Hemimean clan: Difference between revisions

{{Technical data page}}

The '''hemimean clan''' [[Tempering out|tempers out]] the hemimean comma, [[3136/3125]], with [[monzo]] {{monzo| 6 0 -5 2 }}, such that [[7/4]] is split into five steps, of which two make [[5/4]] and three make [[7/5]]; this defines the [[2.5.7 subgroup]] temperament [[didacus]], generated by a tempered hemithird of [[28/25]].

[[Tp tuning #T2 tuning|RMS error]]: 0.2138 cents

= Strong extensions =

|-

! rowspan="2" | Extension !! colspan="2" | 5-limit re-restriction !! rowspan="2" | Mapping of 3 !! rowspan="2" | Tuning range*

! Temperament !! 5-limit generator location

|-

== Hemiwürschmidt ==

{{See also| Würschmidt family }}

Mapping generators: ~2, ~25/14

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~28/25 = 193.898

{{Optimal ET sequence|legend=1| 31, 68, 99, 229, 328 }}

=== 11-limit ===

=== Quadrawürschmidt ===

This has been documented in Graham Breed's temperament finder as ''semihemiwürschmidt'', but ''quadrawürschmidt'' arguably makes more sense.  

Subgroup: 2.3.5.7.11

== Hemithirds ==

{{Main| Hemithirds }}

{{Mapping|legend=1| 1 4 2 2 | 0 -15 2 5 }}

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~28/25 = 193.244

* [[7-odd-limit]]: ~28/25 = {{monzo| 1/10 -1/20 0 1/20 }}

: {{monzo list| 1 0 0 0 | 5/2 3/4 0 -3/4 | 11/5 -1/10 0 1/10 | 5/2 -1/4 0 1/4 }}

* [[9-odd-limit]]: ~28/25 = {{monzo| 6/35 -2/35 0 1/35 }}

: {{monzo list| 1 0 0 0 | 10/7 6/7 0 -3/7 | 82/35 -4/35 0 2/35 | 20/7 -2/7 0 1/7 }}

{{Optimal ET sequence|legend=1| 25, 31, 87, 118 }}

== Spell ==

{{See also| Magic family }}

{{Mapping|legend=1| 1 0 2 2 | 0 10 2 5 }}

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~28/25 = 189.927

: mapping generators: ~2, ~75/49

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~75/49 = 735.155

: mapping generators: ~2, ~48/35

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~48/35 = 551.782

{{Mapping|legend=1| 1 6 0 -3 | 0 -19 10 25 }}

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~75/64 = 278.800

{{Mapping|legend=1| 1 1 2 2 | 0 29 16 40 }}

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~686/675 = 24.217

{{Mapping|legend=1| 1 0 0 -3 | 0 15 22 55 }}

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~27/25 = 126.706

{{Mapping|legend=1| 1 -4 -2 -8 | 0 31 24 60 }}

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~4375/3888 = 216.173

{{Mapping|legend=1| 1 0 -2 -8 | 0 11 30 75 }}

[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~448/405 = 172.917

RMS error: 0.5567 cents

=== Tridecimal didacus ===

Optimal ET sequence: {{Optimal ET sequence| 6, 25, 31, 37 }}

==== Mediantone ====

In the full no-3's [[19-limit]], this temperament is a structure common to quite a few temperaments. It is a rank-2 version of [[orion]] with a mapping for primes 11 and 13. It is a no-3's version of 19-limit [[grosstone]] which can be seen as an extension of [[undecimal meantone]] according to the "mediant-tone" logic of this temperament, and which as aforementioned effectively doubles the complexity of the temperament as a result of finding the generator of [[~]][[19/17]][[~]][[28/25]] as ([[~]][[3/2]])²/[[2/1|2]]. It does not work so well as an extension for [[hemiwur]] to the full 19-limit, but if you want to try anyway (at the cost of primes 17 and 19), a notable patent-val tuning is [[37edo]], which finds prime 3 through the [[würschmidt]] mapping so that [[6/1]] is found at 16 generators.

Subgroup: 2.5.7.11.13.17.19

Optimal tuning (CWE): ~2 = 1\1, ~19/17 = 194.927

==== Roulette ====

Roulette is an alternative no-threes 19-limit extension of tridecimal didacus to mediantone (the two mappings converging at [[37edo]]), equating (8/7)² to [[17/13]] in addition to 13/10, tempering out [[170/169]] and [[833/832]]; in doing so, it also tempers out the micro-comma [[2000033/2000000]] so that ([[50/49]])³ is equated to [[17/16]]. The generator is then equated to 19/17 in the same way as in mediantone.

Subgroup: 2.5.7.11.13.17.19

Optimal ET sequence: {{Optimal ET sequence| 6g, ... 31, 37, 68, 105 }}

== Rectified hebrew ==

[[Category:Temperament clans]]

[[Category:Hemimean clan| ]] <!-- main article -->

[[Category:Hemimean| ]] <!-- key article -->

[[Category:Rank 2]]