Würschmidt family: Difference between revisions

The '''würschmidt family''' (würschmidt is sometimes spelled '''wuerschmidt''') of [[regular temperament|temperaments]] [[tempering out|tempers out]] [[393216/390625]], known as Würschmidt's comma, and named after José Würschmidt.  

== Würschmidt ==

The [[generator]] of würschmidt is a [[5/4|classical major third]], and to get to the interval class of the [[3/1|3rd harmonic]] requires eight of these. In fact, (5/4)⁸ × 393216/390625 = 6.  

Similar to [[meantone]], würschmidt implies that [[3/2]] will be tempered flat and/or 5/4 will be tempered sharp, and therefore [[6/5]] will be tempered flat. Unlike meantone, it is far more accurate. Combining würschmidt with meantone gives [[31edo]] as the first practical tuning with a generator of 10\31, but increasingly good 5-limit edo generators are [[34edo|11\34]] and especially [[65edo|21\65]], which notably is the point where it is combined with [[schismic]] and [[gravity]]. Other edo tunings include [[96edo]], [[99edo]] and [[164edo]]. Another tuning solution is to sharpen the major third by 1/8 of a würschmidt comma, which is to say by 1.43 cents, and thereby achieve pure perfect fifths; this is the [[minimax tuning]] for the [[5-odd-limit]].

[[Mos scale]]s may not be the best approach for würschmidt since they are even more extreme than those of [[magic]]. [[Rothenberg propriety|Rothenberg-proper]] scales do not appear until 28, 31 or even 34 notes, depending on the specific tuning.

[[Comma list]]: 393216/390625

: mapping generators: ~2, ~5/4

* [[WE]]: ~2 = 1199.6942{{c}}, ~5/4 = 387.7005{{c}}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~5/4 = 387.7762{{c}}

{{Optimal ET sequence|legend=1| 3, …, 28, 31, 34, 65, 99, 164, 721c, 885c, 1049cc, 1213ccc }}

The 7-limit extensions can be obtained by adding another comma. Septimal würschmidt adds [[225/224]], worschmidt adds [[126/125]], whirrschmidt adds [[4375/4374]]. These all use the same generator as 5-limit würschmidt.  

Hemiwürschmidt adds [[3136/3125]] and splits the generator in two. This temperament is the best extension available for würschmidt despite its complexity. The details can be found in [[Hemimean clan #Hemiwürschmidt|Hemimean clan]].

Given that würschmidt naturally produces a neutral third at the interval 4 generators up, an obvious extension to prime 11 exists by equating this to [[11/9]], that is by tempering out [[5632/5625]] in addition to [[243/242]].  

With this accuracy level of [[3/2]] available, extensions that add prime [[19/1|19]] exist by tempering out either [[513/512]] or [[1216/1215]] (which meet at 65edo), but they are very complex.

The "free" higher prime for würschmidt outside the 5-limit is in fact [[23/1|23]], via tempering out [[576/575]] ({{S|24}}) and [[12167/12150]] ([[S-expression|S46²⋅S47]]). This is considered immediately below, and the no-11 restriction thereof is considered in [[#Other subgroup extensions]].

Subgroup: 2.3.5.11

Optimal tuning:  

* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 387.6697{{c}}

{{Optimal ET sequence|legend=0| 31, 34, 65 }}

Badness (Sintel): 0.477

Subgroup: 2.3.5.11.23

Optimal tuning:  

* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 387.6770{{c}}

{{Optimal ET sequence|legend=0| 31, 34, 65 }}

Badness (Sintel): 0.300

Septimal würschmidt, aside from the commas listed above, also tempers out [[225/224]]. [[31edo]] or [[127edo]] can be used as tunings. It extends naturally to an 11-limit version which also tempers out [[99/98]], [[176/175]] and 243/242. 127edo is again an excellent tuning for 11-limit würschmidt, as well as for [[minerva]], the 11-limit rank-3 temperament tempering out 99/98 and 176/175.

The S-expression-based comma list of the 11-limit würschmidt discussed here is {[[176/175|S8/S10]], [[243/242|S9/S11]], [[225/224|S15]]}. Tempering out [[81/80|S9]] or [[121/120|S11]] results in [[31edo]], and in complementary fashion, tempering out [[64/63|S8]] or [[100/99|S10]] results in [[34edo]] in the 34d [[val]], where we accept [[17edo]]'s mapping of prime 7. Their val sum, 31 + 34d = 65d, thus observes all of these [[square superparticular]]s by tempering them together. As a result, [[65edo]] is especially structurally natural for this temperament, though high damage on the 7; even so, it is fairly close to the optimal tuning already if you are fine with a significantly flat ~9/7, which has the advantage of ~14/11 more in tune. However, as 31edo is relatively in-tune already, [[96edo]] (= 65d + 31) is also a reasonable choice, as it has the advantage of being a [[patent val]] in the 11-limit, though it uses a different (more accurate) mapping for 13.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 225/224, 8748/8575

{{Mapping|legend=1| 1 -1 2 -3 | 0 8 1 18 }}

[[Optimal tuning]]s:  

: error map: {{val| 0.000 -2.907 +1.067 +4.031 }}

{{Optimal ET sequence|legend=1| 31, 96, 127 }}

[[Badness]] (Sintel): 1.28

Subgroup: 2.3.5.7.11

Comma list: 99/98, 176/175, 243/242

{{Mapping|legend=0| 1 -1 2 -3 -3 | 0 8 1 18 20 }}

Optimal tunings:  

* WE: ~2 = 1199.9618{{c}}, ~5/4 = 387.4347{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 387.4446{{c}}

{{Optimal ET sequence|legend=0| 31, 65d, 96, 127 }}

Subgroup: 2.3.5.7.11.13

Comma list: 99/98, 144/143, 176/175, 275/273

{{Mapping|legend=0| 1 -1 2 -3 -3 5 | 0 8 1 18 20 -4 }}

Optimal tunings:  

* WE: ~2 = 1199.0325{{c}}, ~5/4 = 387.3137{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 387.5893{{c}}

{{Optimal ET sequence|legend=0| 31, 65d }}

Subgroup: 2.3.5.7.11.13

Commas: 66/65, 99/98, 105/104, 243/242

{{Mapping|legend=0| 1 -1 2 -3 -3 -5 | 0 8 1 18 20 27 }}

Optimal tunings:  

* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 387.1252{{c}}

{{Optimal ET sequence|legend=0| 3def, 28def, 31 }}

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 {{val| 127 201 295 '''356''' }} (127d) and not {{val| 127 201 295 '''357''' }} as with würschmidt. In practice, of course, both mappings could be used ambiguously, which might be an interesting avenue for someone to explore.

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 126/125, 33075/32768

{{Mapping|legend=1| 1 -1 2 7 | 0 8 1 -13 }}

* [[WE]]: ~2 = 1200.000{{c}}, ~5/4 = 387.406{{c}}

: [[error map]]: {{val| +0.763 -1.610 +2.851 -2.786 }}

: error map: {{val| 0.000 -2.795 +1.081 -4.960 }}

{{Optimal ET sequence|legend=1| 31, 96d, 127d }}

[[Badness]] (Sintel): 1.64

=== 11-limit ===

Comma list: 126/125, 243/242, 385/384

{{Mapping|legend=0| 1 -1 2 7 -3 | 0 8 1 -13 20 }}

Optimal tunings:  

* WE: ~2 = 1200.7911{{c}}, ~5/4 = 387.6624{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 387.4189{{c}}

{{Optimal ET sequence|legend=0| 31, 65, 96d, 127d }}

Badness (Sintel): 1.11

== Whirrschmidt ==

[[Subgroup]]: 2.3.5.7

[[Comma list]]: 4375/4374, 393216/390625

{{Mapping|legend=1| 1 -1 2 -14 | 0 8 1 52 }}

[[Optimal tuning]]s:  

* [[WE]]: ~2 = 1199.5703{{c}}, ~5/4 = 387.7422{{c}}

: [[error map]]: {{val| -0.430 +0.412 +0.569 -0.216 }}

* [[CWE]]: ~2 = 1200.0000{{c}}, ~5/4 = 387.8729{{c}}

: error map: {{val| 0.000 +1.029 +1.559 +0.567 }}

Subgroup: 2.3.5.7.11

Comma list: 243/242, 896/891, 4375/4356

{{Mapping|legend=0| 1 -1 2 -14 -3 | 0 8 1 52 20 }}

Optimal tunings:  

* WE: ~2 = 1199.2839{{c}}, ~5/4 = 387.6507{{c}}

* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 387.8682{{c}}

{{Optimal ET sequence|legend=0| 34d, 65, 99e }}

Extensions for harmonics [[47/1|47]] and [[49/1|49]] are also available at +11 and +5 generator steps respectively, equalizing the 45::50 [[harmonic series segment|segment]] of the [[harmonic series]]. If we derive a mapping of prime [[7/1|7]] from this mapping of 49, then we get the weak extension [[hemiwürschmidt]].

Subgroup: 2.3.5.23

Comma list: 576/575, 12167/12150

{{Mapping|legend=2| 1 -1 2 0 | 0 8 1 14 }}

Optimal tunings:  

* CWE: ~2 = 1200.0000{{c}}, ~5/4 = 387.7807{{c}}

{{Optimal ET sequence|legend=0| 31, 34, 65, 99, 164 }}

Badness (Sintel): 0.216

[[Category:Temperament families]]

[[Category:Würschmidt family| ]] <!-- main article -->

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

[[Category:Catalogs of rank-2 temperaments]]