User:Eufalesio/Harmonic Segments Along Fifths: Difference between revisions
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Harmonic Segments Along Fifths (HSA5) are a type of [[periodic scale]] | {{Editable user page}} | ||
Harmonic Segments Along Fifths (HSA5) [sing. ''a Harmonic Segment along Fifths''] are a type of [[periodic scale]] built by taking a specific [[harmonic series]] segment that ends in 4/3 and repeating times 3/2. They are a unique mix of [[tetrachord]]<nowiki/>s and [[overtone scales]] that exhibit [[omnitetrachordality]] to some extent, and a type of [[NEJI]] (Near-equivalent Just Intonation). How to approach them musically is not within the scope of this article. | |||
== Definitions == | == Definitions == | ||
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Except the first, the harmonic segments conforming HSA5s up to the 13th are [[strictly proper]] [[constant structure]]<nowiki/>s. | Except the first, the harmonic segments conforming HSA5s up to the 13th are [[strictly proper]] [[constant structure]]<nowiki/>s. | ||
The HSA5s beyond the 3rd will have noticeable gaps between 4/3 and 3/2. | The HSA5s beyond the 3rd will have noticeable gaps between 4/3 and 3/2. A HSA5 has the 9/8 step between 4/3 and 3/2 that never shrinks in size, creating very lopsided scales with fairly complex HSA5s down the line. The gap between 4/3 and 3/2 can be filled in and this can be done two ways: | ||
* Insert intervals within the subgroup of the harmonic segment (Subgroup-extended HSA5) [seHSA5] so that the step variety is minimized. This is an arbitrary choice that depends on the HSA5. | * Insert intervals within the subgroup of the harmonic segment (Subgroup-extended HSA5) [seHSA5] so that the step variety is minimized. This is an arbitrary choice that depends on the HSA5. | ||
* Extend the main harmonic segment beyond 4/3 (Harmonically-extended HSA5) [heHSA5] until the segment between 4/3 and 3/2 has fairly well spaced steps. When the HSA5s is even, this is trivial. | * Extend the main harmonic segment beyond 4/3 (Harmonically-extended HSA5) [heHSA5] until the segment between 4/3 and 3/2 has fairly well spaced steps. When the HSA5s is even, this is trivial. | ||
These are called ''extensions''. They are fairly arbitrary and not an absolute necessity, so the use of one over the other is not something that is entirely justifiable; it may be as much as an artistic choice as the choice of HSA5 itself. | |||
== List of HSA5s == | == List of HSA5s == | ||
=== HSA5 1 - Pythagorean Trial === | === HSA5 1 - Pythagorean Trial === | ||
'''3::4'''. First meaningful HSA5 albeit a trivial case, and the only HSA5 that is also a [[MOS scale]] ([[2L 1s]]), and thus [[achiral]]. | '''3::4'''. First meaningful HSA5 albeit a trivial case, and the only HSA5 that is also a [[MOS scale]] ([[2L 1s]]), and thus [[achiral]]. | ||
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=== HSA5 4 - ''Ngwóoghe'' === | === HSA5 4 - ''Ngwóoghe'' === | ||
'''12::16'''. It is first composite HSA5, containing the 2nd as a subset, and the first one to need extensions. It is naturally a [[10edo]] or detemper in the [[2.3.5.7.13 subgroup]], or otherwise a [[negri]] detemper, [[1L 8s]] 4|4 scale. The only valid heSAH5 is [[17/12]]. Reasonable seHSA5s include 7/5, 13/9, 45/32. | '''12::16'''. It is the first composite HSA5, containing the 2nd as a subset, and the first one to need extensions. It is naturally a [[10edo]] or detemper in the [[2.3.5.7.13 subgroup]], or otherwise a [[negri]] detemper, [[1L 8s]] 4|4 scale. The only valid heSAH5 is [[17/12]]. Reasonable seHSA5s include 7/5, 13/9, 45/32. | ||
{| class="wikitable" data-darkreader-inline-color="" | {| class="wikitable" data-darkreader-inline-color="" | ||
! colspan="9" |HSA5 4 | ! colspan="9" |HSA5 4 | ||