Xenharmonic Wiki

Trisedodge family: Difference between revisions

← Older edit
VisualWikitext
Xenwolf (talk | contribs)
autopatrolled, Bureaucrats, confirmaccount-notify, Interface administrators, Sysops
8,705 edits
 
(32 intermediate revisions by 8 users not shown)
Line 1: Line 1:
The '''trisedodge family''' tempers out the trisedodge comma, 30958682112/30517578125 = {{monzo| 19 10 -15 }}.
{{Technical data page}}
The '''trisedodge family''' tempers out the [[trisedodge comma]], 30958682112/30517578125 = {{monzo| 19 10 -15 }}.


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


Comma list: 30958682112/30517578125
== 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.


Mapping: [{{val| 5 7 11 }}, {{val| 0 3 2 }}]
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.


POTE generator: ~25/24 = 74.0077
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.


{{Val list|legend=1| 15, 50, 65, 340c, 405c, 470c, 535c, 600c }}
[[Subgroup]]: 2.3.5


== 7-limit  ==
[[Comma list]]: 30958682112/30517578125


Subgroup: 2.3.5.7
{{Mapping|legend=1| 5 1 7 | 0 3 2 }}
 
: mapping generators: ~144/125, ~864/625
 
[[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, 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: {{mapping| 5 1 7 15 | 0 3 2 1 }}
 
[[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 }}
 
RMS error: 0.3198 cents
 
== Septimal trisedodge ==
[[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:
* [[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)


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


=== 11-limit  ===
Badness: 0.137695


=== 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:
* 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|legend=1| 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: {{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: {{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.


POTE generator: ~25/24 = 74.9401
Subgroup: 2.3.5.7.11.13.17


{{Val list|legend=1| 15, 50d, 65d, 80 }}
Comma list: 176/175, 351/350, 1040/1029, 1331/1323, 256/255


== Coblack  ==
Mapping: {{mapping| 5 1 7 21 15 37 32 | 0 3 2 -3 1 -8 -5 }}
{{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.
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)


Subgroup: 2.3.5.7
Optimal ET sequence: {{Optimal ET sequence| 15f, 50dfg, 65d, 80, 145d }}


[[Comma list]]: 126/125, 16807/16384
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 }}


[[Mapping]]: [{{val| 5 1 7 14 }}, {{val| 0 3 2 0 }}]
Badness: ?


[[POTE generator]]: ~21/20 = 73.044
Badness (Sintel): 1.463


{{Val list|legend=1| 15, 35, 50, 65, 115d }}
==== 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


[[Badness]]: 0.1073
Subgroup: 2.3.5.7.11.13.17.19.23.29


=== 11-limit  ===
Comma list: 176/175, 351/350, 1040/1029, 1331/1323, 256/255, 190/189, 253/252, 232/231


Subgroup: 2.3.5.7.11
Mapping: {{mapping| 5 1 7 21 15 37 32 12 18 22 | 0 3 2 -3 1 -8 -5 4 2 1 }}


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


Mapping: [{{val| 5 1 7 14 15 }}, {{val| 0 3 2 0 1 }}]
Optimal ET sequence: {{Optimal ET sequence| 15f, 65d, 80 }}


POTE generator: ~21/20 = 73.264
Badness: ?


{{Val list|legend=1| 15, 35, 50, 65, 115d }}
Badness (Sintel): 1.380


== 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:
* 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 ==
{{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]]<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:
* [[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 }}
 
[[Badness]]: 0.107282
 
=== 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:
* 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)


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


POTE generator: ~66/65 = 27.0764
Badness: 0.045070


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


[[Category:Regular temperament theory]]
[[Category:Temperament families]]
[[Category:Temperament family]]
[[Category:Pages with mostly numerical content]]
[[Category:Trisedodge]]
[[Category:Trisedodge family| ]] <!-- main article -->
[[Category:Trisedodge| ]] <!-- key article -->
[[Category:Rank 2]]
[[Category:Rank 2]]
Retrieved from "https://en.xen.wiki/w/Trisedodge_family"