User:Eufalesio/Harmonic Segments Along Fifths: Difference between revisions

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.

These scales require the harmonic segment to start with a threeven number and end with that number times 4/3, so HSA5 are a countable set of scales, and thus can be indexed.

 

 

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.

* 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. When the HSA5 is odd, you have two "valid" choices.

 

 

Simple HSA5s can be visualized as edo or MOS detempers for simple HSA5s, which makes them particularly useful either for NEJI targets or essential tempering.  

== List of HSA5s 1-8 ==

For every provided HSA5, the [[EFR]] and used segment are written out explicitly.

=== HSA5 1 - Pythagorean Trial ===

First meaningful HSA5 albeit a trivial case, and the only HSA5 that is also a [[MOS scale]] ([[2L 1s]]), and thus [[achiral]].

=== HSA5 2 - ''Zontatonic'' ===

<code>12:14:16:18:21:24</code>. Harmonic segment: '''6::8'''.  

 

First usable HSA5, the first prime HSA5, and the only one that is [[Strict variety]] 3 and a [[generator sequence]].

 

It is naturally a [[5edo]] detemper in the [[2.3.7 subgroup]], and either a semaphore ([[1L 4s]] 2|2) or [[archy]] detemper ([[2L 3s]] 3|1).

=== HSA5 3 - ''Íegmul'' ===

<code>18:20:22:24:27:30:33:36</code>. Harmonic segment: '''9::12'''.  

 

Second prime HSA5, and the last one to not need extensions.  

 

It is naturally a [[7edo]] detemper in the [[2.3.5.11 subgroup]], and either a [[porcupine]] ([[1L 6s]] 3|3) or [[dicot]]/[[mothra]] detemper ([[3L 4s]] 5|1 #5).

=== HSA5 4 - ''Ngwóoghe'' ===

<code>24:26:28:30:32:36:39:42:45:48</code>. Harmonic segment: '''12::16'''.

 

First composite HSA5, containing the 2nd as a subset, and the first one to need extensions. The only valid heSAH5 is [[17/12]]. Reasonable seHSA5s include 7/5, 13/9, 45/32.

 

The raw scale is naturally a [[10edo]] detemper in the [[2.3.5.7.13 subgroup]], and a [[negri]] detemper ([[1L 8s]] 4|4). The extensions are naturally a 10edo detemper and a [[pajarous]] detemper, 2L 8s 6|2 #7 scale.

! colspan="10" |heHSA5 4 (17/12)

!'''17/12'''

|<small>18/13</small>

|<small>39/28</small>

|<small>7/5</small>

|<small>45/32</small>

|<small>24/17</small>

|<small>13/9</small>

|<small>56/39</small>

|<small>10/7</small>

|<small>64/45</small>

 

'''15::20'''. It is the third prime HSA5, and the first HSA5 not to be a constant structure without extensions. The only valid heSAH5 is with [[7/5]]. A very natural and reasonable seHSA5 is 17/12, which fits almost perfectly inside 4/3 and 3/2, though other usable seHSA5s are 17/12, 45/32, 64/45.

 

Its extension is naturally a very close [[12edo]] detemper in the [[2.3.5.17.19 subgroup]], and a [[diaschismic]] detemper ([[10L 2s]] 8|2 b4) scale.

! colspan="12" |heHSA5 5 (7/5)

<code>36:38:40:42:44:46:48:54:57:60:63:66:69:72</code>. Harmonic segment: '''18::24'''.

 

Last HSA5 to be a constant structure without extensions. It and its heHSA5 are [[15edo]] detempers in the 2.3.5.7.11.19.23 subgroup, and [[nautilus]] detempers ([[1L 12s]] 6|6 and [[14L 1s]] 6|8).

! colspan="15" |heHSA5 6 (25/18, 13/9)