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Hemimean clan: Difference between revisions

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{{Technical data page}}
{{Technical data page}}
<div style="float: right;">
[[File:Didacus.png|alt=Didacus.png|thumb|600x560px|A chart of the tuning spectrum of undecimal didacus, showing the offsets of prime harmonics 5, 7, and 11, as a function of the generator; all EDO tunings are shown with vertical lines whose length indicates the EDO's tolerance, i.e. half of its step size in either direction of just, and some small EDOs supporting the temperament are labeled.]]
</div>
The '''hemimean clan''' [[Tempering out|tempers out]] the hemimean comma, [[3136/3125]], with [[monzo]] {{monzo| 6 0 -5 2 }}, such that [[7/4]] is split into five steps, of which two make [[5/4]] and three make [[7/5]]; this defines the [[2.5.7 subgroup]] temperament [[didacus]], generated by a tempered hemithird of [[28/25]].
The '''hemimean clan''' [[Tempering out|tempers out]] the hemimean comma, [[3136/3125]], with [[monzo]] {{monzo| 6 0 -5 2 }}, such that [[7/4]] is split into five steps, of which two make [[5/4]] and three make [[7/5]]; this defines the [[2.5.7 subgroup]] temperament [[didacus]], generated by a tempered hemithird of [[28/25]].


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[[Tp tuning #T2 tuning|RMS error]]: 0.2138 cents
[[Tp tuning #T2 tuning|RMS error]]: 0.2138 cents


[[Badness]] (Dirichlet): 0.091
[[Badness]] (Sintel): 0.091


= Strong extensions =
= Strong extensions =
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Mapping generators: ~2, ~25/14
Mapping generators: ~2, ~25/14
{{Multival|legend=1| 16 2 5 -34 -37 6 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~28/25 = 193.898
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~28/25 = 193.898
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{{Optimal ET sequence|legend=1| 31, 68, 99, 229, 328 }}
{{Optimal ET sequence|legend=1| 31, 68, 99, 229, 328 }}


Badness (Dirichlet): 0.304
Badness (Sintel): 0.304


=== 11-limit ===
=== 11-limit ===
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=== Quadrawürschmidt ===
=== Quadrawürschmidt ===
This has been documented in Graham Breed's temperament finder as ''semihemiwürschmidt'', but ''quadrawürschmidt'' arguably makes more sense.  
This has been documented in Graham Breed's temperament finder as ''semihemiwürschmidt'', but ''quadrawürschmidt'' arguably makes more sense.  
The generator of quadrawürschmidt is essentially a [[septimal meantone]] fifth. However, it is not used to represent [[3/2]], as 3/2 is found at the hemiwürschmidt position, 16 wholetones up. The small comma between the generator and 3/2 is taken to represent [[441/440]].


Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11
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{{Mapping|legend=1| 1 4 2 2 | 0 -15 2 5 }}
{{Mapping|legend=1| 1 4 2 2 | 0 -15 2 5 }}
{{Multival|legend=1| 15 -2 -5 -38 -50 -6 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~28/25 = 193.244
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~28/25 = 193.244
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* [[7-odd-limit]]: ~28/25 = {{monzo| 1/10 -1/20 0 1/20 }}
* [[7-odd-limit]]: ~28/25 = {{monzo| 1/10 -1/20 0 1/20 }}
: {{monzo list| 1 0 0 0 | 5/2 3/4 0 -3/4 | 11/5 -1/10 0 1/10 | 5/2 -1/4 0 1/4 }}
: {{monzo list| 1 0 0 0 | 5/2 3/4 0 -3/4 | 11/5 -1/10 0 1/10 | 5/2 -1/4 0 1/4 }}
: [[Eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.7/3
: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7/3
* [[9-odd-limit]]: ~28/25 = {{monzo| 6/35 -2/35 0 1/35 }}
* [[9-odd-limit]]: ~28/25 = {{monzo| 6/35 -2/35 0 1/35 }}
: {{monzo list| 1 0 0 0 | 10/7 6/7 0 -3/7 | 82/35 -4/35 0 2/35 | 20/7 -2/7 0 1/7 }}
: {{monzo list| 1 0 0 0 | 10/7 6/7 0 -3/7 | 82/35 -4/35 0 2/35 | 20/7 -2/7 0 1/7 }}
: [[Eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.9/7
: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.9/7


{{Optimal ET sequence|legend=1| 25, 31, 87, 118 }}
{{Optimal ET sequence|legend=1| 25, 31, 87, 118 }}
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{{Mapping|legend=1| 1 0 2 2 | 0 10 2 5 }}
{{Mapping|legend=1| 1 0 2 2 | 0 10 2 5 }}
{{Multival|legend=1| 10 2 5 -20 -20 6 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~28/25 = 189.927
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~28/25 = 189.927
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: mapping generators: ~2, ~75/49
: mapping generators: ~2, ~75/49
{{Multival|legend=1| 17 6 15 -30 -24 18 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~75/49 = 735.155
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~75/49 = 735.155
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: mapping generators: ~2, ~48/35
: mapping generators: ~2, ~48/35
{{Multival|legend=1| 27 8 20 -50 -44 24 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~48/35 = 551.782
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~48/35 = 551.782
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{{Mapping|legend=1| 1 6 0 -3 | 0 -19 10 25 }}
{{Mapping|legend=1| 1 6 0 -3 | 0 -19 10 25 }}
{{Multival|legend=1| 19 -10 -25 -60 -93 -30 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~75/64 = 278.800
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~75/64 = 278.800
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{{Mapping|legend=1| 1 1 2 2 | 0 29 16 40 }}
{{Mapping|legend=1| 1 1 2 2 | 0 29 16 40 }}
{{Multival|legend=1| 29 16 40 -42 -18 48 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~686/675 = 24.217
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~686/675 = 24.217
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{{Mapping|legend=1| 1 0 0 -3 | 0 15 22 55 }}
{{Mapping|legend=1| 1 0 0 -3 | 0 15 22 55 }}
{{Multival|legend=1| 15 22 55 0 45 66 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~27/25 = 126.706
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~27/25 = 126.706
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{{Mapping|legend=1| 1 -4 -2 -8 | 0 31 24 60 }}
{{Mapping|legend=1| 1 -4 -2 -8 | 0 31 24 60 }}
{{Multival|legend=1| 31 24 60 -34 8 72 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~4375/3888 = 216.173
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~4375/3888 = 216.173
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{{Mapping|legend=1| 1 0 -2 -8 | 0 11 30 75 }}
{{Mapping|legend=1| 1 0 -2 -8 | 0 11 30 75 }}
{{Multival|legend=1| 11 30 75 22 88 90 }}


[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~448/405 = 172.917
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~448/405 = 172.917
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RMS error: 0.5567 cents
RMS error: 0.5567 cents


Badness (Dirichlet): 0.195
Badness (Sintel): 0.195


=== Tridecimal didacus ===
=== Tridecimal didacus ===
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Optimal ET sequence: {{Optimal ET sequence| 6, 25, 31, 37 }}
Optimal ET sequence: {{Optimal ET sequence| 6, 25, 31, 37 }}


Badness (Dirichlet): 0.324
Badness (Sintel): 0.324


==== Mediantone ====
==== Mediantone ====
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<nowiki />* 117d only appears without prime 19
<nowiki />* 117d only appears without prime 19


Badness (Dirichlet): 0.618
Badness (Sintel): 0.618


==== Roulette ====
==== Roulette ====
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Optimal ET sequence: {{Optimal ET sequence| 6g, ... 31, 37, 68, 105 }}
Optimal ET sequence: {{Optimal ET sequence| 6g, ... 31, 37, 68, 105 }}


Badness (Dirichlet): 0.676
Badness (Sintel): 0.676


== Rectified hebrew ==
== Rectified hebrew ==
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[[Category:Temperament clans]]
[[Category:Temperament clans]]
[[Category:Pages with mostly numerical content]]
[[Category:Hemimean clan| ]] <!-- main article -->
[[Category:Hemimean clan| ]] <!-- main article -->
[[Category:Hemimean| ]] <!-- key article -->
[[Category:Hemimean| ]] <!-- key article -->
[[Category:Rank 2]]
[[Category:Rank 2]]
Retrieved from "https://en.xen.wiki/w/Hemimean_clan"