Xenharmonic Wiki

Würschmidt: Difference between revisions

← Older edit
VisualWikitext
Godtone (talk | contribs)
autopatrolled, emailconfirmed
2,412 edits
m if prime 11 was included (which i think it should be but idk how to calculate minimax), then a 31 note (not 34) MOS would make sense to reach 3 * 11 = 33 otonally. as is, 23 is at 14 gens, so 25 notes is very generous (barely allows 27 = 3^3 to be reached)
Hstraub (talk | contribs)
autopatrolled, Sysops
260 edits
Link to German
Tags: Mobile edit Mobile web edit
 
(11 intermediate revisions by 3 users not shown)
Line 1: Line 1:
{{Infobox Regtemp
{{interwiki
| de = Würschmidt
| en = Würschmidt
}}
{{Infobox regtemp
| Title = Würschmidt
| Title = Würschmidt
| Subgroups = 2.3.5, 2.3.5.23
| Subgroups = 2.3.5, 2.3.5.23
| Comma basis = [[393216/390625]] (2.3.5); <br> [[576/575]], [[12167/12150]] (2.3.5.23)
| Comma basis = [[393216/390625]] (2.3.5); <br>[[576/575]], [[12167/12150]] (2.3.5.23)
| Edo join 1 = 31 | Edo join 2 = 34
| Edo join 1 = 31 | Edo join 2 = 34
| Generator = 5/4 | Generator tuning = 387.734 | Optimization method = CTE
| Mapping = 1; 8 1 14
| Generators = 5/4 | Generators tuning = 387.8 | Optimization method = CWE
| MOS scales = [[3L 1s]], [[3L 4s]], …, [[3L 28s]], [[31L 3s]]
| MOS scales = [[3L 1s]], [[3L 4s]], …, [[3L 28s]], [[31L 3s]]
| Mapping = 1; 8 1 14
| Pergen = (P8, ccP5/8)
| Pergen = (P8, ccP5/8)
| Color name = Saquadbiguti
| Color name = Saquadbiguti
| Odd limit 1 = 5 | Mistuning 1 = 1.43 | Complexity 1 = 19
| Odd limit 1 = 5 | Mistuning 1 = 1.43 | Complexity 1 = 19
| Odd limit 2 = (2.3.5.23) 25 | Mistuning 2 = 2.86 | Complexity 2 = 25
| Odd limit 2 = 2.3.5.23 25 | Mistuning 2 = 2.86 | Complexity 2 = 25
}}
}}
'''Würschmidt''' is a [[rank-2 temperament|rank-2]] [[regular temperament|temperament]] and parent of the [[würschmidt family]], characterized by tempering out the [[würschmidt comma]] ([[ratio]]: 393216/390625, {{monzo|legend=1| 17 1 -8 }}). It can be treated as analogous to [[schismatic]] with the roles of the primes 3 and 5 reversed, since würschmidt is [[generator|generated]] by a [[5/4|classical major third (5/4)]], very slightly sharpened so that eight of them make the sixth harmonic ([[6/1]]), giving [[3/2]] the same complexity [[5/4]] does in schismatic, but with comparable accuracy on the part of the generator. Four generators, therefore, reach the interval [[625/512]], which is equated to [[768/625]] and functions as a neutral third.
'''Würschmidt''' is a [[rank-2 temperament|rank-2]] [[regular temperament|temperament]] and parent of the [[würschmidt family]], characterized by tempering out the [[würschmidt comma]] ([[ratio]]: 393216/390625, {{monzo|legend=1| 17 1 -8 }}). It can be treated as analogous to [[schismic]] with the roles of the primes 3 and 5 reversed, since würschmidt is [[generator|generated]] by a [[5/4|classical major third (5/4)]], very slightly sharpened so that eight of them make the sixth harmonic ([[6/1]]), giving [[3/2]] the same complexity [[5/4]] does in schismic, but with comparable accuracy on the part of the generator. Four generators, therefore, reach the interval [[625/512]], which is equated to [[768/625]] and functions as a neutral third.


{{Tdlink|Würschmidt family #Würschmidt}}
{{Tdlink|Würschmidt family #Würschmidt}}
Line 19: Line 23:
Another useful interpretation of the würschmidt comma is that it makes the interval of [[25/24]] equal to two-thirds the size of [[16/15]]. This can be exploited, as 16/15 factorizes into near-2:1 parts as {{nowrap|([[24/23]]) &times; ([[46/45]])}}, and therefore it is illogical ''not'' to set 25/24 equal to 24/23 (and [[128/125]] equal to 46/45) as well and set the remainder, 46/45, equal to a third of 16/15, by tempering {{nowrap|S24 {{=}} [[576/575]]}} and {{nowrap|S46<sup>2</sup> &times; S47 {{=}} [[12167/12150]]}} in the 2.3.5.23 [[subgroup]]. A perhaps more direct way of seeing why equating 25/24 with 24/23 is natural is that würschmidt's generator is a slightly sharpened 5/4 with a slightly flat 3/2 in an optimised tuning, so that 25/24 is sharpened and 24/n is flattened, so equating it with 24/23 takes advantage of the natural tempering tendency. 14 generators turn out to stack to [[23/1]], and notably, 6/1 stacked 7 times and 23/1 stacked four times (at 56 generators) differ only by the 0.59-cent comma [[279936/279841]] = {{nowrap|[[2401/2400|S49]] / ([[25921/25920|S161]]<sup>2</sup>)}}.
Another useful interpretation of the würschmidt comma is that it makes the interval of [[25/24]] equal to two-thirds the size of [[16/15]]. This can be exploited, as 16/15 factorizes into near-2:1 parts as {{nowrap|([[24/23]]) &times; ([[46/45]])}}, and therefore it is illogical ''not'' to set 25/24 equal to 24/23 (and [[128/125]] equal to 46/45) as well and set the remainder, 46/45, equal to a third of 16/15, by tempering {{nowrap|S24 {{=}} [[576/575]]}} and {{nowrap|S46<sup>2</sup> &times; S47 {{=}} [[12167/12150]]}} in the 2.3.5.23 [[subgroup]]. A perhaps more direct way of seeing why equating 25/24 with 24/23 is natural is that würschmidt's generator is a slightly sharpened 5/4 with a slightly flat 3/2 in an optimised tuning, so that 25/24 is sharpened and 24/n is flattened, so equating it with 24/23 takes advantage of the natural tempering tendency. 14 generators turn out to stack to [[23/1]], and notably, 6/1 stacked 7 times and 23/1 stacked four times (at 56 generators) differ only by the 0.59-cent comma [[279936/279841]] = {{nowrap|[[2401/2400|S49]] / ([[25921/25920|S161]]<sup>2</sup>)}}.


Strong extensions to the [[7-limit]] include [[würschmidt family#septimal würschmidt|septimal würschmidt]] (tempering out [[225/224]], finding 7 at +18 generator steps), [[worschmidt]] (tempering out [[126/125]], finding 7 at -13 generator steps), and [[whirrschmidt]] (tempering out [[4375/4374]], finding 7 at +52 generator steps), but these are either considerably higher-damage or much higher-complexity than 5-limit würschmidt. In fact, the best septimal extension may be the weak extension [[hemiwürschmidt]], which splits the ~5/4 generator into two ~[[28/25]]'s by tempering out [[3136/3125]] alongside [[6144/6125]] (notably, in the 2.3.5.7.23 subgroup, this is the extension that tempers out the tiny comma {{nowrap|S161 {{=}} [[25921/25920]]}}).  
Strong extensions to the [[7-limit]] include [[würschmidt family#septimal würschmidt|septimal würschmidt]] (tempering out [[225/224]], finding 7 at +18 generator steps), [[worschmidt]] (tempering out [[126/125]], finding 7 at -13 generator steps), and [[whirrschmidt]] (tempering out [[4375/4374]], finding 7 at +52 generator steps), but these are either considerably higher-damage or much higher-complexity than 5-limit würschmidt. In fact, the best septimal extension may be the weak extension [[hemiwürschmidt]], which splits the ~5/4 generator into two ~[[28/25]]'s by tempering out [[3136/3125]] alongside [[6144/6125]] and [[2401/2400]] (notably, in the 2.3.5.7.23 subgroup, this is the extension that tempers out the tiny comma {{nowrap|S161 {{=}} [[25921/25920]]}}).  


Therefore, it may be advisable to consider würschmidt a no-sevens system, specifically in the 2.3.5.11 subgroup, where an extension that equates 128/125 with [[45/44]] and therefore 625/512 with [[11/9]] (by tempering out [[243/242]] and [[5632/5625]]), finding the 11th harmonic at 20 generators up, is highly natural in addition to the aforementioned extension to prime 23.
Therefore, it may be advisable to consider würschmidt a no-sevens system, specifically in the 2.3.5.11 subgroup, where an extension that equates 128/125 with [[45/44]] and therefore 625/512 with [[11/9]] (by tempering out [[243/242]] and [[5632/5625]]), finding the 11th harmonic at 20 generators up, is highly natural in addition to the aforementioned extension to prime 23.


== Interval chain ==
== Interval chain ==
In the below, octave-reduced harmonics 1–125 are indicated in '''bold'''. All intervals are in the 625-[[odd limit]].
In the below, octave-reduced harmonics 1–33 are indicated in '''bold'''. All intervals are in the 165-[[odd limit]].


{| class="wikitable center-1 right-2"
{| class="wikitable center-all right-2"
|-
|-
! rowspan="2" | &#35; !! rowspan="2" | Cents* !! colspan="2" | Approximate ratios
! rowspan="2" | &#35; !! rowspan="2" | Cents* !! colspan="2" | Approximate ratios
Line 36: Line 40:
| 1 || 387.652 || '''5/4''', 144/115 || 69/55
| 1 || 387.652 || '''5/4''', 144/115 || 69/55
|-
|-
| 2 || 775.304 || '''25/16''', 36/23 || 69/44, 352/225
| 2 || 775.304 || '''25/16''', 36/23 || 69/44
|-
|-
| 3 || 1162.956 || 45/23, '''125/64''', 736/375 || 88/45, 108/55
| 3 || 1162.956 || 45/23, 125/64 || 88/45, 108/55
|-
|-
| 4 || 350.608 || 92/75, 225/184, 625/512 || 11/9, 27/22
| 4 || 350.608 || 92/75 || 11/9, 27/22
|-
|-
| 5 || 738.260 || 23/15, 192/125 || 55/36, 135/88
| 5 || 738.260 || 23/15, 192/125 || 55/36, 135/88
Line 50: Line 54:
| 8 || 701.216 || '''3/2''' || 121/81
| 8 || 701.216 || '''3/2''' || 121/81
|-
|-
| 9 || 1088.868 || '''15/8''', 216/115 || 207/110, 253/135
| 9 || 1088.868 || '''15/8''', 216/115 ||  
|-
|-
| 10 || 276.520 || 27/23, '''75/64''' || 88/75, 207/176
| 10 || 276.520 || 27/23, 75/64 || 88/75
|-
|-
| 11 || 664.172 || 184/125, 135/92, 375/256 || 22/15, 81/55
| 11 || 664.172 || 184/125, 135/92 || 22/15, 81/55
|-
|-
| 12 || 1051.824 || 46/25, 675/368 || 11/6, 81/44
| 12 || 1051.824 || 46/25 || 11/6, 81/44
|-
|-
| 13 || 239.476 || 23/20, 144/125 || 55/48, 132/115
| 13 || 239.476 || 23/20, 144/125 || 55/48, 132/115
Line 62: Line 66:
| 14 || 627.128 || '''23/16''', 36/25 || 33/23
| 14 || 627.128 || '''23/16''', 36/25 || 33/23
|-
|-
| 15 || 1014.780 || 9/5, '''115/64''' || 165/92, 242/135
| 15 || 1014.780 || 9/5, 115/64 || 165/92, 242/135
|-
|-
| 16 || 202.432 || '''9/8''' || 121/108
| 16 || 202.432 || '''9/8''' || 121/108
|-
|-
| 17 || 590.084 || '''45/32''', 162/115 || 253/180
| 17 || 590.084 || 45/32, 162/115 ||  
|-
|-
| 18 || 977.736 || 81/46, 225/128 || 44/25
| 18 || 977.736 || 81/46 || 44/25
|-
|-
| 19 || 165.388 || 138/125, 405/368 || 11/10, 243/220
| 19 || 165.388 || 138/125 || 11/10
|-
|-
| 20 || 553.040 || 69/50, 864/625 || '''11/8''', 243/176
| 20 || 553.040 || 69/50 || '''11/8'''
|-
|-
| 21 || 940.692 || 69/40, 216/125 || '''55/32'''
| 21 || 940.692 || 69/40, 216/125 || 55/32
|-
|-
| 22 || 128.344 || 27/25, '''69/64''' || 99/92
| 22 || 128.344 || 27/25, 69/64 || 99/92
|-
|-
| 23 || 515.996 || 27/20, 345/256 ||  
| 23 || 515.996 || 27/20 ||  
|-
|-
| 24 || 903.648 || '''27/16''' || 253/150
| 24 || 903.648 || '''27/16''' ||  
|-
|-
| 25 || 91.300 || 135/128, 243/230 || 132/125, 253/240
| 25 || 91.300 || 135/128 || 132/125
|-
|-
| 26 || 478.952 || 243/184, 828/625 || 33/25, 253/192
| 26 || 478.952 || || 33/25
|-
|-
| 27 || 866.604 || 207/125 || 33/20
| 27 || 866.604 || 207/125 || 33/20
|-
|-
| 28 || 54.256 || 207/200, 648/625 || '''33/32'''
| 28 || 54.256 || || '''33/32'''
|-
|-
| 29 || 441.908 || 162/125, 207/160 || 165/128
| 29 || 441.908 || 162/125 || 165/128
|-
|-
| 30 || 829.560 || 81/50, 207/128 || 121/75
| 30 || 829.560 || 81/50 || 121/75
|-
|-
| 31 || 17.212 || 81/80 || 121/120
| 31 || 17.212 || 81/80 || 121/120
Line 101: Line 105:
=== Optimized tunings ===
=== Optimized tunings ===
{| class="wikitable mw-collapsible mw-collapsed"
{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | Prime-optimized tunings
|+ style="font-size: 105%; white-space: nowrap;" | Norm-based tunings
|-
|-
! rowspan="2" | Weight-skew\Order !! colspan="2" | Euclidean
! rowspan="2" | Weight-skew\Order !! colspan="2" | Euclidean
Line 143: Line 147:
{| class="wikitable center-all left-4"
{| class="wikitable center-all left-4"
! Edo<br />generator
! Edo<br />generator
! [[Eigenmonzo|Eigenmonzo<br />(unchanged-interval)]]*
! [[Eigenmonzo|Eigenmonzo<br />(unchanged interval)]]*
! Generator (¢)
! Generator (¢)
! Comments
! Comments
Line 150: Line 154:
|  
|  
| 385.7143
| 385.7143
| 28ei val
| 28ei val, major thirds slightly flatter than this fall under 25&28 or [[magic]]
|-
|-
|  
|  
Line 350: Line 354:
|  
|  
| '''400.0000'''
| '''400.0000'''
| '''Upper bound of 2.3.5.23-subgroup 25-odd-limit diamond monotone'''
| '''Upper bound of 2.3.5.23-subgroup 25-odd-limit diamond monotone''', major thirds slightly sharper than this fall under [[smate_family|smate]]
|}
|}
<nowiki />* Besides the octave
<nowiki />* Besides the octave
Line 365: Line 369:
* ''Extrospection'' (2013) – [https://web.archive.org/web/20201127013550/http://micro.soonlabel.com/gene_ward_smith/Others/Freivald/Wurschmidt{{lbrack}}16{{rbrack}}-out.mp3 play] | [https://soundcloud.com/jdfreivald/extrospection SoundCloud] – Würschmidt[16] in 31edo tuning.
* ''Extrospection'' (2013) – [https://web.archive.org/web/20201127013550/http://micro.soonlabel.com/gene_ward_smith/Others/Freivald/Wurschmidt{{lbrack}}16{{rbrack}}-out.mp3 play] | [https://soundcloud.com/jdfreivald/extrospection SoundCloud] – Würschmidt[16] in 31edo tuning.


[[Category:Temperaments]]
[[Category:Würschmidt| ]] <!-- main article -->
[[Category:Rank-2 temperaments]]
[[Category:Würschmidt family]]
[[Category:Würschmidt family]]
Retrieved from "https://en.xen.wiki/w/Würschmidt"