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

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 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.
+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 ==
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 <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]], or otherwise an [[archy]] detemper, [[2L 3s]] 3|1 scale.
+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).
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 !7/6
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 <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]], or otherwise a [[dicot]]/[[mothra]] detemper, [[3L 4s]] 5|1 #6 scale. (1-indexed).
+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).
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 !'''10/9'''
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 <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. 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.
+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.
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 ! colspan="9" |HSA5 4
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 <code>30:32:34:36:38:40:45:48:51:54:57:60</code>. Harmonic segment: '''12::16'''.
-'''15::20'''. It is the third prime HSA5, and the first HSA5 not to be a constant structure without extensions. It is naturally a very close [[12edo]] detemper in the [[2.3.5.17.19 subgroup]]. 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 seHSA5 is 64/45.
+'''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.
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 ! colspan="12" |heHSA5 5 (7/5)
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 <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 is a [[15edo]] detemper in the 2.3.5.7.11.19.23 subgroup.
+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).
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 ! colspan="15" |heHSA5 6 (25/18, 13/9)
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 <code>42:44:46:48:50:52:54:56:63:66:69:72:75:78:81:84</code>'''.''' Harmonic segment: '''21::28'''.
-The fourth prime HSA5. The most reasonable heHSA5 with 29/21 and 10/7 contains 7::14 as a subset. It is naturally a 17edo detemper in the no-17,19-29-limit subgroup. Its chiral version is surprisingly a subharmonic interpolation of HSA5 3.
+The fourth prime HSA5. The most reasonable heHSA5 with 29/21 and 10/7 contains 7::14 as a subset. It is naturally a [[17edo]] detemper in the no-5,17,19,add-25-29-limit subgroup. Its chiral version is surprisingly a subharmonic interpolation of HSA5 3.
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 ! colspan="17" |heHSA5 7 (29/21, 10/7)
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 |}
-=== HSA5 8 ===
+=== HSA5 8 - ''Veljeuwof'' ===
 <code>48:50:52:54:56:58:60:62:64:66:68:70:72:75:78:81:84:87:90:93:96</code>. Harmonic segment: '''24::32.'''
-It contains HSA5 2, HSA5 4 and [[8afdo]] as subsets. As a superset of HSA5 4, it adds many new interesting intervals with odds 9, 11, 15, 21, 25, 29, 31, 35. It is not an easy edo detemper, just a harmonic interpolation of HSA5 4 in the 2.3.5.7.13.29.31 subgroup, with additions of 11 and 17 with the heHSA5.
+It contains HSA5 2, HSA5 4 and as subsets, and the heHSA5 also contains [[8afdo]]. As a superset of HSA5 4, it adds many new interesting intervals with odds 9, 11, 15, 21, 25, 29, 31, 35 along its modes.
+The raw scale or its extension could be seen as [[wilsec]] detempers (negri weak extension), [[1L 16]] 8|8 and [[19L 1s]] 10|9 respectively, but it is a very innacurate realization that only [[19edo]] through patent val can faithfully represent. Thus, it's better seen as just a linear interpolation of HSA5 4 in the 2.3.5.7.13.29.31 subgroup, with additions of 11 and 17 with the heHSA5.
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 ! colspan="20" |heHSA5 8 (11/8, 17/12, 35/24)