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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
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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 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
! 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
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| 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
}}
|}
|}
</div>
<nowiki>*</nowiki> In 2.3.5-targeted [[DKW theory|DKW]] tuning


== Tunings ==
== Tunings ==
Retrieved from "https://en.xen.wiki/w/Würschmidt"