Non-over-1 temperament: Difference between revisions

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A '''non-over-2 temperament''', or '''non-/2 temperament''', is a [[regular temperament]] that tempers a [[subgroup]] corresponding to a harmonic series chord r:n1:...:nk where r ≠ 2, but is not meant to approximate a chord of the form 2:m1:...:mk. Non-over-2 temperaments give regular-temperament interpretations to edos that approximate over-2 chords such as 4:5:6:7:11 poorly, such as [[14edo]], [[18edo]], [[23edo]] and [[29edo]].

== ~~Example~~ ==

== Examples ==

=== Tridec ===

In the broad sense, Tridec can be viewed as any oneirotonic tuning that equates three oneirotonic large steps to a [[4/3]] perfect fourth. [This identification may come in handy since many altered oneirotonic modes have three consecutive large steps.] Based on the JI interpretations of the [[29edo]] and [[37edo]] tunings, it can in fact be viewed as a 2.3.7/5.11/5.13/5 temperament, i.e. a [[Non-over-2 temperament|non-over-2 temperament]] that approximates the chord 5:7:11:13:15. The optimal generator is 455.2178¢, which is very close to 29edo's 11\29 (455.17¢), but we could accept any generator between 17\45 (453.33¢) and 8\21 (457.14¢), if we stipulate that the 3/2 has to be between [[7edo]]'s fifth and [[5edo]]'s fifth.

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]]; however, provided you restrict yourself to the 8-note MOS, you're using it as a non-over-2 temperament.

The sizes of the generator, large step and small step of oneirotonic are as follows in various tridec tunings.

{| class="wikitable right-2 right-3 right-4 right-5"

|-

!

! [[21edo]]

! [[29edo]]

! [[37edo]]

! Optimal ([[POTE]]) tuning

! JI intervals represented (2.3.7/5.11/5.13/5 subgroup)

|-

| generator (g)

| 8\21, 457.14

| 11\29, 455.17

| 14\37, 454.05

| 455.22

| 13/10

|-

| L (3g - octave)

| 3\21, 171.43

| 4\29, 165.52

| 5\37, 162.16

| 165.65

| 11/10

|-

| s (-5g + 2 octaves)

| 2\21, 114.29

| 3\29, 124.14

| 4\37, 129.73

| 123.91

| 14/13, 15/14

|}

=== Sensi ===

[[Sensi]] is ''effectively'' a non-over-2 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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 A '''non-over-2 temperament''', or '''non-/2 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 ≠ 2, but is not meant to approximate a chord of the form 2:m<sub>1</sub>:...:m<sub>k</sub>. Non-over-2 temperaments give regular-temperament interpretations to edos that approximate over-2 chords such as 4:5:6:7:11 poorly, such as [[14edo]], [[18edo]], [[23edo]] and [[29edo]].
-== Example ==
+== Examples ==
+=== Tridec ===
+In the broad sense, Tridec can be viewed as any oneirotonic tuning that equates three oneirotonic large steps to a [[4/3]] perfect fourth. [This identification may come in handy since many altered oneirotonic modes have three consecutive large steps.] Based on the JI interpretations of the [[29edo]] and [[37edo]] tunings, it can in fact be viewed as a 2.3.7/5.11/5.13/5 temperament, i.e. a [[Non-over-2 temperament|non-over-2 temperament]] that approximates the chord 5:7:11:13:15. The optimal generator is 455.2178¢, which is very close to 29edo's 11\29 (455.17¢), but we could accept any generator between 17\45 (453.33¢) and 8\21 (457.14¢), if we stipulate that the 3/2 has to be between [[7edo]]'s fifth and [[5edo]]'s fifth.
+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]]; however, provided you restrict yourself to the 8-note MOS, you're using it as a non-over-2 temperament.
+The sizes of the generator, large step and small step of oneirotonic are as follows in various tridec tunings.
+{| class="wikitable right-2 right-3 right-4 right-5"
+|-
+!
+! [[21edo]]
+! [[29edo]]
+! [[37edo]]
+! Optimal ([[POTE]]) tuning
+! JI intervals represented (2.3.7/5.11/5.13/5 subgroup)
+|-
+| generator (g)
+| 8\21, 457.14
+| 11\29, 455.17
+| 14\37, 454.05
+| 455.22
+| 13/10
+|-
+| L (3g - octave)
+| 3\21, 171.43
+| 4\29, 165.52
+| 5\37, 162.16
+| 165.65
+| 11/10
+|-
+| s (-5g + 2 octaves)
+| 2\21, 114.29
+| 3\29, 124.14
+| 4\37, 129.73
+| 123.91
+| 14/13, 15/14
+|}
+=== Sensi ===
 [[Sensi]] is ''effectively'' a non-over-2 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].)