Non-over-1 temperament: Difference between revisions

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A '''non-over-1 temperament''' is a [[regular temperament]] that tempers a [[subgroup]] corresponding to a harmonic series chord r:n1:...:nk where r ≠ 1, but is not meant to approximate a chord of the form 1:m1:...:mk. 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:n1:...:nk where r ≠ 1, but is not meant to approximate a chord of the form 1:m1:...:mk. 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 ==

=== 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.

* Two generators represents 6/5.

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=== 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.

=== 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:m1:...:mk (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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: * in 2.3.5.7.13 POTE tuning

: † 2.3.5.7.13 ratio interpretations

=== 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 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]]

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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 ==
 === 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.
 * Two generators represents 6/5.
@@ Line 10: / Line 11: @@
 === 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.
 === 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].)
@@ Line 56: / Line 59: @@
 : <sup>*</sup> in 2.3.5.7.13 POTE tuning
 : <sup>†</sup> 2.3.5.7.13 ratio interpretations
 === 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 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]]