6edϕ: Difference between revisions

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'''6 equal divisions of [[acoustic phi]]''' ('''6edϕ''') is the [[nonoctave]] [[tuning system]] derived by dividing [[acoustic phi]] into 6 equal steps of 138.8 [[cent]]s each~~, or the sixth root of the golden ratio~~. It is ~~best known as the equal~~-~~tempered version~~ of ~~the~~ [[~~Bohlen-Pierce~~]] ~~scale~~.

'''6 equal divisions of [[acoustic phi]]''' ('''6edϕ''') is the [[nonoctave]] [[tuning system]] derived by dividing [[acoustic phi]] into 6 equal steps of 138.8 [[cent]]s each. It is well-approximated by every third step of [[26edo]], and somewhat less accurately by [[5edf]].

It serves as a tempered approximation for the [[metallic harmonic series]], and the associated [[metallic intonation]]. The first metallic harmonic (the golden ratio) is represented exactly by the sixth step, while the second and third metallic harmonics are ~~represented~~ by the 11th and 15th steps respectively.

It serves as a basic tempered approximation for the [[metallic harmonic series]], and the associated [[metallic intonation]]. The first metallic harmonic (acoustic phi or the golden ratio) is represented exactly by the sixth step, while the second and third metallic harmonics are approximated by the 11th and 15th steps respectively, which reduce to the fifth and third steps if acoustic phi is taken as an equivalence. The fourth metallic harmonic is also represented exactly because it is the cube of acoustic phi. However, the fifth metallic harmonic is poorly approximated, falling about halfway between two and three steps when reduced.

== Interval table ==

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 {{Infobox ET}}
-'''6 equal divisions of [[acoustic phi]]''' ('''6edϕ''') is the [[nonoctave]] [[tuning system]] derived by dividing [[acoustic phi]] into 6 equal steps of 138.8 [[cent]]s each, or the sixth root of the golden ratio. It is best known as the equal-tempered version of the [[Bohlen-Pierce]] scale.
+'''6 equal divisions of [[acoustic phi]]''' ('''6edϕ''') is the [[nonoctave]] [[tuning system]] derived by dividing [[acoustic phi]] into 6 equal steps of 138.8 [[cent]]s each. It is well-approximated by every third step of [[26edo]], and somewhat less accurately by [[5edf]].
-It serves as a tempered approximation for the [[metallic harmonic series]], and the associated [[metallic intonation]]. The first metallic harmonic (the golden ratio) is represented exactly by the sixth step, while the second and third metallic harmonics are represented by the 11th and 15th steps respectively.
+It serves as a basic tempered approximation for the [[metallic harmonic series]], and the associated [[metallic intonation]]. The first metallic harmonic (acoustic phi or the golden ratio) is represented exactly by the sixth step, while the second and third metallic harmonics are approximated by the 11th and 15th steps respectively, which reduce to the fifth and third steps if acoustic phi is taken as an equivalence. The fourth metallic harmonic is also represented exactly because it is the cube of acoustic phi. However, the fifth metallic harmonic is poorly approximated, falling about halfway between two and three steps when reduced.
 == Interval table ==
 {{Interval table}}
+{{Stub}}