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Trisedodge family: Difference between revisions

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→Septimal trisedodge: add extensions
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Badness: 0.043508
Badness: 0.043508


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


Comma list: 176/175, 325/324, 364/363, 640/637
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.
 
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 (Dirichlet): 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 (Dirichlet): 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 (Dirichlet): 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: {{mapping| 5 1 7 21 15 0 | 0 3 2 -3 1 8 }}
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: ~55/48 = 1\5, ~11/8 = 554.7405 (~25/24 = 74.7405)
* CTE: ~55/48 = 1\5, ~11/8 = 554.673 (~25/24 = 74.673)
* CWE: ~55/48 = 1\5, ~11/8 = 555.1626 (~25/24 = 75.1626)
* CWE: ~55/48 = 1\5, ~11/8 = 554.684 (~25/24 = 74.684)
 
Optimal ET sequence: {{Optimal ET sequence| 15f, 65d, 80 }}
 
Badness: ?


Optimal ET sequence: {{Optimal ET sequence| 15, 80, 175bcde, 255bcdde }}
Badness (Dirichlet): 1.380


Badness: 0.0380
==== 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.


==== Dodgy ====
Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13


Comma list: 176/175, 351/350, 1040/1029, 1331/1323
Comma list: 176/175, 325/324, 364/363, 640/637


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


Optimal tunings:
Optimal tunings:
* CTE: ~55/48 = 1\5, ~11/8 = 554.6802 (~25/24 = 74.6802)
* CTE: ~55/48 = 1\5, ~11/8 = 554.7405 (~25/24 = 74.7405)
* CWE: ~55/48 = 1\5, ~11/8 = 554.6627 (~25/24 = 74.6627)
* CWE: ~55/48 = 1\5, ~11/8 = 555.1626 (~25/24 = 75.1626)


Optimal ET sequence: {{Optimal ET sequence| 15f, 50df, 65d, 80, 145d }}
Optimal ET sequence: {{Optimal ET sequence| 15, 80, 175bcde, 255bcdde }}


Badness: 0.0446
Badness: 0.0380


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