148edo: Difference between revisions

(8 intermediate revisions by 5 users not shown)

Line 1:

'''148edo''' is the [[equal division of the octave]] into 148 equal parts of 8.108 cents each, near a [[kleisma|kleisma]]. It provides the [[Optimal_patent_val|optimal patent val]] for 11-limit [[Diaschismic_family|echidnic temperament]], the 10&46 temperament. It has a fifth on the sharp side, 3.45 cents sharp. It tempers out 2048/2025 in the 5-limit, making it a diaschismic system. In the 7-limit, the [[Patent_val|patent val]] tempers out 686/675 and 1029/1024, but an alternative mapping <148 235 344 416| with a sharp rather than a flat 7 tempers out 3136/3125 instead, and provides a better tuning than the patent val tuning of [[80edo|80edo]] for 7- and 13- limit [[Diaschismic_family|bidia temperament]], the 12&68 temperament. In the 11-limit, the patent val tempers out 385/384 and 441/440, and the alternative mapping with the sharp 7 tempers out 176/175, 896/891 and 1375/1372 instead. In the 13-limit, the patent val tempers out 325/324 and 364/363, and the alternative val 325/324 again, as well as 640/637 and 847/845.

148 = 4 * 37, ~~with divisors~~ 2, 4, 37, 74.

148edo's closest fifth is on the very sharp side, 3.45 cents sharp of just. With better approximations of [[9/1|9]], [[11/1|11]], [[15/1|15]], [[17/1|17]], and [[21/1|21]], it commends itself as a 2.9.15.21.11.17 [[subgroup]] system.

The 5-limit [[patent val]] still makes sense, and it tempers out [[2048/2025]], making it a [[diaschismic]] system. In the 7-limit, the [[patent val]] tempers out [[686/675]] and [[1029/1024]], but the alternative mapping {{val| 148 235 344 '''416''' }} (148d) with a sharp rather than a flat 7 tempers out [[3136/3125]] instead, and provides a better tuning than the patent val tuning of [[80edo]] for 7-, 13-, 17- and 19-limit [[bidia]], the 68 & 80 temperament. In the 11-limit, the patent val tempers out [[385/384]] and [[441/440]], and the alternative mapping with the sharp 7 tempers out [[176/175]], [[896/891]] and [[1375/1372]] instead. In the 13-limit, the patent val tempers out [[325/324]] and [[364/363]], and the alternative val 325/324 again, as well as [[640/637]] and [[847/845]]. It provides the [[optimal patent val]] for [[echidnic]], the 46 & 102 temperament, in the 11-limit, and the 148f val is an excellent tuning for echidnic in the 13- and 17-limit.

=== Harmonics ===

{{Harmonics in equal|148|columns=9|start=10|title=Approximation of odd harmonics in 148edo (continued)}}

=== Subsets and supersets ===

Since 148 = 4 × 37, 148edo has subset edos {{EDOs| 2, 4, 37, and 74 }}.

[[Category:Echidnic]]

[[Category:Bidia]]

@@ Line 1: / Line 1: @@
-'''148edo''' is the [[equal division of the octave]] into 148 equal parts of 8.108 cents each, near a [[kleisma|kleisma]]. It provides the [[Optimal_patent_val|optimal patent val]] for 11-limit [[Diaschismic_family|echidnic temperament]], the 10&amp;46 temperament. It has a fifth on the sharp side, 3.45 cents sharp. It tempers out 2048/2025 in the 5-limit, making it a diaschismic system. In the 7-limit, the [[Patent_val|patent val]] tempers out 686/675 and 1029/1024, but an alternative mapping &lt;148 235 344 416| with a sharp rather than a flat 7 tempers out 3136/3125 instead, and provides a better tuning than the patent val tuning of [[80edo|80edo]] for 7- and 13- limit [[Diaschismic_family|bidia temperament]], the 12&amp;68 temperament. In the 11-limit, the patent val tempers out 385/384 and 441/440, and the alternative mapping with the sharp 7 tempers out 176/175, 896/891 and 1375/1372 instead. In the 13-limit, the patent val tempers out 325/324 and 364/363, and the alternative val 325/324 again, as well as 640/637 and 847/845.
+{{Infobox ET}}
+{{ED intro}}
-= 4 * 37, with divisors 2, 4, 37, 74.
+edo's closest fifth is on the very sharp side, 3.45 cents sharp of just. With better approximations of [[9/1|9]], [[11/1|11]], [[15/1|15]], [[17/1|17]], and [[21/1|21]], it commends itself as a 2.9.15.21.11.17 [[subgroup]] system.
+The 5-limit [[patent val]] still makes sense, and it tempers out [[2048/2025]], making it a [[diaschismic]] system. In the 7-limit, the [[patent val]] tempers out [[686/675]] and [[1029/1024]], but the alternative mapping {{val| 148 235 344 '''416''' }} (148d) with a sharp rather than a flat 7 tempers out [[3136/3125]] instead, and provides a better tuning than the patent val tuning of [[80edo]] for 7-, 13-, 17- and 19-limit [[bidia]], the 68 &amp; 80 temperament. In the 11-limit, the patent val tempers out [[385/384]] and [[441/440]], and the alternative mapping with the sharp 7 tempers out [[176/175]], [[896/891]] and [[1375/1372]] instead. In the 13-limit, the patent val tempers out [[325/324]] and [[364/363]], and the alternative val 325/324 again, as well as [[640/637]] and [[847/845]]. It provides the [[optimal patent val]] for [[echidnic]], the 46 &amp; 102 temperament, in the 11-limit, and the 148f val is an excellent tuning for echidnic in the 13- and 17-limit.
+=== Harmonics ===
+{{Harmonics in equal|148|columns=9}}
+{{Harmonics in equal|148|columns=9|start=10|title=Approximation of odd harmonics in 148edo (continued)}}
+=== Subsets and supersets ===
+Since 148 = 4 × 37, 148edo has subset edos {{EDOs| 2, 4, 37, and 74 }}.
+[[Category:Echidnic]]
+[[Category:Bidia]]