara[12] (sLLLLLsLLLLL) can be easily used, which has the same number of notes as 12edo's chromatic scale, but with two of the semitones from 12edo replaced with quartertones. This gives major and minor thirds separate interval categories (allowing both to be played on certain scale degrees), and the 11th harmonic can be found as the major 5-step. The MODMOS [LLLsLLLLsLLL] of Pajara[12] is the Tellurian scale; it may be derived by splitting a pentatonic scale ([LL][LsL][LL][LsL][LL]), or by splitting each whole tone of MOS diatonic ([LL][LL]s~~[LL]~~[LL][LL][LL]s), which makes for a more xenharmonic way of generalizing 12edo music than simply retuning the chain of fifths. ("When the 12edo goes chromatic, equally divide the whole tone!")		To extend to the 11-limit, Pajara[12] (sLLLLLsLLLLL) can be easily used, which has the same number of notes as 12edo's chromatic scale, but with two of the semitones from 12edo replaced with quartertones. This gives major and minor thirds separate interval categories (allowing both to be played on certain scale degrees), and the 11th harmonic can be found as the major 5-step. The MODMOS [LLLsLLLLsLLL] of Pajara[12] is the Tellurian scale; it may be derived by splitting a pentatonic scale ([LL][LsL][LL][LsL][LL]), or by splitting each whole tone of MOS diatonic ([LL][LL]s[LL][LL][LL]s), which makes for a more xenharmonic way of generalizing 12edo music than simply retuning the chain of fifths. ("When the 12edo goes chromatic, equally divide the whole tone!")

	==== Degrees of pajara[10] ====		==== Degrees of pajara[10] ====

Vector at 08:02, 18 May 2026

2026-05-18T08:02:42Z

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	'''Pajara''', 22 & 32, is a regular temperament wherein the octave is split into two tritone periods, and the generator is a fifth (3/2). A fifth minus a tritone is 16/15 ([[Diaschismic]] tempering), and therefore the 5/4 major third is found two generators below the tritone. Pajara makes the further equivalence of 5/4 plus a period to 7/4 ([[Jubilismic]] tempering) and therefore twice 4/3 is 7/4 ([[Archy]] tempering). The result is a 10-form system generated by a fifth tuned somewhere around 710 cents. There are five patent tunings of Pajara: 12, 22, 54, 32, and 10 (which is also the 20edo val for the 7-limit).		'''Pajara''', 22 & 32, is a regular temperament wherein the octave is split into two tritone periods, and the generator is a fifth (3/2). A fifth minus a tritone is 16/15 ([[Diaschismic]] tempering), and therefore the 5/4 major third is found two generators below the tritone. Pajara makes the further equivalence of 5/4 plus a period to 7/4 ([[Jubilismic]] tempering) and therefore twice 4/3 is 7/4 ([[Archy]] tempering). The result is a 10-form system generated by a fifth tuned somewhere around 710 cents. There are five patent tunings of Pajara: 12, 22, 54, 32, and 10 (which is also the 20edo val for the 7-limit); of these 22, 54, and 32 are considered "reasonable" and 22edo is the generally assumed tuning.

	== Extensions ==		== Extensions ==

Vector at 21:24, 16 May 2026

2026-05-16T21:24:08Z

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	- the fifth is divided into two thirds		- the fifth is divided into two thirds

			Additionally, there is a MODMOS sssLsssssL, which is considered the "pentachordal" pajara MOS due to being constructed from an sssL [[Tetrachord#Pentachord\|pentachord]].

	To extend to the 11-limit, Pajara[12] (sLLLLLsLLLLL) can be easily used, which has the same number of notes as 12edo's chromatic scale, but with two of the semitones from 12edo replaced with quartertones. This gives major and minor thirds separate interval categories (allowing both to be played on certain scale degrees), and the 11th harmonic can be found as the major 5-step. The MODMOS [LLLsLLLLsLLL] of Pajara[12] is the Tellurian scale; it may be derived by splitting a pentatonic scale ([LL][LsL][LL][LsL][LL]), or by splitting each whole tone of MOS diatonic ([LL][LL]s[LL][LL][LL][LL]s), which makes for a more xenharmonic way of generalizing 12edo music than simply retuning the chain of fifths. ("When the 12edo goes chromatic, equally divide the whole tone!")		To extend to the 11-limit, Pajara[12] (sLLLLLsLLLLL) can be easily used, which has the same number of notes as 12edo's chromatic scale, but with two of the semitones from 12edo replaced with quartertones. This gives major and minor thirds separate interval categories (allowing both to be played on certain scale degrees), and the 11th harmonic can be found as the major 5-step. The MODMOS [LLLsLLLLsLLL] of Pajara[12] is the Tellurian scale; it may be derived by splitting a pentatonic scale ([LL][LsL][LL][LsL][LL]), or by splitting each whole tone of MOS diatonic ([LL][LL]s[LL][LL][LL][LL]s), which makes for a more xenharmonic way of generalizing 12edo music than simply retuning the chain of fifths. ("When the 12edo goes chromatic, equally divide the whole tone!")

Vector at 20:55, 16 May 2026

2026-05-16T20:55:19Z

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	\| -		\| -
	\|}		\|}

			==== Notation ====
			Both degrees and notes in 10-form pajara should be notated with 0-indexed numerals in text: the tonic is always 0, and absolute pitch should be specified in relation to standard diatonic notation. 0-indexing is used so that systems such as figured bass that depend on numerals being a single symbol each still work (if 1-indexing was used, the number 10 would indicate a degree). Additionally, Roman numeral analysis in this case would use N for zero.

			As for notating the 10-form on the staff, there are a few different approaches. The first adds an extra line to each staff so that an octave can span 11 staff positions, but comes at the cost of losing intuition for people used to reading intervals from standard notation. The second uses the mosdiatonic notation for pentic, but uses an extra symbol to mark an alteration by a 109c semitone, allowing the full range of notes in pajara to be provided at the cost of potential overloading on symbols as opposed to visual distance to denote pitch. Meanwhile, the third option is simply to notate it starting from mosdiatonic as a base, with ups and downs notation.

	==== 10-tone functional harmony ====		==== 10-tone functional harmony ====

Vector at 20:43, 16 May 2026

2026-05-16T20:43:10Z

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	==== 10-tone functional harmony ====		==== 10-tone functional harmony ====
	The antilatus and unilatus become the varicant and subvaricant, which sit between the mediant/submediant and supertonic/subtonic in terms of stability (and feature as elements of chthonic chords like 6:7:8, an alternative to standard diatonic chords available in the 10-form). Additionally, the tritone acquires the antitonic function. While the dominant serves as a stable "structural anchor" in diatonic, here the antitonic serves as an ''unstable'' structural anchor - the opposite of the tonic both in placement and stability. Note that in the 10-tone system, we return to having two distinct interval qualities down from four, so we go back to having two different keys. 10-tone harmony is also useful in modal music. Also, note that ~~109~~-cent leading tones comprise the majority of the intervals in pajara[10], so their impact may be reduced and in fact one might depend more on the few larger nearmajor seconds that exist in the scale, or skip steps entirely and use subsets.		The antilatus and unilatus become the varicant and subvaricant, which sit between the mediant/submediant and supertonic/subtonic in terms of stability (and feature as elements of chthonic chords like 6:7:8, an alternative to standard diatonic chords available in the 10-form). Additionally, the tritone acquires the antitonic function. While the dominant serves as a stable "structural anchor" in diatonic, here the antitonic serves as an ''unstable'' structural anchor - the opposite of the tonic both in placement and stability. Note that in the 10-tone system, we return to having two distinct interval qualities down from four, so we go back to having two different keys. 10-tone harmony is also useful in modal music. Also, note that ~100-cent leading tones comprise the majority of the intervals in pajara[10], so their impact may be reduced and in fact one might depend more on the few larger nearmajor seconds that exist in the scale, or skip steps entirely and use subsets.
	[[File:Chthonic_harmony_demonstration.mp3\|thumb\|10-form harmony demonstration in 22edo]]		[[File:Chthonic_harmony_demonstration.mp3\|thumb\|10-form harmony demonstration in 22edo]]
	Another thing to note about the 10-tone system is that it is possible to constrain oneself entirely to chthonic harmony, in which case a lot of the familiar functional harmony language somewhat breaks. The role of the traditional dominant with respect to the tonic disappears completely (even if, for instance, the root position of a chord is assumed to be 4:6:7), with instead the chords on the tritone and the sixth including a leading tone up to the tonic (in fact, the dominant in this system becomes a ''stable'' chord rather than a tense one, assuming a 6:7:8 root position).		Another thing to note about the 10-tone system is that it is possible to constrain oneself entirely to chthonic harmony, in which case a lot of the familiar functional harmony language somewhat breaks. The role of the traditional dominant with respect to the tonic disappears completely (even if, for instance, the root position of a chord is assumed to be 4:6:7), with instead the chords on the tritone and the sixth including a leading tone up to the tonic (in fact, the dominant in this system becomes a ''stable'' chord rather than a tense one, assuming a 6:7:8 root position).

Vector at 19:51, 16 May 2026

2026-05-16T19:51:35Z

Show changes

Inthar: /* List of patent vals */

2026-04-21T01:34:55Z

List of patent vals

← Older revision		Revision as of 01:34, 21 April 2026
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	{\| class="wikitable"		{\| class="wikitable"
	!EDO		!EDO
	!Extension		!Extension to 11
	!Generator tuning		!Generator tuning
	!7/4 tuning		!7/4 tuning

Inthar: /* Extensions */

2026-04-20T15:37:02Z

Extensions

← Older revision		Revision as of 15:37, 20 April 2026
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	== Extensions ==		== Extensions ==
	There are two main extensions of Pajara to the 11-limit: ~~pajarous~~ (10 & 22) and a temperament called "~~undecimal pajara~~" (12 & 22, which is supported by only those two patent vals) which is pending a rename. Undecimal ~~pajara~~ is best flat of 22edo; ~~pajarous~~ is best sharp of 22edo.		There are two main extensions of Pajara to the 11-limit: Pajarous (10 & 22) and a temperament called "Undecimal Pajara" (12 & 22, which is supported by only those two patent vals) which is pending a rename. Undecimal Pajara is best flat of 22edo; Pajarous is best sharp of 22edo.

	== Tuning considerations ==		== Tuning considerations ==