Hemififths - Revision history

FloraC: /* Tunings */ note the vals

2026-09-13T15:02:43Z

Tunings: note the vals

← Older revision		Revision as of 15:02, 13 September 2026
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	\|		\|
	\| 350.000		\| 350.000
	\| ~~Lower~~ bound of 7- and 9-odd-limit diamond monotone		\| 24c val, lower bound of 7- and 9-odd-limit diamond monotone
	\|-		\|-
	\|		\|
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	\|		\|
	\| 351.220		\| 351.220
	\| Lower bound of 11- to 15-odd-limit<br>and (13-limit) 21-odd-limit diamond monotone		\| Lower bound of 11- to 15-odd-limit, <br>and 13-limit 21-odd-limit diamond monotone
	\|-		\|-
	\|		\|
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	\|		\|
	\| 351.429		\| 351.429
	\|		\| 140ef val
	\|-		\|-
	\|		\|
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	\| 25/24		\| 25/24
	\| 351.472		\| 351.472
	\| Very close to [[~~Argent tuning\|~~argent tuning]] with neutral intervals (351.47186 cents)		\| Very close to [[argent tuning]] with neutral intervals (351.47186 cents)
	\|-		\|-
	\|		\|
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	\|		\|
	\| 351.515		\| 351.515
	\|		\| 99ef val
	\|-		\|-
	\|		\|
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	\| 15/13		\| 15/13
	\| 351.705		\| 351.705
	\| 15-odd-limit and (13-limit) 21-odd-limit minimax		\| 15-odd-limit and 13-limit 21-odd-limit minimax
	\|-		\|-
	\| [[58edo\|17\58]]		\| [[58edo\|17\58]]
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	\|		\|
	\| 352.000		\| 352.000
	\|		\| 75ce val
	\|-		\|-
	\|		\|
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	\|		\|
	\| 352.941		\| 352.941
	\| ~~Upper~~ bound of 7- to 15-odd-limit<br>and (13-limit) 21-odd-limit diamond monotone		\| 17c val, upper bound of 7- to 15-odd-limit, <br>and 13-limit 21-odd-limit diamond monotone
	\|-		\|-
	\|		\|

FloraC: /* Notation */ - the 24-form notation, since this temp is pretty naturally a 17-form, as is established in the section above

2026-09-13T14:58:28Z

Notation: - the 24-form notation, since this temp is pretty naturally a 17-form, as is established in the section above

← Older revision		Revision as of 14:58, 13 September 2026
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	== Notation ==		== Notation ==
	Hemififths can be notated in [[neutral chain-of-fifths notation]], in which case 5/4 is represented by a sesqui-augmented second (C–D{{sesquisharp2}}), and 7/4 by a semi-augmented sixth (C–A{{demisharp2}}). In the 13-limit extension, 11/8 is represented by the semi-augmented fourth (C–F{{demisharp2}}), and 13/8 by the neutral sixth (C–A{{demiflat2}}). This, of course, defies the tradition of tertian harmony~~. The~~ just major triad on C is ~~{{dash\|C, D~~{{sesquisharp2}}~~, G\|med}}~~, for example~~. One~~ may want to adopt one or more additional modules of accidentals such as arrows or +/- signs to represent the ~~comma~~ steps~~. There are two notable comma steps:~~		Hemififths can be notated in [[neutral chain-of-fifths notation]], in which case 5/4 is represented by a sesqui-augmented second (C–D{{sesquisharp2}}), and 7/4 by a semi-augmented sixth (C–A{{demisharp2}}). In the 13-limit extension, 11/8 is represented by the semi-augmented fourth (C–F{{demisharp2}}), and 13/8 by the neutral sixth (C–A{{demiflat2}}). This, of course, defies the tradition of tertian harmony, as the [[just major triad]] on C is C–D{{sesquisharp2}}–G, for example, so one may want to adopt one or more additional modules of accidentals such as arrows or +/- signs to represent the commatic steps (-17 generator steps, a semidiminished second).
	~~# The syntonic~septimal comma~~ (-17 ~~gensteps~~, semidiminished second);
	~~# The Pythagorean comma~~ (~~+24 gensteps, inverse diminished second~~).		Below is tabulated how to notate each prime harmonic with an arrow representing a commatic step (thus ↑C = D{{sesquiflat2}}).

	~~Below is tabulated how to notate the prime harmonics with an arrow representing a syntonic~septimal comma (thus ^C = Ddb).~~
	{\| class="wikitable center-1 center-3"		{\| class="wikitable center-1 center-3"
	\|+ style="font-size: 105%;" \| Hemififths nomenclature<br>for selected intervals		\|+ style="font-size: 105%;" \| Hemififths nomenclature<br>for selected intervals
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	\| 5/4		\| 5/4
	\| Down major third		\| Down major third
	\| ~~C–vE~~		\| C–↓E
	\|-		\|-
	\| 7/4		\| 7/4
	\| Down minor seventh		\| Down minor seventh
	\| ~~C–vBb~~		\| C–↓B♭
	\|-
	~~\| 11/8~~
	~~\| Semi-augmented fourth~~
	~~\| C–Ft~~
	\|-
	~~\| 13/8~~
	~~\| Neutral sixth~~
	~~\| C–Ad~~
	\|}

	~~Below is tabulated how to notate the prime harmonics with an arrow representing a Pythagorean comma (thus ^C = B#).~~
	~~{\| class="wikitable center-1 center-3"~~
	~~\|+ style="font-size: 105%;" \| Hemififths nomenclature<br>for selected intervals~~
	\|-
	~~! Ratio~~
	~~! Nominal~~
	~~! Example~~
	\|-
	~~\| 3/2~~
	~~\| Perfect fifth~~
	~~\| C–G~~
	\|-
	~~\| 5/4~~
	~~\| Up neutral third~~
	~~\| C–^Ed~~
	\|-
	~~\| 7/4~~
	~~\| Up semidiminished seventh~~
	~~\| C–^Bdb~~
	\|-		\|-
	\| 11/8		\| 11/8
	\| Semi-augmented fourth		\| Semi-augmented fourth
	\| ~~C–Ft~~		\| C–F{{demisharp2}}
	\|-		\|-
	\| 13/8		\| 13/8
	\| Neutral sixth		\| Neutral sixth
	\| ~~C–Ad~~		\| C–A{{demiflat2}}
	\|}		\|}

FloraC: /* As a detemperament of 17et */ list the ratios represented by the commatic step

2026-09-13T14:50:06Z

As a detemperament of 17et: list the ratios represented by the commatic step

← Older revision		Revision as of 14:50, 13 September 2026
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	[[File: Hemififths 17et Detempering.png\|thumb\|Hemififths as a 58-tone 17et detempering]]		[[File: Hemififths 17et Detempering.png\|thumb\|Hemififths as a 58-tone 17et detempering]]

	Hemififths is very naturally considered as a [[detemperament]] of the [[17edo\|17 equal temperament]]. The diagram on the right shows a 58-tone detempered scale, with a generator range of -28 to +29. 58 is the largest number of tones for a mos where intervals in the 17 categories do not overlap. Each category may be further divided into "sub", "plain" and "super" qualities, separated by -17 generator steps, which represents ~~the syntonic~septimal comma~~. Combining this division with the minor, neutral, and major qualities of the 17 equal temperament, hemififths gives us at least ''nine'' qualities for each diatonic category: subminor, minor, supraminor, subneutral, neutral, supraneutral, submajor, major, and supermajor.		Hemififths is very naturally considered as a [[detemperament]] of the [[17edo\|17 equal temperament]]. The diagram on the right shows a 58-tone detempered scale, with a generator range of -28 to +29. 58 is the largest number of tones for a mos where intervals in the 17 categories do not overlap. Each category may be further divided into "sub", "plain" and "super" qualities, separated by the commatic step of -17 generator steps, which represents [[56/55]], [[64/63]], [[66/65]], [[78/77]], [[81/80]], [[91/90]], [[99/98]], [[121/120]], and [[169/168]]. Combining this division with the minor, neutral, and major qualities of the 17 equal temperament, hemififths gives us at least ''nine'' qualities for each diatonic category: subminor, minor, supraminor, subneutral, neutral, supraneutral, submajor, major, and supermajor.

	Notice also the little interval between the largest of a category and the smallest of the next. This interval separates supraminor from subneutral and supraneutral from submajor, and spans 41 generator steps. 41edo tempers it out so that it conflates supraminor with subneutral and supraneutral with submajor, whereas 58edo exaggerates it to the size of the ~~syntonic~septimal~~ comma. 99edo tunes it to one half the size of the ~~syntonic~septimal comma~~, which can be seen as a good compromise.		Notice also the little interval between the largest of a category and the smallest of the next. This interval separates supraminor from subneutral and supraneutral from submajor, and spans 41 generator steps. 41edo tempers it out so that it conflates supraminor with subneutral and supraneutral with submajor, whereas 58edo exaggerates it to the size of the comma. 99edo tunes it to one half the size of the commatic step, which can be seen as a good compromise.

	== Notation ==		== Notation ==

FloraC: Text replacement - "Category:Hemifamity temperaments" to "Category:Aberschismic temperaments"

2026-06-06T12:39:41Z

Text replacement - "Category:Hemifamity temperaments" to "Category:Aberschismic temperaments"

← Older revision		Revision as of 12:39, 6 June 2026
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	[[Category:Rank-2 temperaments]]		[[Category:Rank-2 temperaments]]
	[[Category:Breedsmic temperaments]]		[[Category:Breedsmic temperaments]]
	[[Category:~~Hemifamity~~ temperaments]]		[[Category:Aberschismic temperaments]]
	[[Category:Hemimage temperaments]]		[[Category:Hemimage temperaments]]

FloraC: Cleanup on hatnote & infobox

2026-02-09T09:18:34Z

Cleanup on hatnote & infobox

← Older revision		Revision as of 09:18, 9 February 2026
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	~~: ''This page is about~~ the regular temperament~~. For~~ the irrational interval of a hemififth~~, see [[sqrt~~(3/2)~~]].''~~		{{About\|the regular temperament\|the irrational interval of a hemififth\|Sqrt(3/2)}}
	{{Infobox regtemp		{{Infobox regtemp
	\| Title = Hemififths		\| Title = Hemififths
	\| Subgroups = 2.3.5.7, 2.3.5.7.11, 2.3.5.7.11.13		\| Subgroups = 2.3.5.7, 2.3.5.7.11, 2.3.5.7.11.13
	\| Comma basis = [[2401/2400]], [[5120/5103]] (7-limit); <br> [[243/242]], [[441/440]], [[896/891]] (11-limit); <br>[[144/143]], [[196/195]], [[243/242]], [[364/363]] (13-limit)		\| Comma basis = [[2401/2400]], [[5120/5103]] (7-limit); <br> [[243/242]], [[441/440]], [[896/891]] (11-limit); <br>[[144/143]], [[196/195]], [[243/242]], [[364/363]] (13-limit)
	\| ~~Generator~~ = ~~49/40~~		\| Edo join 1 = 41 \| Edo join 2 = 58
	\| Mapping = 1; 2 25 13 5 -1		\| Mapping = 1; 2 25 13 5 -1
			\| Generators = 49/40
			\| Generators tuning = 351.5
			\| Optimization method = CWE
	\| Pergen = (P8, P5/2)		\| Pergen = (P8, P5/2)
	~~\| Edo join 1 = 41 \| Edo join 2 = 58~~
	~~\| Optimization method = CTE~~
	~~\| Generator tuning = 351.4~~
	\| MOS scales = [[3L 4s]], [[7L 3s]], [[7L 10s]], [[17L 7s]], [[17L 24s]]		\| MOS scales = [[3L 4s]], [[7L 3s]], [[7L 10s]], [[17L 7s]], [[17L 24s]]
	\| Odd limit 1 = 9 \| Mistuning 1 = 1.90 \| Complexity 1 = 41		\| Odd limit 1 = 9 \| Mistuning 1 = 1.90 \| Complexity 1 = 41
	\| Odd limit 2 = 13-limit 21 \| Mistuning 2 = 7.77 \| Complexity 2 = 41		\| Odd limit 2 = 13-limit 21 \| Mistuning 2 = 7.77 \| Complexity 2 = 41
	}}		}}

	'''Hemififths''' is a [[regular temperament\|temperament]] that uses a neutral third as a [[generator]], just as the name suggests. A stack of 13 generators represents [[7/4]] and a stack of 25 generators represents [[5/4]], [[tempering out]] the breedsma, [[2401/2400]], and the argent comma, [[5120/5103]].		'''Hemififths''' is a [[regular temperament\|temperament]] that uses a neutral third as a [[generator]], just as the name suggests. A stack of 13 generators represents [[7/4]] and a stack of 25 generators represents [[5/4]], [[tempering out]] the breedsma, [[2401/2400]], and the argent comma, [[5120/5103]].

FloraC: Make shorter

2026-01-30T06:31:40Z

Make shorter

← Older revision		Revision as of 06:31, 30 January 2026
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	\| Title = Hemififths		\| Title = Hemififths
	\| Subgroups = 2.3.5.7, 2.3.5.7.11, 2.3.5.7.11.13		\| Subgroups = 2.3.5.7, 2.3.5.7.11, 2.3.5.7.11.13
	\| Comma basis = [[2401/2400]], [[5120/5103]] (~~2.3.5.~~7); <br> [[243/242]], [[441/440]], [[896/891]] (~~2.3.5.7.~~11); <br>[[144/143]], [[196/195]], [[243/242]], [[364/363]] (~~2.3.5.7.11.~~13)		\| Comma basis = [[2401/2400]], [[5120/5103]] (7-limit); <br> [[243/242]], [[441/440]], [[896/891]] (11-limit); <br>[[144/143]], [[196/195]], [[243/242]], [[364/363]] (13-limit)
	\| Generator = 49/40		\| Generator = 49/40
	\| Mapping = 1; 2 25 13 5 -1		\| Mapping = 1; 2 25 13 5 -1

Overthink: style

2026-01-30T04:37:11Z

style

← Older revision		Revision as of 04:37, 30 January 2026
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	\| Title = Hemififths		\| Title = Hemififths
	\| Subgroups = 2.3.5.7, 2.3.5.7.11, 2.3.5.7.11.13		\| Subgroups = 2.3.5.7, 2.3.5.7.11, 2.3.5.7.11.13
	\| Comma basis = [[2401/2400]], [[5120/5103]] (L7); <br> [[243/242]], [[441/440]], [[896/891]] (~~L11~~); <br>[[144/143]], [[196/195]], [[243/242]], [[364/363]] (~~L13~~)		\| Comma basis = [[2401/2400]], [[5120/5103]] (2.3.5.7); <br> [[243/242]], [[441/440]], [[896/891]] (2.3.5.7.11); <br>[[144/143]], [[196/195]], [[243/242]], [[364/363]] (2.3.5.7.11.13)
	\| Generator = 49/40		\| Generator = 49/40
	\| Mapping = 1; 2 25 13 5 -1		\| Mapping = 1; 2 25 13 5 -1
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	\| MOS scales = [[3L 4s]], [[7L 3s]], [[7L 10s]], [[17L 7s]], [[17L 24s]]		\| MOS scales = [[3L 4s]], [[7L 3s]], [[7L 10s]], [[17L 7s]], [[17L 24s]]
	\| Odd limit 1 = 9 \| Mistuning 1 = 1.90 \| Complexity 1 = 41		\| Odd limit 1 = 9 \| Mistuning 1 = 1.90 \| Complexity 1 = 41
	\| Odd limit 2 = (13-limit) 21 \| Mistuning 2 = 7.77 \| Complexity 2 = 41		\| Odd limit 2 = 13-limit 21 \| Mistuning 2 = 7.77 \| Complexity 2 = 41
	}}		}}

FloraC: /* Interval chain */ - the 17-form detemperament table cuz it's not easy to read. The diagram is just way better

2025-12-07T10:44:03Z

Interval chain: - the 17-form detemperament table cuz it's not easy to read. The diagram is just way better

Show changes

FloraC: /* Chords */ -> chords and harmony

2025-12-07T10:33:08Z

Chords: -> chords and harmony

← Older revision		Revision as of 10:33, 7 December 2025
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	<nowiki/>* In 7-limit CWE tuning, octave reduced		<nowiki/>* In 7-limit CWE tuning, octave reduced

	== Chords ==		== Chords and harmony ==
	{{~~Main~~\| Chords of hemififths }}		{{See also\| Chords of hemififths }}

	== Scales ==		== Scales ==

FloraC: Move ups and downs notation to the notation section

2025-12-07T10:32:03Z

Move ups and downs notation to the notation section

Show changes