|
|
| Line 47: |
Line 47: |
| [[#Sensible|Sensible]] is a less sparse subgroup extension present in smaller edo tunings like [[111edo]] at the cost of a little accuracy. [[#Sendai|Sendai]] is more accurate but the subgroup is more sparse. Combining both with 2.3.5.7.13-subgroup sensi (called sensation) yields a high-limit temperament that is very constrained in tuning, for which [[65edo]] is an approximate tuning. | | [[#Sensible|Sensible]] is a less sparse subgroup extension present in smaller edo tunings like [[111edo]] at the cost of a little accuracy. [[#Sendai|Sendai]] is more accurate but the subgroup is more sparse. Combining both with 2.3.5.7.13-subgroup sensi (called sensation) yields a high-limit temperament that is very constrained in tuning, for which [[65edo]] is an approximate tuning. |
|
| |
|
| === Sensible ===
| |
| {{See also| Sensipent #Sensible interval table }}
| |
|
| |
| Sensible is an extension of sensipent with prime 11 of dubious canonicity but significantly higher accuracy than [[sensi]]. It interprets the generator as [[165/128]]~[[128/99]] by tempering out [[8019/8000]] so that [[11/8]] is reached as ([[10/9]])<sup>3</sup>. This extension is very strong as supported by the [[optimal ET sequence]] going very far and as supported by another observation that it also tempers out the [[semiporwellisma]], which is equal to [[S-expression|S31⋅S32<sup>2</sup>]] (thus forming the S-expression-based comma list). The vanish of the semiporwellisma, a [[lopsided comma]], implies that this temperament equates ([[33/32]])<sup>2</sup> with [[16/15]] as well as that a natural extension to prime 31 exists through {[[961/960]] ({{s|31}}), [[1024/1023]] ({{s|32}})}, which we will see is very accurate, but this itself suggests that an extension with prime 17 is reasonably accurate through tempering out [[1089/1088]] ({{s|33}}) so that a slightly sharp ~[[22/17]] is equated with the generator.
| |
|
| |
| The aforementioned extension with prime 17 through tempering out 1089/1088 implies tempering out [[256/255]] ({{s|16}}), as {{nowrap| 256/255 {{=}} (22/17)/(165/128) }}.
| |
|
| |
| Sensible uses the accurate mapping of prime 31 in sensipent, so that the sensible generator serves many roles in subgroup harmony, but it is not ~[[9/7]] or ~[[13/10]] which would incur more damage. Its [[S-expression]]-based comma list {{nowrap| is {([[8019/8000|S9/S10]], [[256/255|S16]],) [[529/528|S23]], [[576/575|S24]], [[961/960|S31]], [[1024/1023|S32]], [[1089/1088|S33]]} }} implying also tempering out [[496/495]] (S31⋅S32) and [[528/527]] (S32⋅S33) as well as [[16337/16335]] (S31/S33) = ([[17/15]])/([[33/31]])<sup>2</sup>. A notable [[patent val]] tuning not appearing in the optimal ET sequence is [[157edo]]. A notable non-patent val is [[84edo|84g]].
| |
|
| |
| Subgroup: 2.3.5.11
| |
|
| |
| Comma list: 8019/8000, 16384/16335
| |
|
| |
| Subgroup-val mapping: {{mapping| 1 -1 -1 9 | 0 7 9 -15 }}
| |
|
| |
| : mapping generators: ~2, ~128/99
| |
|
| |
| Optimal tunings:
| |
| * WE: ~2 = 1199.6725{{c}}, ~128/99 = 443.0183{{c}}
| |
| * CWE: ~2 = 1200.0000{{c}}, ~128/99 = 443.1341{{c}}
| |
|
| |
| {{Optimal ET sequence|legend=0| 19, 46, 65, 176, 241, 306 }}
| |
|
| |
| Badness (Sintel): 0.728
| |
|
| |
| ==== 2.3.5.11.17 subgroup ====
| |
| Subgroup: 2.3.5.11.17
| |
|
| |
| Comma list: 256/255, 1089/1088, 1377/1375
| |
|
| |
| Subgroup-val mapping: {{mapping| 1 -1 -1 9 10 | 0 7 9 -15 -16 }}
| |
|
| |
| : mapping generators: ~2, ~22/17
| |
|
| |
| Optimal tunings:
| |
| * WE: ~2 = 1199.5016{{c}}, ~22/17 = 443.0038{{c}}
| |
| * CWE: ~2 = 1200.0000{{c}}, ~22/17 = 443.1878{{c}}
| |
|
| |
| {{Optimal ET sequence|legend=0| 19, 46, 65, 111, 176g }}
| |
|
| |
| Badness (Sintel): 0.639
| |
|
| |
| ==== 2.3.5.11.17.23 subgroup ====
| |
| Subgroup: 2.3.5.11.17.23
| |
|
| |
| Comma list: 256/255, 576/575, 1089/1088, 1377/1375
| |
|
| |
| Subgroup-val mapping: {{mapping| 1 -1 -1 9 10 6 | 0 7 9 -15 -16 -4 }}
| |
|
| |
| Optimal tunings:
| |
| * WE: ~2 = 1199.6207{{c}}, ~22/17 = 443.0400{{c}}
| |
| * CWE: ~2 = 1200.0000{{c}}, ~22/17 = 443.1808{{c}}
| |
|
| |
| {{Optimal ET sequence|legend=0| 19, 46, 65, 111, 176g }}
| |
|
| |
| Badness (Sintel): 0.555
| |
|
| |
| ==== 2.3.5.11.17.23.31 subgroup ====
| |
| Subgroup: 2.3.5.11.17.23.31
| |
|
| |
| Comma list: 256/255, 576/575, 961/960, 1089/1088, 1377/1375
| |
|
| |
| Subgroup-val mapping: {{mapping| 1 -1 -1 9 10 6 2 | 0 7 9 -15 -16 -4 8 }}
| |
|
| |
| Optimal tunings:
| |
| * WE: ~2 = 1199.6623{{c}}, ~22/17 = 443.0616{{c}}
| |
| * CWE: ~2 = 1200.0000{{c}}, ~22/17 = 443.1858{{c}}
| |
|
| |
| {{Optimal ET sequence|legend=0| 19, 46, 65, 111, 176g }}
| |
|
| |
| Badness (Sintel): 0.490
| |
|
| |
| === Overview to extensions ===
| |
| ==== Full 7-limit extensions ==== | | ==== Full 7-limit extensions ==== |
| The second comma of the comma list determines which 7-limit family member we are looking at. Sensi adds [[126/125]]. Sensei adds [[225/224]]. Warrior adds [[5120/5103]]. These are all strong extensions that use the same period and generator as sensipent. | | The second comma of the comma list determines which 7-limit family member we are looking at. Sensi adds [[126/125]]. Sensei adds [[225/224]]. Warrior adds [[5120/5103]]. These are all strong extensions that use the same period and generator as sensipent. |