Non-over-1 temperament - Revision history

Overthink: /* Sensi */ power

2026-04-24T00:47:28Z

Sensi: power

← Older revision		Revision as of 00:47, 24 April 2026
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	=== Sensi ===		=== Sensi ===
	{{See also\|Sensipent family #Sensi\|Chromatic pairs #Sensi}}		{{See also\|Sensipent family #Sensi\|Chromatic pairs #Sensi}}
	[[Sensi]] is ''effectively'' a non-over-1 temperament provided you restrict yourself to the sensi[8] MOS. The sensi[8] MOS only has a 5:6:7:9:13 chord, but no chord of the form 2:m<sub>1</sub>:...:m<sub>k</sub> (except 2:3). Thus sensi can be viewed as a 2.6/5.7/5.9/5.13/10 or 2.3.6/5.7/5.13/10 temperament. (See [http://x31eq.com/cgi-bin/rt.cgi?limit=2_6%2F5_7%2F5_9%2F5_13%2F10&ets=19_27&tuning=po&subgroup=on x31eq data page].)		[[Sensi]] is ''effectively'' a non-over-1 temperament provided you restrict yourself to the sensi[8] MOS. The sensi[8] MOS only has a 5:6:7:9:13 chord, but no chord of the form 2<sup>n</sup>:m<sub>1</sub>:...:m<sub>k</sub> (except 2:3). Thus sensi can be viewed as a 2.6/5.7/5.9/5.13/10 or 2.3.6/5.7/5.13/10 temperament. (See [http://x31eq.com/cgi-bin/rt.cgi?limit=2_6%2F5_7%2F5_9%2F5_13%2F10&ets=19_27&tuning=po&subgroup=on x31eq data page].)

	{\| class="wikitable"		{\| class="wikitable"

Inthar at 20:25, 23 April 2026

2026-04-23T20:25:15Z

← Older revision		Revision as of 20:25, 23 April 2026
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	A '''non-over-1 temperament''' is a [[regular temperament]] that tempers a [[subgroup]] corresponding to a harmonic series chord r:n<sub>1</sub>:...:n<sub>k</sub> where r ≠ 1, but is not meant to approximate a chord of the form 1:m<sub>1</sub>:...:m<sub>k</sub>. Assuming octave equivalence, this means a subgroup of the form 2.k<sub>1</sub>/f.[...].k<sub>r</sub>/f, where f:k<sub>1</sub>:...:k<sub>r</sub> is a JI chord none of whose notes is a power of 2.		A '''non-over-1 temperament''' is a [[regular temperament]] that tempers a [[subgroup]] corresponding to a harmonic series chord r:n<sub>1</sub>:...:n<sub>k</sub> where r ≠ 1, but is not meant to approximate a chord of the form 1:m<sub>1</sub>:...:m<sub>k</sub>. Assuming octave equivalence, this means a subgroup of the form 2.m<sub>1</sub>/f.[...].m<sub>r</sub>/f, where f:m<sub>1</sub>:...:m<sub>r</sub> is a JI chord none of whose notes is a power of 2.

	Non-over-1 temperaments give regular-temperament interpretations to edos that approximate [[Overtone scales\|over-1]] chords (assuming [[octave equivalence]]) such as [[4:5:6:7:11]] poorly, such as [[14edo]], [[18edo]], [[23edo]] and [[29edo]], thus may have much xenharmonic potential. Many of these temperaments have an [[octatonic]] structure, as [[8edo]] represents non-over-1 intervals well for its size as far as the [[17-limit]].		Non-over-1 temperaments give regular-temperament interpretations to edos that approximate [[Overtone scales\|over-1]] chords (assuming [[octave equivalence]]) such as [[4:5:6:7:11]] poorly, such as [[14edo]], [[18edo]], [[23edo]] and [[29edo]], thus may have much xenharmonic potential. Many of these temperaments have an [[octatonic]] structure, as [[8edo]] represents non-over-1 intervals well for its size as far as the [[17-limit]].

Inthar at 20:24, 23 April 2026

2026-04-23T20:24:46Z

← Older revision		Revision as of 20:24, 23 April 2026
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	A '''non-over-1 temperament''' is a [[regular temperament]] that tempers a [[subgroup]] corresponding to a harmonic series chord r:n<sub>1</sub>:...:n<sub>k</sub> where r ≠ 1, but is not meant to approximate a chord of the form 1:m<sub>1</sub>:...:m<sub>k</sub>. Assuming octave equivalence, this means a subgroup of the form 2.k<sub>1</sub>/f.[...].k<sub>1</sub>/f, where f:k<sub>1</sub>:...:k<sub>r</sub> is a JI chord none of whose notes is a power of 2.		A '''non-over-1 temperament''' is a [[regular temperament]] that tempers a [[subgroup]] corresponding to a harmonic series chord r:n<sub>1</sub>:...:n<sub>k</sub> where r ≠ 1, but is not meant to approximate a chord of the form 1:m<sub>1</sub>:...:m<sub>k</sub>. Assuming octave equivalence, this means a subgroup of the form 2.k<sub>1</sub>/f.[...].k<sub>r</sub>/f, where f:k<sub>1</sub>:...:k<sub>r</sub> is a JI chord none of whose notes is a power of 2.

	Non-over-1 temperaments give regular-temperament interpretations to edos that approximate [[Overtone scales\|over-1]] chords (assuming [[octave equivalence]]) such as [[4:5:6:7:11]] poorly, such as [[14edo]], [[18edo]], [[23edo]] and [[29edo]], thus may have much xenharmonic potential. Many of these temperaments have an [[octatonic]] structure, as [[8edo]] represents non-over-1 intervals well for its size as far as the [[17-limit]].		Non-over-1 temperaments give regular-temperament interpretations to edos that approximate [[Overtone scales\|over-1]] chords (assuming [[octave equivalence]]) such as [[4:5:6:7:11]] poorly, such as [[14edo]], [[18edo]], [[23edo]] and [[29edo]], thus may have much xenharmonic potential. Many of these temperaments have an [[octatonic]] structure, as [[8edo]] represents non-over-1 intervals well for its size as far as the [[17-limit]].

Inthar at 20:04, 23 April 2026

2026-04-23T20:04:48Z

← Older revision		Revision as of 20:04, 23 April 2026
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	A '''non-over-1 temperament''' is a [[regular temperament]] that tempers a [[subgroup]] corresponding to a harmonic series chord r:n<sub>1</sub>:...:n<sub>k</sub> where r ≠ 1, but is not meant to approximate a chord of the form 1:m<sub>1</sub>:...:m<sub>k</sub>. Non-over-1 temperaments give regular-temperament interpretations to edos that approximate [[Overtone scales\|over-1]] chords (assuming [[octave equivalence]]) such as [[4:5:6:7:11]] poorly, such as [[14edo]], [[18edo]], [[23edo]] and [[29edo]], thus may have much xenharmonic potential. Many of these temperaments have an [[octatonic]] structure, as [[8edo]] represents non-over-1 intervals well for its size as far as the [[17-limit]].		A '''non-over-1 temperament''' is a [[regular temperament]] that tempers a [[subgroup]] corresponding to a harmonic series chord r:n<sub>1</sub>:...:n<sub>k</sub> where r ≠ 1, but is not meant to approximate a chord of the form 1:m<sub>1</sub>:...:m<sub>k</sub>. Assuming octave equivalence, this means a subgroup of the form 2.k<sub>1</sub>/f.[...].k<sub>1</sub>/f, where f:k<sub>1</sub>:...:k<sub>r</sub> is a JI chord none of whose notes is a power of 2.

			Non-over-1 temperaments give regular-temperament interpretations to edos that approximate [[Overtone scales\|over-1]] chords (assuming [[octave equivalence]]) such as [[4:5:6:7:11]] poorly, such as [[14edo]], [[18edo]], [[23edo]] and [[29edo]], thus may have much xenharmonic potential. Many of these temperaments have an [[octatonic]] structure, as [[8edo]] represents non-over-1 intervals well for its size as far as the [[17-limit]].

	== Examples ==		== Examples ==

Currywurst44: Clarified over-1 meaning

2026-04-23T13:52:52Z

Clarified over-1 meaning

← Older revision		Revision as of 13:52, 23 April 2026
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	A '''non-over-1 temperament''' is a [[regular temperament]] that tempers a [[subgroup]] corresponding to a harmonic series chord r:n<sub>1</sub>:...:n<sub>k</sub> where r ≠ 1, but is not meant to approximate a chord of the form 1:m<sub>1</sub>:...:m<sub>k</sub>. Non-over-1 temperaments give regular-temperament interpretations to edos that approximate [[Overtone scales\|over-1]] chords such as [[4:5:6:7:11]] poorly, such as [[14edo]], [[18edo]], [[23edo]] and [[29edo]], thus may have much xenharmonic potential. Many of these temperaments have an [[octatonic]] structure, as [[8edo]] represents non-over-1 intervals well for its size as far as the [[17-limit]].		A '''non-over-1 temperament''' is a [[regular temperament]] that tempers a [[subgroup]] corresponding to a harmonic series chord r:n<sub>1</sub>:...:n<sub>k</sub> where r ≠ 1, but is not meant to approximate a chord of the form 1:m<sub>1</sub>:...:m<sub>k</sub>. Non-over-1 temperaments give regular-temperament interpretations to edos that approximate [[Overtone scales\|over-1]] chords (assuming [[octave equivalence]]) such as [[4:5:6:7:11]] poorly, such as [[14edo]], [[18edo]], [[23edo]] and [[29edo]], thus may have much xenharmonic potential. Many of these temperaments have an [[octatonic]] structure, as [[8edo]] represents non-over-1 intervals well for its size as far as the [[17-limit]].

	== Examples ==		== Examples ==

Overthink: /* Petrtri */ links

2026-02-05T04:23:07Z

Petrtri: links

← Older revision		Revision as of 04:23, 5 February 2026
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	=== Petrtri ===		=== Petrtri ===
	{{See also\|Subgroup temperaments #Petrtri\|Chromatic pairs #Petrtri}}		{{See also\|Subgroup temperaments #Petrtri\|Chromatic pairs #Petrtri}}
	Define petrtri as the 2.9/5.11/5.13/5 subgroup temperament supported by 13edo and 21edo. Then petrtri mainly approximates 5:9:11:13, and this chord is found twice in the [[oneirotonic]] MOS petrtri[8]. Both 13edo and 21edo support the -7 generators = 5/4 mapping, so petrtri extends to the 2.5.9.11.13 subgroup; however, a 4:5:9:11:13 chord must span 10 generators (11 notes) thus must go outside the 8-note MOS. Thus it is fair to say that approximations of 5:9:11:13 in the 8-note MOS use this non-over-1 temperament.		Define petrtri as the 2.9/5.11/5.13/5 subgroup temperament supported by [[13edo]] and [[21edo]]. Then petrtri mainly approximates 5:9:11:13, and this chord is found twice in the [[oneirotonic]] MOS petrtri[8]. Both 13edo and 21edo support the -7 generators = 5/4 mapping, so petrtri extends to the 2.5.9.11.13 subgroup; however, a 4:5:9:11:13 chord must span 10 generators (11 notes) thus must go outside the 8-note MOS. Thus it is fair to say that approximations of 5:9:11:13 in the 8-note MOS use this non-over-1 temperament.

	=== Sensi ===		=== Sensi ===

Overthink: see alsos, a few smaller changes

2026-02-05T04:19:19Z

see alsos, a few smaller changes

← Older revision		Revision as of 04:19, 5 February 2026
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	A '''non-over-1 temperament''' is a [[regular temperament]] that tempers a [[subgroup]] corresponding to a harmonic series chord r:n<sub>1</sub>:...:n<sub>k</sub> where r ≠ 1, but is not meant to approximate a chord of the form 1:m<sub>1</sub>:...:m<sub>k</sub>. Non-over-1 temperaments give regular-temperament interpretations to edos that approximate [[Overtone scales\|over-1]] chords such as 4:5:6:7:11 poorly, such as [[14edo]], [[18edo]], [[23edo]] and [[29edo]], thus may have much xenharmonic potential. Many of these temperaments have an ~~8-tone~~ structure, as [[8edo]] represents non-over-1 intervals ~~relatively~~ well as far as the [[17-limit]].		A '''non-over-1 temperament''' is a [[regular temperament]] that tempers a [[subgroup]] corresponding to a harmonic series chord r:n<sub>1</sub>:...:n<sub>k</sub> where r ≠ 1, but is not meant to approximate a chord of the form 1:m<sub>1</sub>:...:m<sub>k</sub>. Non-over-1 temperaments give regular-temperament interpretations to edos that approximate [[Overtone scales\|over-1]] chords such as [[4:5:6:7:11]] poorly, such as [[14edo]], [[18edo]], [[23edo]] and [[29edo]], thus may have much xenharmonic potential. Many of these temperaments have an [[octatonic]] structure, as [[8edo]] represents non-over-1 intervals well for its size as far as the [[17-limit]].

	== Examples ==		== Examples ==
	=== Greeley ===		=== Greeley ===
	Greeley is the [[23edo\|23]]&[[31edo\|31]] temperament on the 2.5/3.7/3.11/3 subgroup, with a MOS generator size close to [[porcupine]] but smaller (the POL2 generator is around 155.7756¢).		{{See also\|Subgroup temperaments #Greeley\|Chromatic pairs #Greeley}}
			Greeley is the [[23edo\|23]]&[[31edo\|31]] temperament on the 2.5/3.7/3.11/3 subgroup, with a [[MOS]] [[generator]] size close to [[porcupine]] but smaller (the POL2 generator is around 155.7756¢).
	* One generator represents 12/11 and 11/10.		* One generator represents 12/11 and 11/10.
	* Two generators represents 6/5.		* Two generators represents 6/5.
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	=== Petrtri ===		=== Petrtri ===
			{{See also\|Subgroup temperaments #Petrtri\|Chromatic pairs #Petrtri}}
	Define petrtri as the 2.9/5.11/5.13/5 subgroup temperament supported by 13edo and 21edo. Then petrtri mainly approximates 5:9:11:13, and this chord is found twice in the [[oneirotonic]] MOS petrtri[8]. Both 13edo and 21edo support the -7 generators = 5/4 mapping, so petrtri extends to the 2.5.9.11.13 subgroup; however, a 4:5:9:11:13 chord must span 10 generators (11 notes) thus must go outside the 8-note MOS. Thus it is fair to say that approximations of 5:9:11:13 in the 8-note MOS use this non-over-1 temperament.		Define petrtri as the 2.9/5.11/5.13/5 subgroup temperament supported by 13edo and 21edo. Then petrtri mainly approximates 5:9:11:13, and this chord is found twice in the [[oneirotonic]] MOS petrtri[8]. Both 13edo and 21edo support the -7 generators = 5/4 mapping, so petrtri extends to the 2.5.9.11.13 subgroup; however, a 4:5:9:11:13 chord must span 10 generators (11 notes) thus must go outside the 8-note MOS. Thus it is fair to say that approximations of 5:9:11:13 in the 8-note MOS use this non-over-1 temperament.

	=== Sensi ===		=== Sensi ===
			{{See also\|Sensipent family #Sensi\|Chromatic pairs #Sensi}}
	[[Sensi]] is ''effectively'' a non-over-1 temperament provided you restrict yourself to the sensi[8] MOS. The sensi[8] MOS only has a 5:6:7:9:13 chord, but no chord of the form 2:m<sub>1</sub>:...:m<sub>k</sub> (except 2:3). Thus sensi can be viewed as a 2.6/5.7/5.9/5.13/10 or 2.3.6/5.7/5.13/10 temperament. (See [http://x31eq.com/cgi-bin/rt.cgi?limit=2_6%2F5_7%2F5_9%2F5_13%2F10&ets=19_27&tuning=po&subgroup=on x31eq data page].)		[[Sensi]] is ''effectively'' a non-over-1 temperament provided you restrict yourself to the sensi[8] MOS. The sensi[8] MOS only has a 5:6:7:9:13 chord, but no chord of the form 2:m<sub>1</sub>:...:m<sub>k</sub> (except 2:3). Thus sensi can be viewed as a 2.6/5.7/5.9/5.13/10 or 2.3.6/5.7/5.13/10 temperament. (See [http://x31eq.com/cgi-bin/rt.cgi?limit=2_6%2F5_7%2F5_9%2F5_13%2F10&ets=19_27&tuning=po&subgroup=on x31eq data page].)

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	: <sup>*</sup> in 2.3.5.7.13 POTE tuning		: <sup>*</sup> in 2.3.5.7.13 POTE tuning
	: <sup>†</sup> 2.3.5.7.13 ratio interpretations		: <sup>†</sup> 2.3.5.7.13 ratio interpretations


	=== Tridec ===		=== Tridec ===
			{{See also\|Subgroup temperaments #Tridec\|Chromatic pairs #Tridec}}
	Tridec is a temperament generated by a generator near 455.2178¢ (for example, [[29edo\|11\29]] or [[37edo\|14\37]]) and has an 8-note MOS. If you restrict to the 8-note MOS, Tridec is an 2.7/5.11/5.13/5 temperament that tempers the chord 5:7:11:13; one generator represents a 13/10, three generators represent a 11/10, -4 generators represent a 7/5.		Tridec is a temperament generated by a generator near 455.2178¢ (for example, [[29edo\|11\29]] or [[37edo\|14\37]]) and has an 8-note MOS. If you restrict to the 8-note MOS, Tridec is an 2.7/5.11/5.13/5 temperament that tempers the chord 5:7:11:13; one generator represents a 13/10, three generators represent a 11/10, -4 generators represent a 7/5.

	Tridec essentially contains all the notes of 2.3.5 [[porcupine]] temperament and satisfies all its relations; hence it is essentially the same as 13-limit [[Ammonite]].		Tridec essentially contains all the notes of 2.3.5 [[porcupine]] temperament and satisfies all its relations; hence it is essentially the same as 13-limit [[Ammonite]].

	[[Category:Regular temperament theory]]		[[Category:Regular temperament theory]]

Overthink: /* Greeley */ grammar

2025-12-23T01:06:44Z

Greeley: grammar

← Older revision		Revision as of 01:06, 23 December 2025
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	== Examples ==		== Examples ==
	=== Greeley ===		=== Greeley ===
	Greeley is a [[23edo\|23]]&[[31edo\|31]] temperament on the 2.5/3.7/3.11/3 subgroup, with a MOS generator size close to [[porcupine]] but smaller (the POL2 generator is around 155.7756¢).		Greeley is the [[23edo\|23]]&[[31edo\|31]] temperament on the 2.5/3.7/3.11/3 subgroup, with a MOS generator size close to [[porcupine]] but smaller (the POL2 generator is around 155.7756¢).
	* One generator represents 12/11 and 11/10.		* One generator represents 12/11 and 11/10.
	* Two generators represents 6/5.		* Two generators represents 6/5.
	* Four generators ~~represent~~ 10/7.		* Four generators represents 10/7.
	* Six generators ~~represent~~ 12/7.		* Six generators represents 12/7.

	=== Petrtri ===		=== Petrtri ===

Overthink: + note on 8edo

2025-11-26T19:27:39Z

+ note on 8edo

← Older revision		Revision as of 19:27, 26 November 2025
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	A '''non-over-1 temperament''' is a [[regular temperament]] that tempers a [[subgroup]] corresponding to a harmonic series chord r:n<sub>1</sub>:...:n<sub>k</sub> where r ≠ 1, but is not meant to approximate a chord of the form 1:m<sub>1</sub>:...:m<sub>k</sub>. Non-over-1 temperaments give regular-temperament interpretations to edos that approximate [[Overtone scales\|over-1]] chords such as 4:5:6:7:11 poorly, such as [[14edo]], [[18edo]], [[23edo]] and [[29edo]], thus may have much xenharmonic potential.		A '''non-over-1 temperament''' is a [[regular temperament]] that tempers a [[subgroup]] corresponding to a harmonic series chord r:n<sub>1</sub>:...:n<sub>k</sub> where r ≠ 1, but is not meant to approximate a chord of the form 1:m<sub>1</sub>:...:m<sub>k</sub>. Non-over-1 temperaments give regular-temperament interpretations to edos that approximate [[Overtone scales\|over-1]] chords such as 4:5:6:7:11 poorly, such as [[14edo]], [[18edo]], [[23edo]] and [[29edo]], thus may have much xenharmonic potential. Many of these temperaments have an 8-tone structure, as [[8edo]] represents non-over-1 intervals relatively well as far as the [[17-limit]].

	== Examples ==		== Examples ==

Fredg999 category edits: Removing from Category:Temperament using Cat-a-lot

2022-03-01T02:38:28Z

Removing from Category:Temperament using Cat-a-lot

← Older revision		Revision as of 02:38, 1 March 2022
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	Tridec essentially contains all the notes of 2.3.5 [[porcupine]] temperament and satisfies all its relations; hence it is essentially the same as 13-limit [[Ammonite]].		Tridec essentially contains all the notes of 2.3.5 [[porcupine]] temperament and satisfies all its relations; hence it is essentially the same as 13-limit [[Ammonite]].
	~~[[Category:Temperament]]~~
	[[Category:Regular temperament theory]]		[[Category:Regular temperament theory]]