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

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Harmonic Segments Along Fifths (HSAF) [sing. ''a Harmonic Segment along Fifths''] are a type of octave-[[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).

For a scale to be a HSAF, it must satisfy the following:

* Its [[period]] is the octave.

* Its prime mode always contains 4/3 and 3/2.

* Its prime mode is built like a harmonic series segment up to 4/3, and that same harmonic segment is repeated transposed by 3/2 until it reaches the octave.

* HSAF 1 is 6:8:9:12 with [[delta signature]] +2,+1,+2. Except for HSAF 1, its delta signature is +2,+2,+2,+2,[…],+3,+3,+3,[…]. If the HSAF is even-indexed, the +3's start right at 3/2. If the HSAF is odd-indexed, at the step right before 3/2.

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

* Harmonic segments conforming HSAFs up to the 13th* are [[strictly proper]] [[constant structure]]<nowiki/>s.

* HSAF k is a subset of the 6k::12k harmonic segment.

* Because they are formed from harmonic series segments, they are chiral scales*, whose left-handed counterparts are built from [[subharmonic series]] segments. The left-handed versions are '''not''' HSAFs per se. They are only the chiral version of the HSAF, which you might call a SSAF, for Subharmonic Segment [...].

<nowiki>*</nowiki>except the first

== List of HSAFs 1-8 ==

 

=== HSAF/SSAF 1 - Pythagorean Trial ===

<code>6:8:9:12</code>. Harmonic segment: '''3:4'''.  

 

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

=== HSAF 2 - Septimal minor pentatonic ===

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

 

First usable HSAF, the first prime HSAF, 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).

=== HSAF 3 ===

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

 

Second prime HSAF. 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).  

=== HSAF 4 ===

<code>24:26:28:30:32:34:36:39:42:45:48</code>. Harmonic segment: '''12::16'''.

 

First composite HSAF, containing HSAF 2 as a subset. It is naturally a [[10edo]] detemper in the [[2.3.5.7.13 subgroup]], a [[negri]] detemper ([[1L 8s]] 4|4) if you exclude 17/12, and a [[pajarous]] detemper, 2L 8s 6|2 #7 scale.

!'''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>

=== HSAF 5 ===

 

'''15::20'''. It is the third prime HSAF. 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.

=== HSAF 6 ===

<code>36:38:40:42:44:46:48:50:52:54:57:60:63:66:69:72</code>. Harmonic segment: '''18::24'''.

 

Last HSAF to be a constant structure, and first HSAF to contain two different (non-1) HSAFs as subsets. It is a [[15edo]] detemper in the 2.3.5.7.11.19.23 subgroup, and [[nautilus]] detemper ([[1L 12s]] 6|6 and [[14L 1s]] 6|8).