31edo: Difference between revisions

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The fact that 31edo has meantone diatonic and chromatic scales is well-known, but some other [[MOS]]es and MOS chains are also useful:

The fact that 31edo has meantone diatonic and chromatic scales is well-known, but some other [[MOS]]es and MOS chains{{clarify}} are also useful:

* ~~31edo's~~ 9\31 neutral third generator generates [[~~Step ratio|~~ultrasoft]] [[~~3L 4s|~~mosh]] and [[~~Step ratio|~~superhard]] [[~~7L 3s|dicoid~~]] MOSes.

* 9\31 neutral third generator generates [[ultrasoft]] [[mosh]] and [[superhard]] [[dicotonic]] MOSes.

* ~~Its~~ 12\31 generator generates a [[~~Step ratio|~~semihard]] [[oneirotonic]] scale, similar to the 5L 3s scale in [[13edo]] but with the 9/8, 5/4 and 7/6 better in tune and with the flat fifth closer to just.

* 12\31 generator generates a [[semihard]] [[oneirotonic]] scale, similar to the 5L 3s scale in [[13edo]] but with the 9/8, 5/4 and 7/6 better in tune and with the flat fifth closer to just.

* A chain of 5\31 whole tones is exceptionally rich in 4:5:7 chords, which are approximated very well in 31edo.

* If you're fond of orwell tetrads (which are also found in 31edo's oneirotonic), you will like the 7\31 (271.0¢) subminor third generator. The [[~~Step ratio|~~ultrasoft]] 9-tone orwelloid ([[4L 5s]]) MOS could be treated as a 9-tone well temperament.

* If you're fond of orwell tetrads (which are also found in 31edo's oneirotonic), you will like the 7\31 (271.0¢) subminor third generator. The [[ultrasoft]] 9-tone [[4L 5s|orwelloid (4L 5s)]] MOS could be treated as a 9-tone well temperament.

* It ~~supports~~ [[6edf]] (miracle) and [[9edf]] (Carlos ~~alpha~~), fifth-equivalent equal ~~temperaments~~ that hit many good JI approximations.

* It has close approximations to [[6edf]] (→ [[miracle]]) and [[9edf]] (→ [[Carlos Alpha]]), fifth-equivalent equal divisions that hit many good JI approximations.

See [[~~31edo~~#~~MOSes and rank~~-2 temperaments]] for a table of MOSes and some temperament interpretations.

See [[#Rank-2 temperaments]] for a table of MOSes and some temperament interpretations.

=== Harmonic Scale ===

31edo approximates Mode 8 of the [[~~OverToneSeries|~~harmonic series]] ~~O.K.~~, but many intervals between the harmonics aren't distinguished, most importantly 9/8 (major tone) and 10/9 (minor tone), as 31EDO is a meantone temperament. The interval between the 8th and 11th harmonics is approximated ~~O.K.~~, but the intervals between the 11th harmonic and closer harmonics such as the 12th and 9th harmonics are approximated even better. 31's version of 13/8 is quite wide and only vaguely suggests the [[13-limit]].

31edo approximates Mode 8 of the [[harmonic series]] okay, but many intervals between the harmonics aren't distinguished, most importantly 9/8 (major tone) and 10/9 (minor tone), as 31EDO is a meantone temperament. The interval between the 8th and 11th harmonics is approximated okay, but the intervals between the 11th harmonic and closer harmonics such as the 12th and 9th harmonics are approximated even better. 31's version of 13/8 is quite wide and only vaguely suggests the [[13-limit]].

{| class="wikitable"

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=== ~~MOSes and rank~~-2 temperaments ===

=== Rank-2 temperaments ===

* [[List of 31et rank two temperaments by badness]]

* [[List of edo-distinct 31et rank two temperaments]]

@@ Line 738: / Line 738: @@
 {{main| 31edo MOS scales }}
-The fact that 31edo has meantone diatonic and chromatic scales is well-known, but some other [[MOS]]es and MOS chains are also useful:
+The fact that 31edo has meantone diatonic and chromatic scales is well-known, but some other [[MOS]]es and MOS chains{{clarify}} are also useful:
-* 31edo's 9\31 neutral third generator generates [[Step ratio|ultrasoft]] [[3L 4s|mosh]] and [[Step ratio|superhard]] [[7L 3s|dicoid]] MOSes.
+* 9\31 neutral third generator generates [[ultrasoft]] [[mosh]] and [[superhard]] [[dicotonic]] MOSes.
-* Its 12\31 generator generates a [[Step ratio|semihard]] [[oneirotonic]] scale, similar to the 5L 3s scale in [[13edo]] but with the 9/8, 5/4 and 7/6 better in tune and with the flat fifth closer to just.
+* 12\31 generator generates a [[semihard]] [[oneirotonic]] scale, similar to the 5L 3s scale in [[13edo]] but with the 9/8, 5/4 and 7/6 better in tune and with the flat fifth closer to just.
 * A chain of 5\31 whole tones is exceptionally rich in 4:5:7 chords, which are approximated very well in 31edo.
-* If you're fond of orwell tetrads (which are also found in 31edo's oneirotonic), you will like the 7\31 (271.0¢) subminor third generator. The [[Step ratio|ultrasoft]] 9-tone orwelloid ([[4L 5s]]) MOS could be treated as a 9-tone well temperament.
+* If you're fond of orwell tetrads (which are also found in 31edo's oneirotonic), you will like the 7\31 (271.0¢) subminor third generator. The [[ultrasoft]] 9-tone [[4L 5s|orwelloid (4L 5s)]] MOS could be treated as a 9-tone well temperament.
-* It supports [[6edf]] (miracle) and [[9edf]] (Carlos alpha), fifth-equivalent equal temperaments that hit many good JI approximations.
+* It has close approximations to [[6edf]] (→ [[miracle]]) and [[9edf]] (→ [[Carlos Alpha]]), fifth-equivalent equal divisions that hit many good JI approximations.
-See [[31edo#MOSes and rank-2 temperaments]] for a table of MOSes and some temperament interpretations.
+See [[#Rank-2 temperaments]] for a table of MOSes and some temperament interpretations.
 === Harmonic Scale ===
-edo approximates Mode 8 of the [[OverToneSeries|harmonic series]] O.K., but many intervals between the harmonics aren't distinguished, most importantly 9/8 (major tone) and 10/9 (minor tone), as 31EDO is a meantone temperament. The interval between the 8th and 11th harmonics is approximated O.K., but the intervals between the 11th harmonic and closer harmonics such as the 12th and 9th harmonics are approximated even better. 31's version of 13/8 is quite wide and only vaguely suggests the [[13-limit]].
+edo approximates Mode 8 of the [[harmonic series]] okay, but many intervals between the harmonics aren't distinguished, most importantly 9/8 (major tone) and 10/9 (minor tone), as 31EDO is a meantone temperament. The interval between the 8th and 11th harmonics is approximated okay, but the intervals between the 11th harmonic and closer harmonics such as the 12th and 9th harmonics are approximated even better. 31's version of 13/8 is quite wide and only vaguely suggests the [[13-limit]].
 {| class="wikitable"
@@ Line 1,094: / Line 1,094: @@
 <references/>
-=== MOSes and rank-2 temperaments ===
+=== Rank-2 temperaments ===
 * [[List of 31et rank two temperaments by badness]]
 * [[List of edo-distinct 31et rank two temperaments]]

31edo: Difference between revisions

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