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{{Technical data page}}
The '''trisedodge family''' tempers out the [[trisedodge comma]], 30958682112/30517578125 = {{monzo| 19 10 -15 }}.
The '''trisedodge family''' tempers out the [[trisedodge comma]], 30958682112/30517578125 = {{monzo| 19 10 -15 }}.


Temperaments discussed elsewhere include [[Trienstonic clan #Quindecic|quindecic]] and [[Stearnsmic clan #Decistearn|decistearn]].
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>.
 
Temperaments discussed elsewhere include [[15th-octave temperaments #Quindecic|quindecic]] and [[Stearnsmic clan #Decistearn|decistearn]]. Considered below are trisedodge and coblack.  


== Trisedodge ==
== Trisedodge ==
The generator of trisedodge is ~864/625 at around 554 cents, which in all 11-limit extensions 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, (144/125)(25/24) = 6/5.
In the 11-limit 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 at 2 gens.
Remarkably, trisedodge admits an extension to the full [[29-limit]] which, except for prime 13, is surprisingly obvious/simple a way to extend the 11-limit representation.
[[Subgroup]]: 2.3.5
[[Subgroup]]: 2.3.5


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: mapping generators: ~144/125, ~864/625
: mapping generators: ~144/125, ~864/625


[[Optimal tuning]] ([[POTE]]): ~144/125 = 1\5, ~25/24 = 74.0077
[[Optimal tuning]]s:
* [[CTE]]: ~144/125 = 1\5, ~864/625 = 553.8249 (~25/24 = 73.8249)
* [[POTE]]: ~144/125 = 1\5, ~864/625 = 554.0077 (~25/24 = 74.0077)


{{Optimal ET sequence|legend=1| 15, 50, 65, 340c, 405c, 470c, 535c, 600c }}
{{Optimal ET sequence|legend=1| 15, 35, 50, 65, 340c, 405c, 470c, 535c, 600c }}


[[Badness]]: 0.252724
[[Badness]]: 0.252724
Line 25: Line 36:
Sval mapping: {{mapping| 5 1 7 15 | 0 3 2 1 }}
Sval mapping: {{mapping| 5 1 7 15 | 0 3 2 1 }}


Optimal tuning (POTE): ~55/48 = 1\5, ~25/24 = 74.1247
[[Optimal tuning]]s:
* CTE: ~55/48 = 1\5, ~11/8 = 553.7951 (~25/24 = 73.7951)
* POTE: ~55/48 = 1\5, ~11/8 = 554.1247 (~25/24 = 74.1247)


{{Optimal ET sequence|legend=1| 15, 35, 50, 65, 210e, 275e, 340ce }}
{{Optimal ET sequence|legend=1| 15, 35, 50, 65, 210e, 275e, 340ce }}
Line 38: Line 51:
{{Mapping|legend=1| 5 1 7 21 | 0 3 2 -3 }}
{{Mapping|legend=1| 5 1 7 21 | 0 3 2 -3 }}


[[Optimal tuning]]s (all have ~144/125 = 1\5):
[[Optimal tuning]]s:
* [[POTE]]: ~25/24 = 74.9480
* [[CTE]]: ~144/125 = 1\5, ~175/128 = 554.5146 (~25/24 = 74.5146)
* [[CTE]]: ~25/24 = 74.515
* [[POTE]]: ~144/125 = 1\5, ~175/128 = 554.9480 (~25/24 = 74.9480)


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


Badness (Dirichlet): 3.485
Badness: 0.137695


=== 11-limit ===
=== 11-limit ===
Line 53: Line 66:
Mapping: {{mapping| 5 1 7 21 15 | 0 3 2 -3 1 }}
Mapping: {{mapping| 5 1 7 21 15 | 0 3 2 -3 1 }}


Optimal tunings: (all have 1\5 = ~55/48)
Optimal tunings:  
* POTE: ~25/24 = 74.9401
* CTE: ~55/48 = 1\5, ~11/8 = 554.4664 (~25/24 = 74.4664)
* CTE: ~25/24 = 74.466
* POTE: ~55/48 = 1\5, ~11/8 = 554.9401 (~25/24 = 74.9401)


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


Badness (Dirichlet): 1.438
Badness: 0.043508


=== 13-limit ===
==== 13-limit ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 176/175, 352/351, 1040/1029, 3025/3024
Comma list: 176/175, 351/350, 1040/1029, 1331/1323


Mapping: {{mapping| 5 1 7 21 15 37 | 0 3 2 -3 1 -8 }}
Mapping: {{mapping| 5 1 7 21 15 37 | 0 3 2 -3 1 -8 }}


Optimal tuning ([[CTE]]): ~25/24 = 74.68
Optimal tunings:
* CTE: ~55/48 = 1\5, ~11/8 = 554.6802 (~25/24 = 74.6802)
* CWE: ~55/48 = 1\5, ~11/8 = 554.6627 (~25/24 = 74.6627)
 
Optimal ET sequence: {{Optimal ET sequence| 15f, 50df, 65d, 80, 145d }}
 
Badness: 0.0446
 
==== 17-limit ====
We extend to prime 17 by using the sharp tendency of prime 5 to justify tempering out ([[16/15]])/([[17/16]]) = [[256/255|S16]]. Note that prime 3 is also tuned sharp (though less than prime 5) in optimized tunings.
 
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 176/175, 351/350, 1040/1029, 1331/1323, 256/255
 
Mapping: {{mapping| 5 1 7 21 15 37 32 | 0 3 2 -3 1 -8 -5 }}
 
Optimal tunings:
* CTE: ~55/48 = 1\5, ~11/8 = 554.722 (~25/24 = 74.722)
* CWE: ~55/48 = 1\5, ~11/8 = 554.614 (~25/24 = 74.614)
 
Optimal ET sequence: {{Optimal ET sequence| 15f, 50dfg, 65d, 80, 145d }}
 
Badness: ?
 
Badness (Sintel): 1.609
 
==== 19-limit ====
We extend to prime 19 by tempering out [[361/360|361/360 = S19]] or equivalently [[400/399|400/399 = S20]], whose naturalness becomes much clearer when we consider it in the 23-limit as the result of tempering out ([[23/19]])/([[11/10|22/20]])<sup>2</sup> = [[2300/2299|S20/S22]], relying on the surprisingly obvious mapping of [[23/16]] as one period above [[5/4]] so that [[~]][[23/20]] = 1\5.
 
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 176/175, 351/350, 1040/1029, 1331/1323, 256/255, 190/189,
 
Mapping: {{mapping| 5 1 7 21 15 37 32 12 | 0 3 2 -3 1 -8 -5 4 }}
 
Optimal tunings:
* CTE: ~55/48 = 1\5, ~11/8 = 554.698 (~25/24 = 74.698)
* CWE: ~55/48 = 1\5, ~11/8 = 554.667 (~25/24 = 74.667)
 
Optimal ET sequence: {{Optimal ET sequence| 15f, 65d, 80 }}
 
Badness: ?
 
Badness (Sintel): 1.542
 
==== 23-limit ====
As mentioned, prime 23 is found as prime 5 plus a period (up to octave-equivalence). This id done by tempering out ([[55/48]])/([[23/20]]) = [[276/275]] which is [[3025/3024]] flat of [[253/252]]. Curiously, the [[CTE]] and [[CWE]] tunings are almost exactly the same here (different by about a hundredth of a cent).
 
Subgroup: 2.3.5.7.11.13.17.19.23
 
Comma list: 176/175, 351/350, 1040/1029, 1331/1323, 256/255, 190/189, 253/252
 
Mapping: {{mapping| 5 1 7 21 15 37 32 12 18 | 0 3 2 -3 1 -8 -5 4 2 }}
 
Optimal tunings:
* CTE: ~23/20 = 1\5, ~11/8 = 554.689 (~24/23 = 74.689)
* CWE: ~23/20 = 1\5, ~11/8 = 554.688 (~24/23 = 74.688)
 
Optimal ET sequence: {{Optimal ET sequence| 15f, 65d, 80 }}
 
Badness: ?
 
Badness (Sintel): 1.463
 
==== 29-limit ====
There's a surprisingly obvious mapping of [[29/16]] as two periods above [[11/8]] so that [[~]][[29/22]] = 2\5 and meaning equating [[~]][[32/29]] with [[~]][[11/10]], the generator. This defines trisedodge as being an unambiguously full 29-limit temperament, with an interesting feature of having two possible mappings of prime 7 and 13; prime 7 can either be mapped the more accurate way as septimal trisedodge does or it can be mapped as it is approximated in [[5edo]], while prime 13 can alternately be found as 8 generators ''up'' instead of down, corresponding to [[#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 MOS 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: 2.3.5.7.11.13.17.19.23.29
 
Comma list: 176/175, 351/350, 1040/1029, 1331/1323, 256/255, 190/189, 253/252, 232/231


{{Optimal ET sequence|legend=1| 15f, 50df, 65d, 80, 145d }}
Mapping: {{mapping| 5 1 7 21 15 37 32 12 18 22 | 0 3 2 -3 1 -8 -5 4 2 1 }}


Badness (Dirichlet): 1.843
Optimal tunings:
* CTE: ~23/20 = 1\5, ~11/8 = 554.673 (~24/23 = 74.673)
* CWE: ~23/20 = 1\5, ~11/8 = 554.684 (~24/23 = 74.684)
 
Optimal ET sequence: {{Optimal ET sequence| 15f, 65d, 80 }}
 
Badness: ?
 
Badness (Sintel): 1.380
 
==== 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
 
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:
* CTE: ~55/48 = 1\5, ~11/8 = 554.7405 (~25/24 = 74.7405)
* CWE: ~55/48 = 1\5, ~11/8 = 555.1626 (~25/24 = 75.1626)
 
Optimal ET sequence: {{Optimal ET sequence| 15, 80, 175bcde, 255bcdde }}
 
Badness: 0.0380


== Coblack ==
== Coblack ==
{{see also| Cloudy clan #Coblack }}
{{See also| Cloudy clan #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.
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 [[blacksmith]]<ref name="naming"/>.  


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 85: Line 193:
{{Mapping|legend=1| 5 1 7 14 | 0 3 2 0 }}
{{Mapping|legend=1| 5 1 7 14 | 0 3 2 0 }}


[[Optimal tuning]] ([[POTE]]): ~8/7 = 1\5, ~21/20 = 73.044
[[Optimal tuning]]s:
* [[CTE]]: ~8/7 = 1\5, ~48/35 = 553.8429 (~21/20 = 73.8429)
* [[POTE]]: ~8/7 = 1\5, ~48/35 = 553.044 (~21/20 = 73.044)


{{Optimal ET sequence|legend=1| 15, 35, 50, 65, 115d }}
{{Optimal ET sequence|legend=1| 15, 35, 50, 65, 115d }}
Line 98: Line 208:
Mapping: {{mapping| 5 1 7 14 15 | 0 3 2 0 1 }}
Mapping: {{mapping| 5 1 7 14 15 | 0 3 2 0 1 }}


Optimal tuning (POTE): ~8/7 = 1\5, ~21/20 = 73.264
Optimal tunings:
* CTE: ~8/7 = 1\5, ~11/8 = 553.7951 (~21/20 = 73.7951)
* POTE: ~8/7 = 1\5, ~11/8 = 553.264 (~21/20 = 73.264)


{{Optimal ET sequence|legend=1| 15, 35, 50, 65, 115d }}
Optimal ET sequence: {{Optimal ET sequence| 15, 35, 50, 65, 115d }}


Badness: 0.045070
Badness: 0.045070
== Notes ==


[[Category:Temperament families]]
[[Category:Temperament families]]
[[Category:Pages with mostly numerical content]]
[[Category:Trisedodge family| ]] <!-- main article -->
[[Category:Trisedodge family| ]] <!-- main article -->
[[Category:Trisedodge| ]] <!-- key article -->
[[Category:Trisedodge| ]] <!-- key article -->
[[Category:Rank 2]]
[[Category:Rank 2]]

Latest revision as of 00:40, 24 June 2025

This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.

The trisedodge family tempers out the trisedodge comma, 30958682112/30517578125 = [19 10 -15⟩.

Named by Petr Pařízek in 2011, trisedodge (originally spelt trisedoge) means that three semidiminished octaves add up to 7/1, and that an octave is made of 5 periods[1].

Temperaments discussed elsewhere include quindecic and decistearn. Considered below are trisedodge and coblack.

Trisedodge

The generator of trisedodge is ~864/625 at around 554 cents, which in all 11-limit extensions 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, (144/125)(25/24) = 6/5.

In the 11-limit 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 at 2 gens.

Remarkably, trisedodge admits an extension to the full 29-limit which, except for prime 13, is surprisingly obvious/simple a way to extend the 11-limit representation.

Subgroup: 2.3.5

Comma list: 30958682112/30517578125

Mapping: [⟨5 1 7], ⟨0 3 2]]

mapping generators: ~144/125, ~864/625

Optimal tunings:

  • CTE: ~144/125 = 1\5, ~864/625 = 553.8249 (~25/24 = 73.8249)
  • POTE: ~144/125 = 1\5, ~864/625 = 554.0077 (~25/24 = 74.0077)

Optimal ET sequence: 15, 35, 50, 65, 340c, 405c, 470c, 535c, 600c

Badness: 0.252724

Countdown

Subgroup: 2.3.5.11

Comma list: 4000/3993, 6912/6875

Sval mapping: [⟨5 1 7 15], ⟨0 3 2 1]]

Optimal tunings:

  • CTE: ~55/48 = 1\5, ~11/8 = 553.7951 (~25/24 = 73.7951)
  • POTE: ~55/48 = 1\5, ~11/8 = 554.1247 (~25/24 = 74.1247)

Optimal ET sequence: 15, 35, 50, 65, 210e, 275e, 340ce

RMS error: 0.3198 cents

Septimal trisedodge

Subgroup: 2.3.5.7

Comma list: 4000/3969, 110592/109375

Mapping: [⟨5 1 7 21], ⟨0 3 2 -3]]

Optimal tunings:

  • CTE: ~144/125 = 1\5, ~175/128 = 554.5146 (~25/24 = 74.5146)
  • POTE: ~144/125 = 1\5, ~175/128 = 554.9480 (~25/24 = 74.9480)

Optimal ET sequence: 15, 50d, 65d, 80

Badness: 0.137695

11-limit

Subgroup: 2.3.5.7.11

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

Mapping: [⟨5 1 7 21 15], ⟨0 3 2 -3 1]]

Optimal tunings:

  • CTE: ~55/48 = 1\5, ~11/8 = 554.4664 (~25/24 = 74.4664)
  • POTE: ~55/48 = 1\5, ~11/8 = 554.9401 (~25/24 = 74.9401)

Optimal ET sequence: 15, 50d, 65d, 80

Badness: 0.043508

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 176/175, 351/350, 1040/1029, 1331/1323

Mapping: [⟨5 1 7 21 15 37], ⟨0 3 2 -3 1 -8]]

Optimal tunings:

  • CTE: ~55/48 = 1\5, ~11/8 = 554.6802 (~25/24 = 74.6802)
  • CWE: ~55/48 = 1\5, ~11/8 = 554.6627 (~25/24 = 74.6627)

Optimal ET sequence: 15f, 50df, 65d, 80, 145d

Badness: 0.0446

17-limit

We extend to prime 17 by using the sharp tendency of prime 5 to justify tempering out (16/15)/(17/16) = S16. Note that prime 3 is also tuned sharp (though less than prime 5) in optimized tunings.

Subgroup: 2.3.5.7.11.13.17

Comma list: 176/175, 351/350, 1040/1029, 1331/1323, 256/255

Mapping: [⟨5 1 7 21 15 37 32], ⟨0 3 2 -3 1 -8 -5]]

Optimal tunings:

  • CTE: ~55/48 = 1\5, ~11/8 = 554.722 (~25/24 = 74.722)
  • CWE: ~55/48 = 1\5, ~11/8 = 554.614 (~25/24 = 74.614)

Optimal ET sequence: 15f, 50dfg, 65d, 80, 145d

Badness: ?

Badness (Sintel): 1.609

19-limit

We extend to prime 19 by tempering out 361/360 = S19 or equivalently 400/399 = S20, whose naturalness becomes much clearer when we consider it in the 23-limit as the result of tempering out (23/19)/(22/20)2 = S20/S22, relying on the surprisingly obvious mapping of 23/16 as one period above 5/4 so that ~23/20 = 1\5.

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 176/175, 351/350, 1040/1029, 1331/1323, 256/255, 190/189,

Mapping: [⟨5 1 7 21 15 37 32 12], ⟨0 3 2 -3 1 -8 -5 4]]

Optimal tunings:

  • CTE: ~55/48 = 1\5, ~11/8 = 554.698 (~25/24 = 74.698)
  • CWE: ~55/48 = 1\5, ~11/8 = 554.667 (~25/24 = 74.667)

Optimal ET sequence: 15f, 65d, 80

Badness: ?

Badness (Sintel): 1.542

23-limit

As mentioned, prime 23 is found as prime 5 plus a period (up to octave-equivalence). This id done by tempering out (55/48)/(23/20) = 276/275 which is 3025/3024 flat of 253/252. Curiously, the CTE and CWE tunings are almost exactly the same here (different by about a hundredth of a cent).

Subgroup: 2.3.5.7.11.13.17.19.23

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

Mapping: [⟨5 1 7 21 15 37 32 12 18], ⟨0 3 2 -3 1 -8 -5 4 2]]

Optimal tunings:

  • CTE: ~23/20 = 1\5, ~11/8 = 554.689 (~24/23 = 74.689)
  • CWE: ~23/20 = 1\5, ~11/8 = 554.688 (~24/23 = 74.688)

Optimal ET sequence: 15f, 65d, 80

Badness: ?

Badness (Sintel): 1.463

29-limit

There's a surprisingly obvious mapping of 29/16 as two periods above 11/8 so that ~29/22 = 2\5 and meaning equating ~32/29 with ~11/10, the generator. This defines trisedodge as being an unambiguously full 29-limit temperament, with an interesting feature of having two possible mappings of prime 7 and 13; prime 7 can either be mapped the more accurate way as septimal trisedodge does or it can be mapped as it is approximated in 5edo, while prime 13 can alternately be found as 8 generators up instead of down, corresponding to #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 MOS 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: 2.3.5.7.11.13.17.19.23.29

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

Mapping: [⟨5 1 7 21 15 37 32 12 18 22], ⟨0 3 2 -3 1 -8 -5 4 2 1]]

Optimal tunings:

  • CTE: ~23/20 = 1\5, ~11/8 = 554.673 (~24/23 = 74.673)
  • CWE: ~23/20 = 1\5, ~11/8 = 554.684 (~24/23 = 74.684)

Optimal ET sequence: 15f, 65d, 80

Badness: ?

Badness (Sintel): 1.380

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

Comma list: 176/175, 325/324, 364/363, 640/637

Mapping: [⟨5 1 7 21 15 0], ⟨0 3 2 -3 1 8]]

Optimal tunings:

  • CTE: ~55/48 = 1\5, ~11/8 = 554.7405 (~25/24 = 74.7405)
  • CWE: ~55/48 = 1\5, ~11/8 = 555.1626 (~25/24 = 75.1626)

Optimal ET sequence: 15, 80, 175bcde, 255bcdde

Badness: 0.0380

Coblack

See also: Cloudy clan § 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 blacksmith[1].

Subgroup: 2.3.5.7

Comma list: 126/125, 16807/16384

Mapping: [⟨5 1 7 14], ⟨0 3 2 0]]

Optimal tunings:

  • CTE: ~8/7 = 1\5, ~48/35 = 553.8429 (~21/20 = 73.8429)
  • POTE: ~8/7 = 1\5, ~48/35 = 553.044 (~21/20 = 73.044)

Optimal ET sequence: 15, 35, 50, 65, 115d

Badness: 0.107282

11-limit

Subgroup: 2.3.5.7.11

Comma list: 126/125, 245/242, 385/384

Mapping: [⟨5 1 7 14 15], ⟨0 3 2 0 1]]

Optimal tunings:

  • CTE: ~8/7 = 1\5, ~11/8 = 553.7951 (~21/20 = 73.7951)
  • POTE: ~8/7 = 1\5, ~11/8 = 553.264 (~21/20 = 73.264)

Optimal ET sequence: 15, 35, 50, 65, 115d

Badness: 0.045070

Notes

  1. ↑ 1.0 1.1 Yahoo! Tuning Group | Suggested names for the unclasified temperaments
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