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Trisedodge family: Difference between revisions

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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>.  
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.
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).  
 
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
Line 21: Line 15:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~144/125 = 240.0000{{c}}, ~864/625 = 553.8249{{c}} (~25/24 = 73.8249{{c}})
* [[WE]]: ~144/125 = 239.9482{{c}}, ~864/625 = 553.8881{{c}} (~25/24 = 73.9917{{c}})
* [[POTE]]: ~144/125 = 240.0000{{c}}, ~864/625 = 554.0077{{c}} (~25/24 = 74.0077{{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 }}
 
{{Optimal ET sequence|legend=1| 15, 35, 50, 65, 340c, 405c, …, 600c }}
 
[[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.


{{Optimal ET sequence|legend=1| 15, 35, 50, 65, 340c, 405c, 470c, 535c, 600c }}
Temperaments discussed elsewhere include [[15th-octave temperaments #Quindecic|quindecic]] and [[Stearnsmic clan #Decistearn|decistearn]]. Considered below are trisedodge and coblack.


[[Badness]] (Smith): 0.252724
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 ===
=== Countdown ===
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Optimal tunings:  
Optimal tunings:  
* CTE: ~55/48 = 240.0000{{c}}, ~11/8 = 553.7951{{c}} (~25/24 = 73.7951{{c}})
* WE: ~55/48 = 240.0000{{c}}, ~11/8 = 553.9041{{c}} (~25/24 = 74.0952{{c}})
* POTE: ~55/48 = 240.0000{{c}}, ~11/8 = 554.1247{{c}} (~25/24 = 74.1247{{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 }}
{{Optimal ET sequence|legend=0| 15, 35, 50, 65, 210e, 275e, 340ce }}


RMS error: 0.3198 cents
Badness (Sintel): 0.794


== Septimal trisedodge ==
== 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
[[Subgroup]]: 2.3.5.7


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[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~144/125 = 240.0000{{c}}, ~175/128 = 554.5146{{c}} (~25/24 = 74.5146{{c}})
* [[WE]]: ~144/125 = 239.7187{{c}}, ~175/128 = 554.2976{{c}} (~25/24 = 74.8601{{c}})
* [[POTE]]: ~144/125 = 240.0000{{c}}, ~175/128 = 554.9480{{c}} (~25/24 = 74.9480{{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 }}


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


Badness (Smith): 0.137695
[[Badness]] (Sintel): 3.48


=== 11-limit ===
=== 11-limit ===
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Optimal tunings:  
Optimal tunings:  
* CTE: ~55/48 = 240.0000{{c}}, ~11/8 = 554.4664{{c}} (~25/24 = 74.4664{{c}})
* WE: ~55/48 = 239.7335{{c}}, ~11/8 = 554.3239{{c}} (~25/24 = 74.8569{{c}})
* POTE: ~55/48 = 240.0000{{c}}, ~11/8 = 554.9401{{c}} (~25/24 = 74.9401{{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 }}
{{Optimal ET sequence|legend=0| 15, 50d, 65d, 80 }}


Badness (Smith): 0.043508
Badness (Sintel): 1.44


==== 13-limit ====
==== 13-limit ====
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Optimal tunings:
Optimal tunings:
* CTE: ~55/48 = 240.0000{{c}}, ~11/8 = 554.6802{{c}} (~25/24 = 74.6802{{c}})
* 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}})
* CWE: ~55/48 = 240.0000{{c}}, ~11/8 = 554.6627{{c}} (~25/24 = 74.6627{{c}})


{{Optimal ET sequence|legend=0| 15f, 50df, 65d, 80, 145d }}
{{Optimal ET sequence|legend=0| 15f, 50df, 65d, 80, 145d }}


Badness (Smith): 0.0446
Badness (Sintel): 1.84


==== 17-limit ====
==== 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
Subgroup: 2.3.5.7.11.13.17


Comma list: 176/175, 351/350, 1040/1029, 1331/1323, 256/255
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 }}
Mapping: {{mapping| 5 1 7 21 15 37 32 | 0 3 2 -3 1 -8 -5 }}


Optimal tunings:
Optimal tunings:
* CTE: ~55/48 = 240.000{{c}}, ~11/8 = 554.722{{c}} (~25/24 = 74.722{{c}})
* CTE: ~55/48 = 239.7935{{c}}, ~11/8 = 554.1209{{c}} (~25/24 = 74.5340{{c}})
* CWE: ~55/48 = 240.000{{c}}, ~11/8 = 554.614{{c}} (~25/24 = 74.614{{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 }}
{{Optimal ET sequence|legend=0| 15f, 50dfg, 65d, 80, 145d }}


Badness (Sintel): 1.609
Badness (Sintel): 1.61


==== 19-limit ====
==== 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
Subgroup: 2.3.5.7.11.13.17.19


Comma list: 176/175, 351/350, 1040/1029, 1331/1323, 256/255, 190/189,
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 }}
Mapping: {{mapping| 5 1 7 21 15 37 32 12 | 0 3 2 -3 1 -8 -5 4 }}


Optimal tunings:
Optimal tunings:
* CTE: ~55/48 = 240.000{{c}}, ~11/8 = 554.698{{c}} (~25/24 = 74.698{{c}})
* WE: ~55/48 = 239.8147{{c}}, ~11/8 = 554.2353{{c}} (~25/24 = 74.6060{{c}})
* CWE: ~55/48 = 240.000{{c}}, ~11/8 = 554.667{{c}} (~25/24 = 74.667{{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 }}
{{Optimal ET sequence|legend=0| 15f, 65d, 80 }}


Badness (Sintel): 1.542
Badness (Sintel): 1.54


==== 23-limit ====
==== 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
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
Comma list: 176/175, 190/189, 253/252, 256/255, 351/350, 1040/1029, 1331/1323


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


Optimal tunings:
Optimal tunings:
* CTE: ~23/20 = 240.0000{{c}}, ~11/8 = 554.689{{c}} (~24/23 = 74.689{{c}})
* WE: ~23/20 = 239.8299{{c}}, ~11/8 = 554.2946{{c}} (~24/23 = 74.6347{{c}})
* CWE: ~23/20 = 240.0000{{c}}, ~11/8 = 554.688{{c}} (~24/23 = 74.688{{c}})
* CWE: ~23/20 = 240.0000{{c}}, ~11/8 = 554.6878{{c}} (~24/23 = 74.6878{{c}})


{{Optimal ET sequence|legend=0| 15f, 65d, 80 }}
{{Optimal ET sequence|legend=0| 15f, 65d, 80 }}


Badness (Sintel): 1.463
Badness (Sintel): 1.46


==== 29-limit ====
==== 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
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
Comma list: 176/175, 190/189, 232/231, 253/252, 256/255, 351/350, 1040/1029, 1331/1323


Mapping: {{mapping| 5 1 7 21 15 37 32 12 18 22 | 0 3 2 -3 1 -8 -5 4 2 1 }}
Mapping: {{mapping| 5 1 7 21 15 37 32 12 18 22 | 0 3 2 -3 1 -8 -5 4 2 1 }}


Optimal tunings:
Optimal tunings:
* CTE: ~23/20 = 240.000{{c}}, ~11/8 = 554.673{{c}} (~24/23 = 74.673{{c}})
* WE: ~23/20 = 239.8259{{c}}, ~11/8 = 554.2822{{c}} (~24/23 = 74.6304{{c}})
* CWE: ~23/20 = 240.000{{c}}, ~11/8 = 554.684{{c}} (~24/23 = 74.684{{c}})
* CWE: ~23/20 = 240.0000{{c}}, ~11/8 = 554.6835{{c}} (~24/23 = 74.6835{{c}})


{{Optimal ET sequence|legend=0| 15f, 65d, 80 }}
{{Optimal ET sequence|legend=0| 15f, 65d, 80 }}


Badness (Sintel): 1.380
Badness (Sintel): 1.38


==== Trisey ====
==== Trisey ====
Line 166: Line 167:


Optimal tunings:
Optimal tunings:
* CTE: ~55/48 = 240.0000{{c}}, ~11/8 = 554.7405{{c}} (~25/24 = 74.7405{{c}})
* 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}})
* 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 }}
{{Optimal ET sequence|legend=0| 15, 80, 175bcde, 255bcdde }}


Badness (Smith): 0.0380
Badness (Sintel): 1.57


== Coblack ==
== Coblack ==
Line 183: Line 184:


[[Optimal tuning]]s:  
[[Optimal tuning]]s:  
* [[CTE]]: ~8/7 = 240.0000{{c}}, ~48/35 = 553.8429{{c}} (~21/20 = 73.8429{{c}})
* [[WE]]: ~8/7 = 240.2499{{c}}, ~48/35 = 553.6203{{c}} (~21/20 = 73.1204{{c}})
* [[POTE]]: ~8/7 = 240.000{{c}}, ~48/35 = 553.044{{c}} (~21/20 = 73.044{{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 }}
{{Optimal ET sequence|legend=1| 15, 35, 50, 65, 115d }}


[[Badness]] (Smith): 0.107282
[[Badness]] (Sintel): 2.71


=== 11-limit ===
=== 11-limit ===
Line 198: Line 201:


Optimal tunings:  
Optimal tunings:  
* CTE: ~8/7 = 240.0000{{c}}, ~11/8 = 553.7951{{c}} (~21/20 = 73.7951{{c}})
* WE: ~8/7 = 240.1524{{c}}, ~11/8 = 553.6154{{c}} (~21/20 = 73.3106{{c}})
* POTE: ~8/7 = 240.000{{c}}, ~11/8 = 553.264{{c}} (~21/20 = 73.264{{c}})
* CWE: ~8/7 = 240.0000{{c}}, ~11/8 = 553.3989{{c}} (~21/20 = 73.3989{{c}})


{{Optimal ET sequence|legend=0| 15, 35, 50, 65, 115d }}
{{Optimal ET sequence|legend=0| 15, 35, 50, 65, 115d }}


Badness (Smith): 0.045070
Badness (Sintel): 1.49


== References ==
== References ==
Retrieved from "https://en.xen.wiki/w/Trisedodge_family"