Würschmidt: Difference between revisions

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I don't see why negative steps are included nor why they're not shown up to the same number as the positive steps. -table note template in favor of plain notes. There's also way too many bolded stuff
Lériendil (talk | contribs)
8,044 edits
unilateral undo; negative steps are shown to -8 as they are simpler than positive steps 21 to 28 for completing the 28-note MOS, and bolding all harmonics in the interval chain is convention
Tag: Undo
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'''Würschmidt''' is a [[rank-2 temperament|rank-2]] [[regular temperament|temperament]] and parent of the [[würschmidt family]], characterized by tempering out the würschmidt comma, [[393216/390625]]. 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, ([[393216/390625]]). 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.


{{Tdlink|Würschmidt family #Würschmidt}}
{{tdlink|Würschmidt family #Würschmidt}}


== Extensions ==
== Extensions ==
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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 chains ==
In the below, octave-reduced harmonics 1–23 are indicated in '''bold'''.
In the below, octave-reduced harmonics below 125 are indicated in '''bold'''.


<div><div style="display: inline-grid; margin-right: 25px;">
{| class="wikitable center-1 right-2"
{| class="wikitable center-1 right-2"
|+ style="font-size: 105%;" | Würschmidt
|-
|-
! rowspan="2" | # !! rowspan="2" | Cents* !! colspan="2" | Approximate Ratios
! rowspan="2" | &#35; !! rowspan="2" | Cents&#42; !! colspan="2" | Approximate Ratios
|-
|-
! 2.3.5.23 Subgroup !! Add-11 Extension
! 2.3.5.23 subgroup !! Add-11 extension
|-
| -8 || 497.59 || 4/3 || 162/121
|-
| -7 || 885.39 || 5/3, 192/115 || 92/55
|-
| -6 || 73.19 || 24/23, 25/24 || 23/22, 288/275
|-
| -5 || 460.99 || 30/23, 125/96 || 72/55, 176/135
|-
| -4 || 848.79 || 75/46, 368/225, 625/384 || 18/11, 44/27
|-
| -3 || 36.60 || 46/45, 128/125 || 45/44, 55/54
|-
| -2 || 424.40 || 23/18, 32/25 || 88/69, 225/176
|-
| -1 || 812.20 || 8/5, 115/72 || 110/69
|-
|-
| 0 || 0.0 || '''1/1''' ||
| 0 || 0.0 || '''1/1''' ||
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| 1 || 387.80 || '''5/4''', 144/115 || 69/55
| 1 || 387.80 || '''5/4''', 144/115 || 69/55
|-
|-
| 2 || 775.60 || 25/16, 36/23 || 69/44, 352/225
| 2 || 775.60 || '''25/16''', 36/23 || 69/44, 352/225
|-
|-
| 3 || 1163.40 || 45/23, 125/64, 736/375 || 88/45, 108/55
| 3 || 1163.40 || 45/23, '''125/64''', 736/375 || 88/45, 108/55
|-
|-
| 4 || 351.21 || 92/75, 225/184, 625/512 || 11/9, 27/22
| 4 || 351.21 || 92/75, 225/184, 625/512 || 11/9, 27/22
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| 9 || 1090.21 || '''15/8''', 216/115 || 207/110, 253/135
| 9 || 1090.21 || '''15/8''', 216/115 || 207/110, 253/135
|-
|-
| 10 || 278.01 || 27/23, 75/64 || 88/75, 207/176
| 10 || 278.01 || 27/23, '''75/64''' || 88/75, 207/176
|-
|-
| 11 || 665.82 || 184/125, 135/92, 375/256 || 22/15, 81/55
| 11 || 665.82 || 184/125, 135/92, 375/256 || 22/15, 81/55
Line 49: Line 67:
| 14 || 629.22 || '''23/16''', 36/25 || 33/23, 275/192
| 14 || 629.22 || '''23/16''', 36/25 || 33/23, 275/192
|-
|-
| 15 || 1017.02 || 9/5, 115/64 || 165/92, 242/135
| 15 || 1017.02 || 9/5, '''115/64''' || 165/92, 242/135
|-
|-
| 16 || 204.82 || '''9/8''' || 121/108
| 16 || 204.82 || '''9/8''' || 121/108
|-
|-
| 17 || 592.62 || 45/32, 162/115 || 253/180
| 17 || 592.62 || '''45/32''', 162/115 || 253/180
|-
|-
| 18 || 980.43 || 81/46, 225/128 || 44/25
| 18 || 980.43 || 81/46, 225/128 || 44/25
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|-
|-
| 20 || 556.03 || 69/50, 864/625 || '''11/8''', 243/176
| 20 || 556.03 || 69/50, 864/625 || '''11/8''', 243/176
{{table notes|cols=4
| In 2.3.5-targeted [[DKW theory|DKW]] tuning
}}
|}
|}
<nowiki>*</nowiki> In 2.3.5-targeted [[DKW theory|DKW]] tuning
</div>


== Tunings ==
== Tunings ==

Revision as of 14:49, 3 October 2024

Würschmidt is a rank-2 temperament and parent of the würschmidt family, characterized by tempering out the würschmidt comma, (393216/390625). It can be treated as analogous to schismatic with the roles of the primes 3 and 5 reversed, since würschmidt is generated by a 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.

For technical data, see Würschmidt family #Würschmidt.

Extensions

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 (24/23)×(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 S24 = 576/575 and S462 × S47 = 12167/12150 in the 2.3.5.23 subgroup. 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.

Strong extensions to the 7-limit include 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 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.

Interval chains

In the below, octave-reduced harmonics below 125 are indicated in bold.

Würschmidt
# Cents* Approximate Ratios
2.3.5.23 subgroup Add-11 extension
-8 497.59 4/3 162/121
-7 885.39 5/3, 192/115 92/55
-6 73.19 24/23, 25/24 23/22, 288/275
-5 460.99 30/23, 125/96 72/55, 176/135
-4 848.79 75/46, 368/225, 625/384 18/11, 44/27
-3 36.60 46/45, 128/125 45/44, 55/54
-2 424.40 23/18, 32/25 88/69, 225/176
-1 812.20 8/5, 115/72 110/69
0 0.0 1/1
1 387.80 5/4, 144/115 69/55
2 775.60 25/16, 36/23 69/44, 352/225
3 1163.40 45/23, 125/64, 736/375 88/45, 108/55
4 351.21 92/75, 225/184, 625/512 11/9, 27/22
5 739.01 23/15, 192/125 55/36, 135/88
6 1126.81 23/12, 48/25 44/23, 275/144
7 314.61 6/5, 115/96 55/46
8 702.41 3/2 121/81
9 1090.21 15/8, 216/115 207/110, 253/135
10 278.01 27/23, 75/64 88/75, 207/176
11 665.82 184/125, 135/92, 375/256 22/15, 81/55
12 1053.62 46/25, 675/368 11/6, 81/44
13 241.42 23/20, 144/125 55/48, 132/115
14 629.22 23/16, 36/25 33/23, 275/192
15 1017.02 9/5, 115/64 165/92, 242/135
16 204.82 9/8 121/108
17 592.62 45/32, 162/115 253/180
18 980.43 81/46, 225/128 44/25
19 168.23 138/125, 405/368 11/10, 243/220
20 556.03 69/50, 864/625 11/8, 243/176

Template:Table notes

Tunings

Optimized tunings

Prime-Optimized Tunings
Weight-skew\Order Euclidean
Weil (2.3.5) CWE: ~5/4 = 387.776¢
Tenney (2.3.5) POTE: ~5/4 = 387.7993¢
Weil (2.3.5.23) CWE: ~6/5 = 387.781¢
Tenney (2.3.5.23) POTE: ~6/5 = 387.8051¢
DR and equal-beating tunings
Optimized chord Generator value Polynomial Further notes
3:4:5 (+1 +1) ~5/4 = 387.4975 g8 + 8g - 16 = 0 1-3-5 equal-beating tuning, close to 3/29-comma
4:5:6 (+1 +1) ~5/4 = 388.1207 g8 - 8g + 8 = 0 1-3-5 equal-beating tuning, close to 3/19-comma
10:12:15 (+2 +3) ~5/4 = 388.2216 g8 - 2g7 + 4 = 0 Close to 1/6-comma
15:18:23 (+3 +5) ~5/4 = 387.9215 4g7 - 3g5 - 10 = 0

Tuning spectrum

The below assumes the 2.3.5.11.23 subgroup extension. Note that "e" and "i" are the warts for primes 11 and 23, respectively.

Edo
Generator 		Eigenmonzo
(Unchanged-interval)* 		Generator (¢) Comments
9\28 385.7143 28ei val
11/9 386.3137 -1/4 vishdel comma
5/4 386.8520 Untempered tuning, lower bound of 5-odd-limit diamond tradeoff
10\31 387.0968 Lower bound of 2.3.5.23-subgroup 25-odd-limit diamond monotone
23/22 387.1739
375/256 387.3542 1/11-comma
41\127 387.4016 127e val
11/6 387.4469
75/64 387.4582 1/10-comma
31\96 387.5000
11/8 387.5659
52\161 387.5776
15/8 387.5854 1/9-comma
73\226 387.6106
11/10 387.6318
45/32 387.6602 2/17-comma
21\65 387.6923
23/12 387.7199
23/16 387.7338
116\359 387.7437 359ee val
3/2 387.7444 1/8-comma
74\229 387.7729 229e val
53\164 387.8049 164e val
23/18 387.8178 1/2 S24
85\263 387.8327 263ee val
9/5 387.8393 2/15-comma
23/20 387.8431
32\99 387.8788 99e val
75\232 387.9310 232eei val
6/5 387.9490 1/7-comma, upper bound of 5-odd-limit diamond tradeoff
43\133 387.9699 133e val
46/25 387.9706
23/15 388.0011
54\167 388.0240 167eei val
48/25 388.2213 1/6-comma, upper bound of 2.3.5.23-subgroup 25-odd-limit diamond tradeoff
11\34 388.2353
192/125 388.6028 1/5-comma
23\71 388.7324 71eei val
768/625 389.1750 1/4-comma
12\37 389.1892 37eei val
1\3 400.0000 Upper bound of 2.3.5.23-subgroup 25-odd-limit diamond monotone

* besides the octave

Other tunings

  • DKW (2.3.5): ~2 = 1\1, ~5/4 = 387.8015
  • 5-odd-limit minimax: ~2 = 1\1, ~5/4 = 387.7444 (eigenmonzo 3/2 aka 1/8-comma, generator = 61/8)

Music

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