Würschmidt family: Difference between revisions

The [[5-limit]] parent comma for the '''würschmidt family''' is [[393216/390625]], known as Würschmidt's comma, and named after José Würschmidt, Its [[monzo]] is |17 1 -8>, and flipping that yields <<8 1 17|| for the wedgie. This tells us the [[generator]] is a major third, and that to get to the interval class of fifths will require eight of these. In fact, (5/4)^8 * 393216/390625 = 6. 10\31, 11\34 or 21\65 are possible generators and other tunings include 96edo, 99edo and 164edo. Another tuning solution is to sharpen the major third by 1/8th of a Würschmidt comma, which is to say by 1.43 cents, and thereby achieve pure fifths; this is the [[minimax tuning]]. Würschmidt is well-supplied with MOS scales, with 10, 13, 16, 19, 22, 25, 28, 31 and 34 note [[MOS]] all possibilities.

== Würschmidt ==

[[Mapping]]: [<1 7 3|, <0 -8 -1|]

[[EDO|Vals]]: {{Val list| 31, 34, 65, 99, 164, 721c, 885c }}

=== Music ===

* [http://micro.soonlabel.com/gene_ward_smith/Others/Freivald/Wurschmidt%5b16%5d-out.mp3 Extrospection] by [https://soundcloud.com/jdfreivald/extrospection Jake Freivald]; Würschmidt[16] tuned in 31et.

=== Seven limit children ===

The second comma of the [[Normal_lists|normal comma list]] defines which 7-limit family member we are looking at. Wurschmidt adds |12 3 -6 -1>, worschmidt adds 65625/65536 = |-16 1 5 1>, whirrschmidt adds 4375/4374 = |-1 -7 4 1> and hemiwuerschmidt adds 6144/6125 = |11 1 -3 -2>.

== Septimal Würschmidt ==

Würschmidt, aside from the commas listed above, also tempers out 225/224. [[31edo]] or [[127edo]] can be used as tunings. Würschmidt has <<8 1 18 -17 6 39|| for a wedgie. It extends naturally to an 11-limit version <<8 1 18 20 ,,,|| which also tempers out 99/98, 176/175 and 243/242. [[127edo]] is again an excellent tuning for 11-limit wurschmidt, as well as for minerva, the 11-limit rank three temperament tempering out 99/98 and 176/175.

[[Mapping]]: [<1 7 3 15|, <0 -8 -1 -18|]

[[EDO|Vals]]: {{Val list| 31, 96, 127, 285bd, 412bbdd }}

=== 11-limit ===

Mapping: [<1 7 3 15 17|, <0 -8 -1 -18 -20|]

=== 13-limit ===

Mapping: [<1 7 3 15 17 1|, <0 -8 -1 -18 -20 4|]

=== Worseschmidt ===

Map: [<1 7 3 15 17 22|, <0 -8 -1 -18 -20 -27|]

Vals: {{EDOs| 3def, 28def, 31 }}

== Worschmidt ==

Worschmidt tempers out 126/125 rather than 225/224, and can use [[31edo]], [[34edo]], or [[127edo]] as a tuning. If 127 is used, note that the val is <127 201 295 356| and not <127 201 295 357| as with würschmidt. The wedgie now is <<8 1 -13 -17 -43 -33|. In practice, of course, both mappings could be used ambiguously, which might be an interesting avenue for someone to explore.

[[Mapping]]: [<1 7 3 -6|, <0 -8 -1 13|]

[[EDO|Vals]]: {{Val list| 31, 65, 96d, 127d }}

=== 11-limit ===

Mapping: [<1 7 3 -6 17|, <0 -8 -1 13 -20|]

== Whirrschmidt ==

[[99edo]] is such a good tuning for whirrschimdt that we hardly need look any farther. Unfortunately, the temperament while accurate is complex, with <<8 1 52 -17 60 118|| for a wedgie.

[[Mapping]]: [<1 7 3 38|, <0 -8 -1 -52|]

[[EDO|Vals]]: {{Val list| 31dd, 34d, 65, 99 }}

== Hemiwürschmidt ==

'''Hemiwürschmidt''' (sometimes spelled '''Hemiwuerschmidt'''), which splits the major third in two and uses that for a generator, is the most important of these temperaments even with the rather large complexity for the fifth. It tempers out [[3136/3125]], 6144/6125 and 2401/2400. [[68edo]], [[99edo]] and [[130edo]] can all be used as tunings, but 130 is not only the most accurate, it shows how hemiwürschmidt extends to a higher limit temperament, <<16 2 5 40 -39 -49 -48 28...

[[Mapping]]: [<1 15 4 7|, <0 -16 -2 -5|]

[[Wedgie]]: <<16 2 5 -34 -37 6||

[[EDO|Vals]]: {{Val list| 31, 68, 99, 229, 328, 557c, 885cc }}

[[Badness]]: 0.0203

=== 11-limit ===

Mapping: [<1 15 4 7 37|, <0 -16 -2 -5 -40|]

==== 13-limit ====

Mapping: [<1 15 4 7 37 -29|, <0 -16 -2 -5 -40 39|]

==== Hemithir ====

Mapping: [<1 15 4 7 37 -3|, <0 -16 -2 -5 -40 8|]

=== Hemiwur ===

Mapping: [<1 15 4 7 11|, <0 -16 -2 -5 -9|]

==== 13-limit ====

Mapping: [<1 15 4 7 11 -3|, <0 -16 -2 -5 -9 8|]

==== Hemiwar ====

Mapping: [<1 15 4 7 11 23|, <0 -16 -2 -5 -9 -23|]

== Relationships to other temperaments ==