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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.
{{Interwiki
| en = Würschmidt
| de = Würschmidt
}}
{{Infobox regtemp
| Title = Würschmidt
| Subgroups = 2.3.5, 2.3.5.11, 2.3.5.11.23
| Comma basis = [[393216/390625]] (2.3.5); <br>[[243/242]], [[5632/5625]] (2.3.5.11);<br>[[243/242]], [[276/275]], [[529/528]] (2.3.5.11.23)
| Edo join 1 = 31 | Edo join 2 = 34
| Mapping = 1; 8 1 20 14
| Generators = 5/4 | Generators tuning = 387.7 | Optimization method = CWE
| MOS scales = [[3L 1s]], [[3L 4s]], …, [[3L 28s]], [[31L 3s]]
| Pergen = (P8, ccP5/8)
| Color name = Saquadbiguti
| Odd limit 1 = 5 | Mistuning 1 = 1.43 | Complexity 1 = 10
| Odd limit 2 = 2.3.5.11.23 23 | Mistuning 2 = 3.12 | Complexity 2 = 22
}}
'''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]] has 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}}
For technical data, see [[Würschmidt family #Würschmidt]].


== Extensions ==
== 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 S46<sup>2</sup> × 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 [[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 [[2401/2400]] and [[6144/6125]].
 
Therefore, it may be advisable to consider würschmidt a no-7's 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.


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. 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.
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]])⋅([[46/45]])}}, and therefore, if one is interested in adding another prime to this temperament, 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 out [[576/575]] ({{S|24}}) and [[12167/12150]] ([[S-expression|S46<sup>2</sup>⋅S47]]). 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 equating it with 24/23 takes advantage of the natural tempering tendency. 14 generators turn out to stack to [[23/1]]. 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]] ([[2401/2400|S49]]/([[25921/25920|S161]]<sup>2</sup>)).


== Interval chains ==
== Interval chain ==
In the below, octave-reduced harmonics below 125 are indicated in '''bold'''.
In the below, octave-reduced harmonics 1–23 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" | &#35; !! rowspan="2" | Cents&#42; !! colspan="2" | Approximate Ratios
! rowspan="2" | # !! rowspan="2" | Cents* !! colspan="2" | Approximate ratios
|-
|-
! 2.3.5.23 subgroup !! Add-11 extension
! 5-limit !! Add-11 add-23 extension
|-
|-
| -8 || 497.59 || '''4/3''' || 162/121
| 0 || 0.0 || '''1/1''' ||
|-
|-
| -7 || 885.39 || 5/3, 192/115 || 92/55
| 1 || 387.8 || '''5/4''' ||  
|-
|-
| -6 || 73.19 || 24/23, 25/24 || 23/22, 288/275
| 2 || 775.6 || 25/16 || 36/23, 69/44
|-
|-
| -5 || 460.99 || 30/23, 125/96 || 72/55, 176/135
| 3 || 1163.3 || 125/64 || 45/23, 88/45, 108/55
|-
|-
| -4 || 848.79 || 75/46, 368/225, 625/384 || 18/11, 44/27
| 4 || 351.1 || 625/512, 768/625 || 11/9, 27/22
|-
|-
| -3 || 36.60 || 46/45, 128/125 || 45/44, 55/54
| 5 || 738.9 || 192/125 || 23/15, 55/36
|-
|-
| -2 || 424.40 || 23/18, 32/25 || 88/69, 225/176
| 6 || 1126.7 || 48/25 || 23/12, 44/23
|-
|-
| -1 || 812.20 || 8/5, 115/72 || 110/69
| 7 || 314.4 || 6/5 ||  
|-
|-
| 0 || 0.0 || '''1/1''' ||
| 8 || 702.2 || '''3/2''' ||
|-
| 9 || 1090.0 || '''15/8''' ||  
|-
| 10 || 277.8 || 75/64 || 27/23, 88/75
|-
| 11 || 665.5 || 375/256 || 22/15, 81/55
|-
|-
| 1 || 387.80 || '''5/4''', 144/115 || 69/55
| 12 || 1053.3 || 1152/625, 1875/1024 || 11/6, 46/25, 81/44
|-
|-
| 2 || 775.60 || '''25/16''', 36/23 || 69/44, 352/225
| 13 || 241.1 || 144/125 || 23/20, 55/48
|-
|-
| 3 || 1163.40 || 45/23, '''125/64''', 736/375 || 88/45, 108/55
| 14 || 628.9 || 36/25 || '''23/16''', 33/23
|-
|-
| 4 || 351.21 || 92/75, 225/184, 625/512 || 11/9, 27/22
| 15 || 1016.6 || 9/5 ||  
|-
|-
| 5 || 739.01 || 23/15, 192/125 || 55/36, 135/88
| 16 || 204.4 || '''9/8''' ||  
|-
|-
| 6 || 1126.81 || 23/12, 48/25 || 44/23, 275/144
| 17 || 592.2 || 45/32 ||  
|-
|-
| 7 || 314.61 || 6/5, 115/96 || 55/46
| 18 || 980.0 || 225/128 || 44/25, 81/46
|-
|-
| 8 || 702.41 || '''3/2''' || 121/81
| 19 || 167.7 || 1125/1024 || 11/10
|-
|-
| 9 || 1090.21 || '''15/8''', 216/115 || 207/110, 253/135
| 20 || 555.5 || 864/625 || '''11/8'''
|-
|-
| 10 || 278.01 || 27/23, '''75/64''' || 88/75, 207/176
| 21 || 943.3 || 216/125 || 55/32
|-
|-
| 11 || 665.82 || 184/125, 135/92, 375/256 || 22/15, 81/55
| 22 || 131.1 || 27/25 || 69/64, 99/92
|-
|-
| 12 || 1053.62 || 46/25, 675/368 || 11/6, 81/44
| 23 || 518.9 || 27/20 ||  
|-
|-
| 13 || 241.42 || 23/20, 144/125 || 55/48, 132/115
| 24 || 906.6 || 27/16 ||  
|-
|-
| 14 || 629.22 || '''23/16''', 36/25 || 33/23, 275/192
| 25 || 94.4 || 135/128 || 132/125
|-
|-
| 15 || 1017.02 || 9/5, '''115/64''' || 165/92, 242/135
| 26 || 482.2 || 675/512 || 33/25
|-
|-
| 16 || 204.82 || '''9/8''' || 121/108
| 27 || 870.0 || 3375/2048, 5184/3125 || 33/20
|-
|-
| 17 || 592.62 || '''45/32''', 162/115 || 253/180
| 28 || 57.7 || 648/625 || 33/32
|-
|-
| 18 || 980.43 || 81/46, 225/128 || 44/25
| 29 || 445.5 || 162/125 || 165/128
|-
|-
| 19 || 168.23 || 138/125, 405/368 || 11/10, 243/220
| 30 || 833.3 || 81/50 || 121/75
|-
|-
| 20 || 556.03 || 69/50, 864/625 || '''11/8''', 243/176
| 31 || 21.1 || 81/80 || 121/120
{{table notes|cols=4
| In 2.3.5-targeted [[DKW theory|DKW]] tuning
}}
|}
|}
</div>
<nowiki/>* In 5-limit [[CWE]] tuning


== Tunings ==
== Tunings ==
=== Optimized tunings ===
=== Norm-based 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;" | 5-limit norm-based tunings
|-
! rowspan="2" |  !! colspan="3" | Euclidean
|-
|-
! Weight-skew\Order !! Euclidean
! Constrained !! Constrained & skewed !! Destretched
|-
|-
| Weil || (2.3.5) CWE: ~5/4 = 387.776¢
! Tenney
| CTE: ~5/4 = 387.734{{c}} || CWE: ~5/4 = 387.776{{c}} || POTE: ~5/4 = 387.7993{{c}}
|-
|-
| Tenney || (2.3.5) POTE: ~5/4 = 387.7993¢
! Equilateral
| CEE: ~5/4 = 387.7224{{c}}<br>(8/65-comma) ||  ||
|}
{| class="wikitable mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | 5-limit add-23 norm-based tunings
|-
|-
| Weil || (2.3.5.23) CWE: ~6/5 = 387.781¢
! rowspan="2" | !! colspan="3" | Euclidean
|-
|-
| Tenney || (2.3.5.23) POTE: ~6/5 = 387.8051¢
! Constrained !! Constrained & skewed !! Destretched
|-
! Tenney
| CTE: ~5/4 = 387.734{{c}} || CWE: ~5/4 = 387.781{{c}} ||POTE: ~5/4 = 387.8051{{c}}
|}
|}


=== Target tunings ===
{| class="wikitable mw-collapsible mw-collapsed"
{| class="wikitable mw-collapsible mw-collapsed"
|+style="font-size: 105%; white-space: nowrap;" | [[Delta-rational chord|DR]] and equal-beating tunings
|+ style="font-size: 105%; white-space: nowrap;" | [[Delta-rational chord|DR]] and equal-beating tunings
|-
|-
! Optimized chord !! Generator value !! Polynomial !! Further notes
! Optimized chord !! Generator value !! Polynomial !! Further notes
|-
|-
| 3:4:5 (+1 +1) || ~5/4 = 387.4975 || ''g''<sup>8</sup> + 8''g'' - 16 = 0 || 1-3-5 equal-beating tuning, close to 3/29-comma
| 3:4:5 (+1 +1) || ~5/4 = 387.4975 || ''g''<sup>8</sup> + 8''g'' &minus; 16 = 0 || {{dash|1, 3, 5|med}} equal-beating tuning, close to 3/29-comma
|-
|-
| 4:5:6 (+1 +1) || ~5/4 = 388.1207 || ''g''<sup>8</sup> - 8''g'' + 8 = 0 || 1-3-5 equal-beating tuning, close to 3/19-comma
| 4:5:6 (+1 +1) || ~5/4 = 388.1207 || ''g''<sup>8</sup> &minus; 8''g'' + 8 = 0 || {{dash|1, 3, 5|med}} equal-beating tuning, close to 3/19-comma
|-
|-
| 10:12:15 (+2 +3) || ~5/4 = 388.2216 || ''g''<sup>8</sup> - 2''g''<sup>7</sup> + 4 = 0 || Close to 1/6-comma
| 10:12:15 (+2 +3) || ~5/4 = 388.2216 || ''g''<sup>8</sup> &minus; 2''g''<sup>7</sup> + 4 = 0 || Close to 1/6-comma
|-
|-
| 15:18:23 (+3 +5) || ~5/4 = 387.9215 || 4''g''<sup>7</sup> - 3''g''<sup>5</sup> - 10 = 0 ||
| 15:18:23 (+3 +5) || ~5/4 = 387.9215 || 4''g''<sup>7</sup> &minus; 3''g''<sup>5</sup> &minus; 10 = 0 ||
|}
 
{| class="wikitable center-all left-5 mw-collapsible mw-collapsed"
|+ style="font-size: 105%; white-space: nowrap;" | Odd-limit-based target tunings
! rowspan="2" | Target
! colspan="2" | Minimax
|-
! Generator
! Eigenmonzo*
|-
| 5-odd-limit
| ~3/2 = 387.7444{{c}}
| 3/2
|}
|}


=== Tuning spectrum ===
=== 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.
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.


{| 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&nbsp;(¢)
! Comments
! Comments
|-
|-
Line 124: Line 168:
|  
|  
| 385.7143
| 385.7143
| 28ei val
| 28ei val, major thirds slightly flatter than this fall under 25 & 28 or [[magic]]
|-
|-
|  
|  
| [[11/9]]
| [[5/4]]
| 386.3137
| 386.3137
| 1/4 [[5632/5625|vishdel comma]]
|-
|
| [[5/4]]
| 386.8520
| Untempered tuning, lower bound of 5-odd-limit diamond tradeoff
| Untempered tuning, lower bound of 5-odd-limit diamond tradeoff
|-
|-
Line 224: Line 263:
| [[3/2]]
| [[3/2]]
| 387.7444
| 387.7444
| 1/8-comma
| 1/8-comma, 5-odd-limit minimax
|-
| [[294edo|95\294]]
|
| 387.7551
| 294e val
|-
|-
| [[229edo|74\229]]
| [[229edo|74\229]]
Line 267: Line 311:
|-
|-
|  
|  
| [[6/5]]
| [[5/3]]
| 387.9490
| 387.9490
| 1/7-comma, upper bound of 5-odd-limit diamond tradeoff
| 1/7-comma, upper bound of 5-odd-limit diamond tradeoff
Line 277: Line 321:
|-
|-
|  
|  
| [[46/25]]
| [[25/23]]
| 387.9706
| 387.9706
|  
|  
Line 292: Line 336:
|-
|-
|  
|  
| [[48/25]]
| [[25/24]]
| 388.2213
| 388.2213
| 1/6-comma, upper bound of 2.3.5.23-subgroup 25-odd-limit diamond tradeoff
| 1/6-comma, upper bound of 2.3.5.23-subgroup 25-odd-limit diamond tradeoff
Line 302: Line 346:
|-
|-
|  
|  
| [[192/125]]
| [[125/96]]
| 388.6028
| 388.6028
| 1/5-comma
| 1/5-comma
Line 312: Line 356:
|-
|-
|  
|  
| [[768/625]]
| [[625/384]]
| 389.1750
| 389.1750
| 1/4-comma
| 1/4-comma
Line 324: Line 368:
|  
|  
| '''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>*</nowiki> besides the octave
<nowiki />* Besides the octave


=== Other tunings ===
=== Other tunings ===
* [[DKW theory|DKW]] (2.3.5): ~2 = 1\1, ~5/4 = 387.8015
* 5-limit [[DKW theory|DKW]]: ~5/4 = 387.8015{{c}}
 
== Music ==
; [[Chris Vaisvil]]
* ''Ancient Stardust'' (2013) – [https://www.chrisvaisvil.com/ancient-stardust-wurschmidt13/ blog] | [https://web.archive.org/web/20201127013456/http://micro.soonlabel.com/jake_freivald/tunings_by_jake_freivald/20130811_wurschmidt{{lbrack}}13{{rbrack}}.mp3 play] – Würschmidt[13] in 5-odd-limit minimax tuning
 
; [[Jake Freivald]]
* ''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]]

Latest revision as of 09:34, 9 September 2026

Würschmidt
Subgroups 2.3.5, 2.3.5.11, 2.3.5.11.23
Comma basis 393216/390625 (2.3.5);
243/242, 5632/5625 (2.3.5.11);
243/242, 276/275, 529/528 (2.3.5.11.23)
Reduced mapping ⟨1; 8 1 20 14]
ET join 31 & 34
Generators (CWE) ~5/4 = 387.7 ¢
MOS scales 3L 1s, 3L 4s, …, 3L 28s, 31L 3s
Ploidacot beta-octacot
Pergen (P8, ccP5/8)
Color name Saquadbiguti
Minimax error 5-odd-limit: 1.43 ¢;
2.3.5.11.23 23-odd-limit: 3.12 ¢
Target scale size 5-odd-limit: 10 notes;
2.3.5.11.23 23-odd-limit: 22 notes

Würschmidt is a rank-2 temperament and parent of the würschmidt family, characterized by tempering out the würschmidt comma (ratio: 393216/390625, monzo: [17 1 -8⟩). It can be treated as analogous to schismic 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 has 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.

For technical data, see Würschmidt family #Würschmidt.

Extensions

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 2401/2400 and 6144/6125.

Therefore, it may be advisable to consider würschmidt a no-7's 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.

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, if one is interested in adding another prime to this temperament, 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 out 576/575 (S24) and 12167/12150 (S462⋅S47). 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 equating it with 24/23 takes advantage of the natural tempering tendency. 14 generators turn out to stack to 23/1. 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 (S49/(S1612)).

Interval chain

In the below, octave-reduced harmonics 1–23 are indicated in bold.

# Cents* Approximate ratios
5-limit Add-11 add-23 extension
0 0.0 1/1
1 387.8 5/4
2 775.6 25/16 36/23, 69/44
3 1163.3 125/64 45/23, 88/45, 108/55
4 351.1 625/512, 768/625 11/9, 27/22
5 738.9 192/125 23/15, 55/36
6 1126.7 48/25 23/12, 44/23
7 314.4 6/5
8 702.2 3/2
9 1090.0 15/8
10 277.8 75/64 27/23, 88/75
11 665.5 375/256 22/15, 81/55
12 1053.3 1152/625, 1875/1024 11/6, 46/25, 81/44
13 241.1 144/125 23/20, 55/48
14 628.9 36/25 23/16, 33/23
15 1016.6 9/5
16 204.4 9/8
17 592.2 45/32
18 980.0 225/128 44/25, 81/46
19 167.7 1125/1024 11/10
20 555.5 864/625 11/8
21 943.3 216/125 55/32
22 131.1 27/25 69/64, 99/92
23 518.9 27/20
24 906.6 27/16
25 94.4 135/128 132/125
26 482.2 675/512 33/25
27 870.0 3375/2048, 5184/3125 33/20
28 57.7 648/625 33/32
29 445.5 162/125 165/128
30 833.3 81/50 121/75
31 21.1 81/80 121/120

* In 5-limit CWE tuning

Tunings

Norm-based tunings

5-limit norm-based tunings
Euclidean
Constrained Constrained & skewed Destretched
Tenney CTE: ~5/4 = 387.734 ¢ CWE: ~5/4 = 387.776 ¢ POTE: ~5/4 = 387.7993 ¢
Equilateral CEE: ~5/4 = 387.7224 ¢
(8/65-comma)
5-limit add-23 norm-based tunings
Euclidean
Constrained Constrained & skewed Destretched
Tenney CTE: ~5/4 = 387.734 ¢ CWE: ~5/4 = 387.781 ¢ POTE: ~5/4 = 387.8051 ¢

Target tunings

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
Odd-limit-based target tunings
Target Minimax
Generator Eigenmonzo*
5-odd-limit ~3/2 = 387.7444 ¢ 3/2

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, major thirds slightly flatter than this fall under 25 & 28 or magic
5/4 386.3137 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, 5-odd-limit minimax
95\294 387.7551 294e val
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
5/3 387.9490 1/7-comma, upper bound of 5-odd-limit diamond tradeoff
43\133 387.9699 133e val
25/23 387.9706
23/15 388.0011
54\167 388.0240 167eei val
25/24 388.2213 1/6-comma, upper bound of 2.3.5.23-subgroup 25-odd-limit diamond tradeoff
11\34 388.2353
125/96 388.6028 1/5-comma
23\71 388.7324 71eei val
625/384 389.1750 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, major thirds slightly sharper than this fall under smate

* Besides the octave

Other tunings

  • 5-limit DKW: ~5/4 = 387.8015 ¢

Music

Chris Vaisvil
  • Ancient Stardust (2013) – blog | play – Würschmidt[13] in 5-odd-limit minimax tuning
Jake Freivald
  • Extrospection (2013) – play | SoundCloud – Würschmidt[16] in 31edo tuning.
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