User:Eufalesio/Harmonic Segments Along Fifths - Revision history

Eufalesio: Ready to delete as it has been published into main space

2026-10-15T21:54:11Z

Ready to delete as it has been published into main space

← Older revision		Revision as of 21:54, 15 October 2026
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	{{~~Editable user page}}{{Idiosyncratic terms~~\|~~"Zontatonic", "Íegmul" and "Ngwóoghe" were coined by Eufalesio and "Veljeuwof" by Lawitalawitaluna~~ to ~~have explicit names for HSAF 2, 3, 4~~, 8.}}		{{Delete\|If main article is to be kept, this is no longer necessary.}}

	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).		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).
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	=== HSAF 2 - Septimal minor pentatonic ~~/ ''Zontatonic''~~ ===		=== HSAF 2 - Septimal minor pentatonic ===
	<code>12:14:16:18:21:24</code>. Harmonic segment: '''6::8'''. ''Zontatonic'' is derived as a portmanteau of [[color notation]] ''zo'' (septimal) and "pentatonic", which in this case refers specifically to HSAF 2 or , though the longer yet more understandable "septimal minor pentatonic" is a possible substitute.		<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]].		First usable HSAF, the first prime HSAF, and the only one that is [[Strict variety]] 3 and a [[generator sequence]].
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	=== HSAF 3 ~~- ''Íegmul''~~ ===		=== HSAF 3 ===
	<code>18:20:22:24:27:30:33:36</code>. Harmonic segment: '''9::12'''~~. Its name is derived from the ''Rraei'' word for ''hearth,'' íegmul [ˈi.egmuɫ]~~.		<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).		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).
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	=== HSAF 4 ~~- ''Ngwóoghe''~~ ===		=== HSAF 4 ===
	<code>24:26:28:30:32:34:36:39:42:45:48</code>. Harmonic segment: '''12::16'''. ~~Its name is derived from the Feñkioó word for ''stencil,'' ngwóoghe [ŋwôːɣe]~~		<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.		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.
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	=== HSAF 8 ~~- ''Veljeuwof''~~ ===		=== HSAF 8 ===
	<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.''' ~~Its name is derived from the ''Eruvan'' expression ''(kauwof) veljeuwof)'' for ''twenty (reference [pitches]), veljeuwof'' [vɛʎɛuwɔɸ].~~		<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 HSAF 2 and HSAF 4. As a superset of HSAF 4, it adds many new interesting intervals with odds 9, 11, 15, 21, 25, 29, 31, 35 along its modes. It is also the last HSAF to fit within the [[31-limit]].		It contains HSAF 2 and HSAF 4. As a superset of HSAF 4, it adds many new interesting intervals with odds 9, 11, 15, 21, 25, 29, 31, 35 along its modes. It is also the last HSAF to fit within the [[31-limit]].

Eufalesio: Lawitalawitaluna's names

2026-10-14T19:16:34Z

Lawitalawitaluna's names

← Older revision		Revision as of 19:16, 14 October 2026
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	=== HSAF 2 - ''Zontatonic'' ===		=== HSAF 2 - Septimal minor pentatonic / ''Zontatonic'' ===
	<code>12:14:16:18:21:24</code>. Harmonic segment: '''6::8'''. ~~Its name~~ is derived as a portmanteau of [[color notation]] ''zo'' (septimal) and "pentatonic", which in this case refers specifically to HSAF 2, though the longer yet more understandable "septimal minor pentatonic" is a possible substitute.		<code>12:14:16:18:21:24</code>. Harmonic segment: '''6::8'''. ''Zontatonic'' is derived as a portmanteau of [[color notation]] ''zo'' (septimal) and "pentatonic", which in this case refers specifically to HSAF 2 or , though the longer yet more understandable "septimal minor pentatonic" is a possible substitute.

	First usable HSAF, the first prime HSAF, and the only one that is [[Strict variety]] 3 and a [[generator sequence]].		First usable HSAF, the first prime HSAF, and the only one that is [[Strict variety]] 3 and a [[generator sequence]].
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	=== HSAF 8 - ''Veljeuwof'' ===		=== HSAF 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.''' Its name is derived ~~form~~ the ~~TBA word~~ for ''twenty, veljeuwof'' [~~TBA~~].		<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.''' Its name is derived from the ''Eruvan'' expression ''(kauwof) veljeuwof)'' for ''twenty (reference [pitches]), veljeuwof'' [vɛʎɛuwɔɸ].

	It contains HSAF 2 and HSAF 4. As a superset of HSAF 4, it adds many new interesting intervals with odds 9, 11, 15, 21, 25, 29, 31, 35 along its modes. It is also the last HSAF to fit within the [[31-limit]].		It contains HSAF 2 and HSAF 4. As a superset of HSAF 4, it adds many new interesting intervals with odds 9, 11, 15, 21, 25, 29, 31, 35 along its modes. It is also the last HSAF to fit within the [[31-limit]].

Eufalesio: Declared idiosyncratic terms and explained etymologies

2026-10-14T19:04:27Z

Declared idiosyncratic terms and explained etymologies

← Older revision		Revision as of 19:04, 14 October 2026
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	{{Editable user page}}		{{Editable user page}}{{Idiosyncratic terms\|"Zontatonic", "Íegmul" and "Ngwóoghe" were coined by Eufalesio and "Veljeuwof" by Lawitalawitaluna to have explicit names for HSAF 2, 3, 4, 8.}}

	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).		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).
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	=== HSAF 2 - ''Zontatonic'' ===		=== HSAF 2 - ''Zontatonic'' ===
	<code>12:14:16:18:21:24</code>. Harmonic segment: '''6::8'''.		<code>12:14:16:18:21:24</code>. Harmonic segment: '''6::8'''. Its name is derived as a portmanteau of [[color notation]] ''zo'' (septimal) and "pentatonic", which in this case refers specifically to HSAF 2, though the longer yet more understandable "septimal minor pentatonic" is a possible substitute.

	First usable HSAF, the first prime HSAF, and the only one that is [[Strict variety]] 3 and a [[generator sequence]].		First usable HSAF, the first prime HSAF, and the only one that is [[Strict variety]] 3 and a [[generator sequence]].
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	=== HSAF 3 - ''Íegmul'' ===		=== HSAF 3 - ''Íegmul'' ===
	<code>18:20:22:24:27:30:33:36</code>. Harmonic segment: '''9::12'''.		<code>18:20:22:24:27:30:33:36</code>. Harmonic segment: '''9::12'''. Its name is derived from the ''Rraei'' word for ''hearth,'' íegmul [ˈi.egmuɫ].

	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).		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).
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	=== HSAF 4 - ''Ngwóoghe'' ===		=== HSAF 4 - ''Ngwóoghe'' ===
	<code>24:26:28:30:32:34:36:39:42:45:48</code>. Harmonic segment: '''12::16'''.		<code>24:26:28:30:32:34:36:39:42:45:48</code>. Harmonic segment: '''12::16'''. Its name is derived from the Feñkioó word for ''stencil,'' ngwóoghe [ŋwôːɣe]

	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.		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.
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	=== HSAF 8 - ''Veljeuwof'' ===		=== HSAF 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.'''		<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.''' Its name is derived form the TBA word for ''twenty, veljeuwof'' [TBA].

	It contains HSAF 2 and HSAF 4. As a superset of HSAF 4, it adds many new interesting intervals with odds 9, 11, 15, 21, 25, 29, 31, 35 along its modes. It is also the last HSAF to fit within the [[31-limit]].		It contains HSAF 2 and HSAF 4. As a superset of HSAF 4, it adds many new interesting intervals with odds 9, 11, 15, 21, 25, 29, 31, 35 along its modes. It is also the last HSAF to fit within the [[31-limit]].

Eufalesio: HSAF 1 is a stickler

2026-10-14T08:41:47Z

HSAF 1 is a stickler

← Older revision		Revision as of 08:41, 14 October 2026
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	* Its prime mode always contains 4/3 and 3/2.		* 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.		* 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.
	* ~~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.		* 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 ~~HSAF~~ are a countable set of scales, and thus can be indexed.		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.

	== Properties ==		== Properties ==

Eufalesio: Tighten en

2026-10-13T16:00:17Z

Tighten en

← Older revision		Revision as of 16:00, 13 October 2026
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	{{Editable user page}}		{{Editable user page}}

	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)~~. How to approach them musically is not within the scope of this article~~.		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).

	== Definitions ==		== Definitions ==

Eufalesio: Tighten

2026-10-13T15:58:31Z

Tighten

Show changes

Eufalesio at 18:36, 11 October 2026

2026-10-11T18:36:16Z

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	* Harmonic segments conforming HSAFs up to the 13th* are [[strictly proper]] [[constant structure]]<nowiki/>s.		* Harmonic segments conforming HSAFs up to the 13th* are [[strictly proper]] [[constant structure]]<nowiki/>s.
	* The kth HSAF is a subset of the 6k::12k harmonic segment.		* The kth HSAF 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.		* 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		<nowiki>*</nowiki>except the first
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	For every provided HSAF, the [[EFR]] and used segment are written out explicitly.		For every provided HSAF, the [[EFR]] and used segment are written out explicitly.

	=== HSAF 1 - Pythagorean Trial ===		=== HSAF/SSAF 1 - Pythagorean Trial ===
	<code>6:8:9:12</code>. Harmonic segment: '''3:4'''.		<code>6:8:9:12</code>. Harmonic segment: '''3:4'''.

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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'''.		<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 HSAF. The most reasonable heHSAF with 29/21 and 10/7 contains 7::14 as a subset. A good seHSAFs is with 11/8 and 13/9. ~~Its chiral version~~ is surprisingly a subharmonic interpolation of HSAF 3.		The fourth prime HSAF. The most reasonable heHSAF with 29/21 and 10/7 contains 7::14 as a subset. A good seHSAFs is with 11/8 and 13/9. SSAF 7 is surprisingly a subharmonic interpolation of HSAF 3.

	It is naturally a [[17edo]] detemper in the no-5,17,19,add-25-29-limit subgroup. Its extension can be seen as a strange liese extension detemper ([[2L 15s]] 15\|1 10b).		It is naturally a [[17edo]] detemper in the no-5,17,19,add-25-29-limit subgroup. Its extension can be seen as a strange liese extension detemper ([[2L 15s]] 15\|1 10b).

Eufalesio: HSA5 to HSAF

2026-10-11T10:53:14Z

HSA5 to HSAF

Show changes

Eufalesio: Hotfixes!

2026-09-14T00:35:35Z

Hotfixes!

← Older revision		Revision as of 00:35, 14 September 2026
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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'''.		<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-5,17,19,add-25-29-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. A good seHSA5s is with 11/8 and 13/9. Its chiral version is surprisingly a subharmonic interpolation of HSA5 3.

	Its extension can be seen as a strange liese extension detemper ([[2L 15s]] 15\|1 10b).		It is naturally a [[17edo]] detemper in the no-5,17,19,add-25-29-limit subgroup. Its extension can be seen as a strange liese extension detemper ([[2L 15s]] 15\|1 10b).
	{\| class="wikitable"		{\| class="wikitable"
	! colspan="17" \|heHSA5 7 (29/21, 10/7)		! colspan="17" \|heHSA5 7 (29/21, 10/7)

Eufalesio: Last HSA5 to link to a MODMOS

2026-09-14T00:31:05Z

Last HSA5 to link to a MODMOS

← Older revision		Revision as of 00:31, 14 September 2026
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	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.		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.

			Its extension can be seen as a strange liese extension detemper ([[2L 15s]] 15\|1 10b).
	{\| class="wikitable"		{\| class="wikitable"
	! colspan="17" \|heHSA5 7 (29/21, 10/7)		! colspan="17" \|heHSA5 7 (29/21, 10/7)