43edo: Difference between revisions

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=== Prime harmonics ===

{{Harmonics in equal|43|start=12|collapsed=true|title=Approximation of prime harmonics in 43edo (continued)}}

{{Harmonics in equal|43|columns=11|start=12|collapsed=true|title=Approximation of prime harmonics in 43edo (continued)}}

=== Subsets and supersets ===

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=== Higher-limit JI ===

Although not [[consistent]], 43edo performs quite well in very high prime limits. It has unambiguous mappings for ~~all~~ prime harmonics up to ''113'' (after which the demands on its pitch resolution finally become too great~~), with the sole~~ exceptions of 23, 71, 89, and 103, ~~making a great~~ [[#Ringer 43|Ringer ~~scale~~]]. Mappings for ratios between these prime harmonics can then be derived from those for the primes themselves, allowing for a complete set of approximations to the first 16 harmonics in the harmonic series and an almost-complete approximation of the first 32 harmonics, although the limited consistency will give some unusual results~~. Indeed, one step of 43edo is very close to the [[64/63|septimal comma (64/63)]]; similarly, two steps is close to [[32/31]], and four steps tunes [[16/15]] almost perfectly~~.

Although not [[consistent]], 43edo performs quite well in very high prime limits. It has unambiguous mappings for most prime harmonics up to ''113'', after which the demands on its pitch resolution finally become too great. The exceptions are 23, 71, 89, and 103, which have more than 35% relative error (10 cents absolute error). This high-limit capability is useful for approaches based on the harmonic series, such as for creating [[#Ringer 43|Ringer scales]]. Mappings for ratios between these prime harmonics can then be derived from those for the primes themselves, allowing for a complete set of approximations to the first 16 harmonics in the harmonic series and an almost-complete approximation of the first 32 harmonics, although the limited consistency will give some unusual results.

~~43edo has less than 35% relative error (less than 10 cents error) on an impressive 17 of the 19 prime~~ harmonics ~~in the [[67-limit]]. The only ones it misses are 23 and 41. So it can be used as a solid full [[19-limit]] tuning~~, ~~or as a solid no-23-or-41 67-limit tuning.~~

Within harmonics 1–63, 43edo approximates harmonics 15, 31, 37, 61, and 63 close to exactly – within less than a cent (less than 3% relative error). Indeed, one step of 43edo is very close to the [[64/63|septimal comma (64/63)]]; similarly, two steps is close to [[32/31]], and four steps tunes [[16/15]] almost perfectly. It approximates 3, 9, 13, 27, 39, 43, 53 and 61 flat. It approximates 5, 7, 11, 17, 19, 21, 25, 29, 33, 47, 49, 51, 57 and 59 sharp. Overall this gives 43edo a slightly sharp tendency/feeling.

It approximates harmonics 31, 37 and 61 close to exactly – within less than a cent (less than 3% relative error). It approximates 3, 13, 43, 53 and 61 ~~slightly~~ flat. It approximates 5, 7, 11, 17, 19, 29, 47, 59 ~~and 67 slightly~~ sharp. Overall this gives 43edo a slightly sharp tendency/feeling~~, though with the major exception of harmonic 3 (the perfect fifth)~~.

== Regular temperament properties ==

@@ Line 15: / Line 15: @@
 === Prime harmonics ===
-{{Harmonics in equal|43}}
+{{Harmonics in equal|43|columns=11}}
-{{Harmonics in equal|43|start=12|collapsed=true|title=Approximation of prime harmonics in 43edo (continued)}}
+{{Harmonics in equal|43|columns=11|start=12|collapsed=true|title=Approximation of prime harmonics in 43edo (continued)}}
 === Subsets and supersets ===
@@ Line 398: / Line 398: @@
 === Higher-limit JI ===
-Although not [[consistent]], 43edo performs quite well in very high prime limits. It has unambiguous mappings for all prime harmonics up to ''113'' (after which the demands on its pitch resolution finally become too great), with the sole exceptions of 23, 71, 89, and 103, making a great [[#Ringer 43|Ringer scale]]. Mappings for ratios between these prime harmonics can then be derived from those for the primes themselves, allowing for a complete set of approximations to the first 16 harmonics in the harmonic series and an almost-complete approximation of the first 32 harmonics, although the limited consistency will give some unusual results. Indeed, one step of 43edo is very close to the [[64/63|septimal comma (64/63)]]; similarly, two steps is close to [[32/31]], and four steps tunes [[16/15]] almost perfectly.
+Although not [[consistent]], 43edo performs quite well in very high prime limits. It has unambiguous mappings for most prime harmonics up to ''113'', after which the demands on its pitch resolution finally become too great. The exceptions are 23, 71, 89, and 103, which have more than 35% relative error (10 cents absolute error). This high-limit capability is useful for approaches based on the harmonic series, such as for creating [[#Ringer 43|Ringer scales]]. Mappings for ratios between these prime harmonics can then be derived from those for the primes themselves, allowing for a complete set of approximations to the first 16 harmonics in the harmonic series and an almost-complete approximation of the first 32 harmonics, although the limited consistency will give some unusual results.
-edo has less than 35% relative error (less than 10 cents error) on an impressive 17 of the 19 prime harmonics in the [[67-limit]]. The only ones it misses are 23 and 41. So it can be used as a solid full [[19-limit]] tuning, or as a solid no-23-or-41 67-limit tuning.
+Within harmonics 1–63, 43edo approximates harmonics 15, 31, 37, 61, and 63 close to exactly – within less than a cent (less than 3% relative error). Indeed, one step of 43edo is very close to the [[64/63|septimal comma (64/63)]]; similarly, two steps is close to [[32/31]], and four steps tunes [[16/15]] almost perfectly. It approximates 3, 9, 13, 27, 39, 43, 53 and 61 flat. It approximates 5, 7, 11, 17, 19, 21, 25, 29, 33, 47, 49, 51, 57 and 59 sharp. Overall this gives 43edo a slightly sharp tendency/feeling.
-It approximates harmonics 31, 37 and 61 close to exactly – within less than a cent (less than 3% relative error). It approximates 3, 13, 43, 53 and 61 slightly flat. It approximates 5, 7, 11, 17, 19, 29, 47, 59 and 67 slightly sharp. Overall this gives 43edo a slightly sharp tendency/feeling, though with the major exception of harmonic 3 (the perfect fifth).
 == Regular temperament properties ==