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If we take a look at the 5-limit version of sensi called [[sensipent]], we find a high-accuracy extension that specifically only requires prime 31, interpreting the generator accurately as [[~]][[40/31]][[~]][[31/24]] (by splitting [[16/15]] into [[~]][[32/31]][[~]][[31/30]]). This can be left as is, or one can extend to other slightly less accurate primes; the main two strategies for doing so are called [[sendai]], focusing on accuracy and adding primes 23 and 29, and [[sensible]], which adds primes 11, 17 and 23 and focuses on adding more primes, in both cases doing so while avoiding the less accurate [[~]][[9/7]] and [[~]][[13/10]] interpretations of the sensi generator. They merge meaningfully (though ''not'' uniquely) in [[65edo]], which can be seen by that [[65edo]] is an amazing no-7's no-13's [[31-limit]] temperament, where we've gained prime 19 through a possible extension of either sendai or sensible. Furthermore, [[65edo]] can also be used as a tuning of 7-limit sensi through the 65d [[val]] (which corresponds to 65edo roughly supporting [[garibaldi]]), though note that if one tries to use its patent but very sharp [[~]][[13/8|13]] (which makes the most sense if one accepts the 65d val) then [[13/10]] is mapped distinctly and sharp of the [[~]][[9/7]] sensi generator.
If we take a look at the 5-limit version of [[sensi]] called [[sensipent]], we find a high-accuracy extension that specifically only requires prime 31, interpreting the generator accurately as [[~]][[40/31]]~[[31/24]] (by splitting [[16/15]] into ~[[32/31]]~[[31/30]]). This can be left as is, or one can extend to other slightly less accurate primes; the main two strategies for doing so are called [[sendai]], focusing on accuracy and adding primes 23 and 29, and [[sensible]], which adds primes 11, 17 and 23 and focuses on adding more primes, in both cases doing so while avoiding the less accurate ~[[9/7]] and ~[[13/10]] interpretations of the sensi generator. They merge meaningfully (though ''not'' uniquely) in [[65edo]], which can be seen by that 65edo is an amazing no-7's no-13's [[31-limit]] temperament, where we've gained prime 19 through a possible extension of either sendai or sensible. Furthermore, 65edo can also be used as a tuning of 7-limit sensi, though the mapping of 13 that identifies the sensi generator as ~13/10 is the 65f val (with a flat 13, which makes better sense than the patent val).


For technical data, see:
For technical data, see:
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=== Sensipent interval table ===
=== Sensipent interval table ===
Amazingly, in the 2.3.5.31-subgroup-limited 155-odd-limit, every interval of every number of generators up to 23 is given at least one interpretation, so that the 27-note MOS ([[19L 8s]]) is surprisingly well-supplied with harmony. The main "holes" are at 24 and 26 gens (as 25 is [[~]][[75/64]]) and that these interpretations tend to be rather complex, requiring a good tuning and a context to justify them. For these reasons, the extensions [[sensible]] and [[sendai]] are likely to be preferred in practice, whose interval tables are thus also documented here, being alternative but higher-accuracy extensions to 2.3.5.31 sensipent.
Amazingly, in the 2.3.5.31-subgroup-limited 155-odd-limit, every interval of every number of generators up to 23 is given at least one interpretation, so that the 27-note [[MOS]] ([[19L 8s]]) is surprisingly well-supplied with harmony. The main "holes" are at 24 and 26 gens (as 25 is ~[[75/64]]) and that these interpretations tend to be rather complex, requiring a good tuning and a context to justify them. For these reasons, the extensions sensible and sendai are likely to be preferred in practice, whose interval tables are thus also documented here, being alternative but higher-accuracy extensions to 2.3.5.31 sensipent.
{| class="wikitable"
{| class="wikitable"
! Gens
! Gens
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|-
|-
| 1
| 1
| 443.05
| 443.047
| [[40/31]], [[31/24]], [[162/125]]
| [[40/31]], [[31/24]], [[162/125]]
|-
|-
| 2
| 2
| 886.1
| 886.095
| [[5/3]]
| [[5/3]]
|-
|-
| 3
| 3
| 129.15
| 129.142
| [[100/93]], [[155/144]], [[27/25]]
| [[100/93]], [[155/144]], [[27/25]]
|-
|-
| 4
| 4
| 572.2
| 572.190
| [[25/18]], [[216/155]]
| [[25/18]], [[216/155]]
|-
|-
| 5
| 5
| 1015.25
| 1015.237
| [[9/5]]
| [[9/5]]
|-
|-
| 6
| 6
| 258.3
| 258.284
| [[125/108]], [[36/31]], [[93/80]]
| [[125/108]], [[36/31]], [[93/80]]
|-
|-
| 7
| 7
| 701.35
| 701.332
| '''[[3/2]]'''
| '''[[3/2]]'''
|-
|-
| 8
| 8
| 1144.4
| 1144.379
| [[60/31]], '''[[31/16]]'''
| [[60/31]], '''[[31/16]]'''
|-
|-
| 9
| 9
| 387.45
| 387.427
| '''[[5/4]]'''
| '''[[5/4]]'''
|-
|-
| 10
| 10
| 830.5
| 830.474
| [[50/31]], [[155/96]], [[81/50]]
| [[50/31]], [[155/96]], [[81/50]]
|-
|-
| 11
| 11
| 73.55
| 73.521
| [[25/24]], [[162/155]]
| [[25/24]], [[162/155]]
|-
|-
| 12
| 12
| 516.6
| 516.569
| [[125/93]], [[27/20]]
| [[125/93]], [[27/20]]
|-
|-
| 13
| 13
| 959.65
| 959.616
| [[125/72]], [[54/31]]
| [[125/72]], [[54/31]]
|-
|-
| 14
| 14
| 202.7
| 202.664
| '''[[9/8]]'''
| '''[[9/8]]'''
|-
|-
| 15
| 15
| 645.75
| 645.711
| [[45/31]], '''[[93/64]]'''
| [[45/31]], '''[[93/64]]'''
|-
|-
| 16
| 16
| 1088.8
| 1088.758
| '''[[15/8]]'''
| '''[[15/8]]'''
|-
|-
| 17
| 17
| 331.85
| 331.806
| [[75/62]], '''[[155/128]]'''
| [[75/62]], '''[[155/128]]'''
|-
|-
| 18
| 18
| 774.9
| 774.853
| '''[[25/16]]'''
| '''[[25/16]]'''
|-
|-
| 19
| 19
| 17.95
| 17.901
| [[125/124]], [[81/80]]
| [[125/124]], [[81/80]]
|-
|-
| 20
| 20
| 461.0
| 460.948
| [[125/96]], [[81/62]]
| [[125/96]], [[81/62]]
|-
|-
| 21
| 21
| 904.05
| 903.995
| '''[[27/16]]'''
| '''[[27/16]]'''
|-
|-
| 22
| 22
| 147.1
| 147.043
| [[135/124]]
| [[135/124]]
|-
|-
| 23
| 23
| 590.15
| 590.090
| '''[[45/32]]'''
| '''[[45/32]]'''
|}
|}
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|-
|-
|1
|1
|442.989338
|442.989
|[[31/24]], [[40/31]]
|[[31/24]], [[40/31]]
|-
|-
|2
|2
|885.978677
|885.979
|[[5/3]]
|[[5/3]]
|-
|-
|3
|3
|128.968015
|128.968
|[[27/25]], [[29/27]]
|[[27/25]], [[29/27]]
|-
|-
|4
|4
|571.957354
|571.957
|[[32/23]], [[25/18]]
|[[32/23]], [[25/18]]
|-
|-
|5
|5
|1014.946692
|1014.947
|[[9/5]]
|[[9/5]]
|-
|-
|6
|6
|257.936031
|257.936
|[[36/31]], [[29/25]]
|[[36/31]], [[29/25]]
|-
|-
|7
|7
|700.925369
|700.925
|[[3/2]]
|[[3/2]]
|-
|-
|8
|8
|1143.914707
|1143.915
|[[31/16]], [[29/15]], [[60/31]]
|[[31/16]], [[29/15]], [[60/31]]
|-
|-
|9
|9
|386.904046
|386.904
|[[5/4]]
|[[5/4]]
|-
|-
|10
|10
|829.893384
|829.893
|[[29/18]], [[50/31]]
|[[29/18]], [[50/31]]
|-
|-
|11
|11
|72.882723
|72.883
|[[24/23]], [[25/24]]
|[[24/23]], [[25/24]]
|-
|-
|12
|12
|515.872061
|515.872
|[[27/20]], [[31/23]]
|[[27/20]], [[31/23]]
|-
|-
|13
|13
|958.8614
|958.861
|[[40/23]], [[54/31]]
|[[40/23]], [[54/31]]
|-
|-
|14
|14
|201.850738
|201.851
|[[9/8]]
|[[9/8]]
|-
|-
|15
|15
|644.840076
|644.840
|[[29/20]]
|[[29/20]]
|-
|-
|16
|16
|1087.829415
|1087.829
|[[15/8]], [[58/31]]
|[[15/8]], [[58/31]]
|-
|-
|17
|17
|330.818753
|330.819
|[[29/24]]
|[[29/24]]
|-
|-
|18
|18
|773.808092
|773.808
|[[25/16]], [[36/23]]
|[[25/16]], [[36/23]]
|}
|}
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|-
|-
| 5
| 5
| 1016.27
| 1016.270
| '''[[115/64]]''', [[124/69]], [[9/5]], [[92/51]]
| '''[[115/64]]''', [[124/69]], [[9/5]], [[92/51]]
|-
|-
Line 230: Line 230:
|-
|-
| 10
| 10
| 832.54
| 832.540
| [[50/31]], [[160/99]], [[186/115]], [[55/34]], [[81/50]], [[138/85]]
| [[50/31]], [[160/99]], [[186/115]], [[55/34]], [[81/50]], [[138/85]]
|-
|-
Line 250: Line 250:
|-
|-
| 15
| 15
| 648.81
| 648.810
| [[100/69]], [[45/31]], '''[[93/64]]''', '''[[16/11]]''', [[99/68]], [[124/85]]
| [[100/69]], [[45/31]], '''[[93/64]]''', '''[[16/11]]''', [[99/68]], [[124/85]]
|-
|-
Line 270: Line 270:
|-
|-
| 20
| 20
| 465.08
| 465.080
| [[30/23]], [[81/62]], [[115/88]], [[72/55]]
| [[30/23]], [[81/62]], [[115/88]], [[72/55]]
|-
|-
Line 290: Line 290:
|-
|-
| 25
| 25
| 281.35
| 281.350
| '''[[75/64]]''', [[27/23]], [[20/17]]
| '''[[75/64]]''', [[27/23]], [[20/17]]
|-
|-
Line 300: Line 300:


{{todo|inline=1|expand}}
{{todo|inline=1|expand}}
[[Category:Temperaments]]
[[Category:Sensipent| ]] <!-- main article -->
[[Category:Sensipent| ]] <!-- main article -->
[[Category:Rank-2 temperaments]]
[[Category:Sensipent family]]
[[Category:Sensipent family]]
Retrieved from "https://en.xen.wiki/w/Sensipent"