148edo: Difference between revisions

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{{~~EDO~~ intro}} 148edo has a fifth on the sharp side, 3.45 cents sharp. The equal temperament tempers out [[2048/2025]] in the 5-limit, 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 }} 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- and 13- limit [[bidia]], 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]]. It provides the [[optimal patent val]] for 11-limit [[echidnic]], the 10 & 46 temperament.

=== ~~Prime harmonics~~ ===

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]]

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 {{Infobox ET}}
-{{EDO intro}} 148edo has a fifth on the sharp side, 3.45 cents sharp. The equal temperament tempers out [[2048/2025]] in the 5-limit, 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 }} 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- and 13- limit [[bidia]], 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]]. It provides the [[optimal patent val]] for 11-limit [[echidnic]], the 10 &amp; 46 temperament.
+{{ED intro}}
-=== Prime harmonics ===
+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.
-{{Harmonics in equal|148}}
+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]]