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 [[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:n1:...:nk where r ≠ 1, but is not meant to approximate a chord of the form 1:m1:...:mk. Assuming octave equivalence, this means a subgroup of the form 2.m1/f.[...].mr/f, where f:m1:...:mr 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 ==

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=== 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].)

[[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 2n: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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-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>. 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]].
 == Examples ==
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 === 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].)
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