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The '''trisedodge family''' tempers out the trisedodge comma, 30958682112/30517578125 = {{monzo| 19 10 -15 }}.
{{Technical data page}}
The '''trisedodge family''' of [[regular temperament|temperaments]] [[tempering out|tempers out]] the [[trisedodge comma]] ({{monzo|legend=1| 19 10 -15 }}, [[ratio]]: 30 958 682 112 / 30 517 578 125).


== Trisedodge  ==
Named by [[Petr Pařízek]] in 2011, ''trisedodge'' (originally spelt ''trisedoge'') means that three semidiminished [[octave]]s add up to [[7/1]], and that an octave is made of 5 [[period]]s<ref name="naming">[https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_101780.html Yahoo! Tuning Group | ''Suggested names for the unclasified temperaments'']</ref>.


Subgroup: 2.3.5
== Trisedodge ==
The generator of trisedodge is [[~]][[864/625]] at around 554 [[cent]]s, which in all [[11-limit]] [[extension]]s is used to represent [[11/8]], and three of them and a period is equal to [[3/1]]. This generator, when reduced to the minimal size, represents [[25/24]]. However, another possible generator is ~[[6/5]], reached by a period plus 25/24, that is, 6/5 = (144/125)⋅(25/24).


Comma list: 30958682112/30517578125
[[Subgroup]]: 2.3.5


Mapping: [{{val| 5 7 11 }}, {{val| 0 3 2 }}]
[[Comma list]]: 30958682112/30517578125


POTE generator: ~25/24 = 74.0077
{{Mapping|legend=1| 5 1 7 | 0 3 2 }}
: mapping generators: ~144/125, ~864/625


{{Val list|legend=1| 15, 50, 65, 340c, 405c, 470c, 535c, 600c }}
[[Optimal tuning]]s:
* [[WE]]: ~144/125 = 239.9482{{c}}, ~864/625 = 553.8881{{c}} (~25/24 = 73.9917{{c}})
: [[error map]]: {{val| -0.259 -0.342 +1.100 }}
* [[CWE]]: ~144/125 = 240.0000{{c}}, ~864/625 = 553.9342{{c}} (~25/24 = 73.9342{{c}})
: error map: {{val| 0.000 -0.152 +1.555 }}


== 7-limit  ==
{{Optimal ET sequence|legend=1| 15, 35, 50, 65, 340c, 405c, …, 600c }}


Subgroup: 2.3.5.7
[[Badness]] (Sintel): 5.93
 
=== Overview to extensions ===
The second comma of the comma list defines which 7-limit family member we are looking at. Among these are septimal trisedodge (65d & 80), which adds [[4000/3993]], and coblack (50 & 65), which adds [[126/125]]. Remarkably, septimal trisedodge admits an extension to the full [[29-limit]], which, except for prime 13, is obvious and simple a way to extend the 11-limit representation.
 
Temperaments discussed elsewhere include [[15th-octave temperaments #Quindecic|quindecic]] and [[Stearnsmic clan #Decistearn|decistearn]]. Considered below are trisedodge and coblack.
 
Septimal trisedodge and coblack have the common [[2.3.5.11-subgroup|2.3.5.11 subgroup]] [[restriction]], called countdown, considered immediately below. In this temperament, the generator can be taken to be ~11/10, reached as a period minus 25/24, that is, (55/48)/(25/24) = 11/10. Therefore, since a period plus a gen is 6/5 and a period minus a gen is 11/10, we reach 12/11 in 2 generator steps.
 
=== Countdown ===
Subgroup: 2.3.5.11
 
Comma list: 4000/3993, 6912/6875
 
Subgroup-val mapping: {{mapping| 5 1 7 15 | 0 3 2 1 }}
 
Optimal tunings:
* WE: ~55/48 = 240.0000{{c}}, ~11/8 = 553.9041{{c}} (~25/24 = 74.0952{{c}})
* CWE: ~55/48 = 240.0000{{c}}, ~11/8 = 554.0109{{c}} (~25/24 = 74.0109{{c}})
 
{{Optimal ET sequence|legend=0| 15, 35, 50, 65, 210e, 275e, 340ce }}
 
Badness (Sintel): 0.794
 
== Septimal trisedodge ==
We can extend trisedodge to the [[17-limit]] by using the sharp tendency of prime 5 to justify tempering out [[256/255]] ({{S|16}}). Note that prime 3 is also tuned sharp (though less than prime 5) in optimized tunings. We can then extend it to the 19-limit by tempering out [[361/360]] ({{S|19}}) or equivalently [[400/399]] ({{S|20}}), whose naturalness becomes much clearer when we consider it in the [[23-limit]], where we equate [[23/19]] with a stack of two [[11/10]]'s, tempering out [[2300/2299]] ([[S-expression|S20/S22]]), relying on the obvious mapping of [[23/16]] as one period above [[5/4]] so that [[~]][[23/20]] is tuned to 1\5. The mapping of 23 also implies tempering out [[276/275]] (the difference between [[55/48]] and [[23/20]]), which is [[3025/3024]] flat of [[253/252]]. Finally, there is an obvious mapping for [[29/16]] as two periods above [[11/8]] so that [[~]][[29/22]] is tuned to 2\5 and that [[~]][[32/29]] is equated with [[~]][[11/10]], the generator.
 
This defines trisedodge as being an unambiguously full [[29-limit]] temperament, with an interesting feature of having possible alternative mappings for primes 7 and 13. Prime 7 can either be mapped the more accurate way as septimal trisedodge does or it can be mapped as in [[#Coblack|coblack]], while prime 13 can alternatively be found as 8 generators ''up'' instead of down, corresponding to [[#Trisey|trisey]], though using both of those mappings simultaneously only really makes sense in [[80edo]], which is a reasonable edo tuning for it and happens to correspond to the 80-note generator chain of trisedodge required for finding every prime relative to the same root, though note that [[11/10]] is practically just there so that intervals of 29 require error cancellation of the oversharp 29th harmonic to help justify harmonically
 
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 4000/3969, 110592/109375
[[Comma list]]: 4000/3969, 110592/109375


[[Mapping]]: [{{val| 5 7 11 15 }}, {{val| 0 3 2 -3 }}]
{{Mapping|legend=1| 5 1 7 21 | 0 3 2 -3 }}


[[POTE generator]]: ~25/24 = 74.9480
[[Optimal tuning]]s:  
* [[WE]]: ~144/125 = 239.7187{{c}}, ~175/128 = 554.2976{{c}} (~25/24 = 74.8601{{c}})
: [[error map]]: {{val| -1.406 +0.656 +0.312 +2.374 }}
* [[CWE]]: ~144/125 = 240.0000{{c}}, ~175/128 = 554.8511{{c}} (~25/24 = 74.8511{{c}})
: error map: {{val| 0.000 +2.598 +3.388 +6.621 }}


{{Val list|legend=1| 15, 50d, 65d, 80 }}
{{Optimal ET sequence|legend=1| 15, 50d, 65d, 80 }}


=== 11-limit  ===
[[Badness]] (Sintel): 3.48


=== 11-limit ===
Subgroup: 2.3.5.7.11
Subgroup: 2.3.5.7.11


Comma list: 176/175, 1331/1323, 2560/2541
Comma list: 176/175, 1331/1323, 2560/2541


Mapping: [{{val| 5 7 11 15 17 }}, {{val| 0 3 2 -3 1 }}]
Mapping: {{mapping| 5 1 7 21 15 | 0 3 2 -3 1 }}
 
Optimal tunings:
* WE: ~55/48 = 239.7335{{c}}, ~11/8 = 554.3239{{c}} (~25/24 = 74.8569{{c}})
* CWE: ~55/48 = 240.0000{{c}}, ~11/8 = 554.8505{{c}} (~25/24 = 74.8505{{c}})
 
{{Optimal ET sequence|legend=0| 15, 50d, 65d, 80 }}
 
Badness (Sintel): 1.44
 
==== 13-limit ====
Subgroup: 2.3.5.7.11.13


POTE generator: ~25/24 = 74.9401
Comma list: 176/175, 351/350, 1040/1029, 1331/1323


{{Val list|legend=1| 15, 50d, 65d, 80 }}
Mapping: {{mapping| 5 1 7 21 15 37 | 0 3 2 -3 1 -8 }}


== Coblack  ==
Optimal tunings:
{{see also| Cloudy clan #Coblack }}
* WE: ~55/48 = 239.7764{{c}}, ~11/8 = 554.1429{{c}} (~25/24 = 74.5902{{c}})
* CWE: ~55/48 = 240.0000{{c}}, ~11/8 = 554.6627{{c}} (~25/24 = 74.6627{{c}})


In addition to 126/125, the coblack temperament tempers out the [[cloudy comma]], 16807/16384, which is the amount by which five septimal supermajor seconds ([[8/7]]) fall short of an octave.
{{Optimal ET sequence|legend=0| 15f, 50df, 65d, 80, 145d }}


Subgroup: 2.3.5.7
Badness (Sintel): 1.84


[[Comma list]]: 126/125, 16807/16384
==== 17-limit ====
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 176/175, 256/255, 351/350, 1040/1029, 1331/1323
 
Mapping: {{mapping| 5 1 7 21 15 37 32 | 0 3 2 -3 1 -8 -5 }}
 
Optimal tunings:
* CTE: ~55/48 = 239.7935{{c}}, ~11/8 = 554.1209{{c}} (~25/24 = 74.5340{{c}})
* CWE: ~55/48 = 240.0000{{c}}, ~11/8 = 554.6141{{c}} (~25/24 = 74.6141{{c}})
 
{{Optimal ET sequence|legend=0| 15f, 50dfg, 65d, 80, 145d }}
 
Badness (Sintel): 1.61
 
==== 19-limit ====
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 176/175, 190/189, 256/255, 351/350, 1040/1029, 1331/1323
 
Mapping: {{mapping| 5 1 7 21 15 37 32 12 | 0 3 2 -3 1 -8 -5 4 }}
 
Optimal tunings:
* WE: ~55/48 = 239.8147{{c}}, ~11/8 = 554.2353{{c}} (~25/24 = 74.6060{{c}})
* CWE: ~55/48 = 240.0000{{c}}, ~11/8 = 554.6675{{c}} (~25/24 = 74.6675{{c}})
 
{{Optimal ET sequence|legend=0| 15f, 65d, 80 }}
 
Badness (Sintel): 1.54
 
==== 23-limit ====
Subgroup: 2.3.5.7.11.13.17.19.23
 
Comma list: 176/175, 190/189, 253/252, 256/255, 351/350, 1040/1029, 1331/1323


[[Mapping]]: [{{val| 5 1 7 14 }}, {{val| 0 3 2 0 }}]
Mapping: {{mapping| 5 1 7 21 15 37 32 12 18 | 0 3 2 -3 1 -8 -5 4 2 }}


[[POTE generator]]: ~21/20 = 73.044
Optimal tunings:
* WE: ~23/20 = 239.8299{{c}}, ~11/8 = 554.2946{{c}} (~24/23 = 74.6347{{c}})
* CWE: ~23/20 = 240.0000{{c}}, ~11/8 = 554.6878{{c}} (~24/23 = 74.6878{{c}})


{{Val list|legend=1| 15, 35, 50, 65, 115d }}
{{Optimal ET sequence|legend=0| 15f, 65d, 80 }}


[[Badness]]: 0.1073
Badness (Sintel): 1.46


=== 11-limit ===
==== 29-limit ====
Subgroup: 2.3.5.7.11.13.17.19.23.29


Subgroup: 2.3.5.7.11
Comma list: 176/175, 190/189, 232/231, 253/252, 256/255, 351/350, 1040/1029, 1331/1323


Comma list: 126/125, 245/242, 385/384
Mapping: {{mapping| 5 1 7 21 15 37 32 12 18 22 | 0 3 2 -3 1 -8 -5 4 2 1 }}


Mapping: [{{val| 5 1 7 14 15 }}, {{val| 0 3 2 0 1 }}]
Optimal tunings:
* WE: ~23/20 = 239.8259{{c}}, ~11/8 = 554.2822{{c}} (~24/23 = 74.6304{{c}})
* CWE: ~23/20 = 240.0000{{c}}, ~11/8 = 554.6835{{c}} (~24/23 = 74.6835{{c}})


POTE generator: ~21/20 = 73.264
{{Optimal ET sequence|legend=0| 15f, 65d, 80 }}


{{Val list|legend=1| 15, 35, 50, 65, 115d }}
Badness (Sintel): 1.38


== Quindecic  ==
==== Trisey ====
Note that trisey can be extended to the full [[29-limit]] by following canonical trisedodge extension path; [[80edo]] is a good tuning for merging trisedodge and trisey.


Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 28/27, 49/48, 55/54, 77/75
Comma list: 176/175, 325/324, 364/363, 640/637
 
Mapping: {{mapping| 5 1 7 21 15 0 | 0 3 2 -3 1 8 }}
 
Optimal tunings:
* WE: ~55/48 = 239.7425{{c}}, ~11/8 = 554.6648{{c}} (~25/24 = 75.1797{{c}})
* CWE: ~55/48 = 240.0000{{c}}, ~11/8 = 555.1626{{c}} (~25/24 = 75.1626{{c}})
 
{{Optimal ET sequence|legend=0| 15, 80, 175bcde, 255bcdde }}
 
Badness (Sintel): 1.57
 
== Coblack ==
In addition to 126/125, the coblack temperament tempers out the [[cloudy comma]], 16807/16384, which is the amount by which five septimal supermajor seconds ([[8/7]]) fall short of an octave. Coblack was also named by Petr Pařízek, who considered it a counterpart of [[blackwood]]<ref name="naming"/>.
 
[[Subgroup]]: 2.3.5.7
 
[[Comma list]]: 126/125, 16807/16384
 
{{Mapping|legend=1| 5 1 7 14 | 0 3 2 0 }}
 
[[Optimal tuning]]s:
* [[WE]]: ~8/7 = 240.2499{{c}}, ~48/35 = 553.6203{{c}} (~21/20 = 73.1204{{c}})
: [[error map]]: {{val| +1.250 -0.844 +2.676 -5.327 }}
* [[CWE]]: ~8/7 = 240.0000{{c}}, ~48/35 = 553.2893{{c}} (~21/20 = 73.2893{{c}})
: error map: {{val| 0.000 -2.087 +0.265 -8.826 }}
 
{{Optimal ET sequence|legend=1| 15, 35, 50, 65, 115d }}
 
[[Badness]] (Sintel): 2.71
 
=== 11-limit ===
Subgroup: 2.3.5.7.11
 
Comma list: 126/125, 245/242, 385/384
 
Mapping: {{mapping| 5 1 7 14 15 | 0 3 2 0 1 }}
 
Optimal tunings:
* WE: ~8/7 = 240.1524{{c}}, ~11/8 = 553.6154{{c}} (~21/20 = 73.3106{{c}})
* CWE: ~8/7 = 240.0000{{c}}, ~11/8 = 553.3989{{c}} (~21/20 = 73.3989{{c}})


Mapping: [{{val| 15 24 35 42 52 0 }}, {{val| 0 0 0 0 0 1 }}]
{{Optimal ET sequence|legend=0| 15, 35, 50, 65, 115d }}


POTE generator: ~66/65 = 27.0764
Badness (Sintel): 1.49


{{Val list|legend=1| 15, 30 }}
== References ==


[[Category:Regular temperament theory]]
[[Category:Trisedodge family| ]] <!-- main article -->
[[Category:Temperament family]]
[[Category:Trisedodge| ]] <!-- key article -->
[[Category:Trisedodge]]
[[Category:Temperament families]]
[[Category:Rank 2]]
[[Category:Catalogs of rank-2 temperaments]]
Retrieved from "https://en.xen.wiki/w/Trisedodge_family"