Harmonisma - Revision history

FloraC: Complete data

2026-02-25T06:40:50Z

Complete data

← Older revision		Revision as of 06:40, 25 February 2026
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	\| ⟨ \|\| 0 \|\| 0 \|\| 0 \|\| 0 \|\| -2 \|\| -3 \|\| ]]		\| ⟨ \|\| 0 \|\| 0 \|\| 0 \|\| 0 \|\| -2 \|\| -3 \|\| ]]
	\|}		\|}
			: mapping generators: ~2, ~3, ~5, ~7, ~44/39

	: ~~mapping generators~~: ~2, ~3, ~5, ~7, ~44/39		[[Optimal tuning]]s:
			* [[WE]]: ~2 = 1199.9974{{c}}, ~3/2 = 701.9619{{c}}, ~5/4 = 386.3189{{c}}, ~7/4 = 968.8378{{c}}, ~44/39 = 208.7715{{c}}
			* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.9615{{c}}, ~5/4 = 386.3168{{c}}, ~7/4 = 968.8365{{c}}, ~44/39 = 208.7728{{c}}

	{{Optimal ET sequence\|legend=1\| 41, 46, 58, 80, 87, 103, 121, 149, 161, 183, 190, 224, 270, 494, 684, 764, 954, 1178, 1236, 1448, 1506, 2190, 2684, 4190, 4771, 6691, 6961, 7455, 9645, 17100 }}		{{Optimal ET sequence\|legend=1\| 41, 46, 58, 80, 87, 103, 121, 149, 161, 183, 190, 224, 270, 494, 684, 764, 954, 1178, 1236, 1448, 1506, 2190, 2684, 4190, 4771, 6691, 6961, 7455, 9645, 17100 }}

			[[Badness]] (Sintel): 0.207

	== Etymology ==		== Etymology ==

FloraC: Misc. cleanup

2026-02-25T06:37:25Z

Misc. cleanup

← Older revision		Revision as of 06:37, 25 February 2026
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	\| Comma = yes		\| Comma = yes
	}}		}}
			'''10648/10647''', the '''harmonisma''', is an [[unnoticeable comma\|unnoticeable]] no-5's [[13-limit]] [[comma]] of about 0.1626 [[cent]]s. It is equal to (([[16/13]])⋅([[11/9]]))/(([[14/11]])⋅([[13/11]])). In terms of other commas, it is ([[352/351]])/([[364/363]]), ([[3025/3024]])/([[4225/4224]]), ([[4096/4095]])/([[6656/6655]]), or ([[9801/9800]])/([[123201/123200]]).
	'''10648/10647''', the '''harmonisma''', is a no-5's [[13-limit]] [[~~unnoticeable~~ comma]] of about 0.1626 [[cent]]s. It is equal to ([[16/13]] × [[11/9]])/([[14/11]] × [[13/11]]). In terms of other commas, it is ([[352/351]])/([[364/363]]), ([[3025/3024]])/([[4225/4224]]), ([[4096/4095]])/([[6656/6655]]), or ([[9801/9800]])/([[123201/123200]]).

	== Temperaments ==		== Temperaments ==
	Tempering out this comma in the full 13-limit gives the rank-5 '''harmonismic temperament'''. Equal ~~temperaments~~ where this comma is tempered with very high accuracy, such as [[764edo]], will have an interval corresponding to a "sharp fifth" of (ideally) 706.7 to 706.9 cents, corresponding to the range of fifths from 13/11 × 14/11 (→ [[182/121]]) on the lower end and 11/9 × 16/13 (→ [[176/117]]) on the higher end, and this interval is not mapped to 3/2. However, such temperaments are generally very precise, so [[224edo]], [[270edo]] and [[311edo]] offer slightly more manageable tunings. For less accurate temperaments still, 10648/10647 is notable as a comma of [[parapyth]].		[[Tempering out]] this comma in the full 13-limit gives the rank-5 '''harmonismic temperament'''. [[Equal temperament]]s where this comma is tempered out with very high accuracy, such as [[764edo]], will have an interval corresponding to a "sharp fifth" of (ideally) 706.7 to 706.9 cents, corresponding to the range of fifths from (13/11)⋅(14/11) = [[182/121]] on the lower end and (11/9)⋅(16/13) = [[176/117]] on the higher end, and this interval is not mapped to [[3/2]]. However, such temperaments are generally very precise, so [[224edo]], [[270edo]] and [[311edo]] offer slightly more manageable tunings. For less accurate temperaments still, 10648/10647 is notable as a comma of [[parapyth]].

	The harmonisma, 10648/10647, plays a striking role in [[Secor29htt\|George Secor's 29-tone high tolerance temperament]] of 1975, the first temperament in the High Tolerance Temperament family. In this tuning, the fifth at 703.579 cents produces an augmented second (+9 fifths) at a just [[63/52]] (9/8 × 14/13), or a diminished seventh (-9 fifths) at [[104/63]], which exceeds three 13/11 thirds by a harmonisma. ~~The~~ 63/52 exceeds the Pythagorean augmented second, 19683/16384 (a 32805/32768 schisma larger than 6/5), by the [[secorian comma]], 28672/28431. Likewise 104/63 is narrower than the Pythagorean 32768/19683 by 28672/28431.		The harmonisma, 10648/10647, plays a striking role in [[Secor29htt\|George Secor's 29-tone high tolerance temperament]] of 1975, the first temperament in the High Tolerance Temperament family. In this tuning, the fifth at 703.579 cents produces an augmented second (+9 fifths) at a just [[63/52]] (equal to ([[9/8]])⋅([[14/13]])), or a diminished seventh (-9 fifths) at [[104/63]], which exceeds three 13/11 thirds by a harmonisma. 63/52 exceeds the Pythagorean augmented second, [[19683/16384]] (a [[32805/32768]] schisma larger than [[6/5]]), by the [[secorian comma]], 28672/28431. Likewise 104/63 is narrower than the Pythagorean diminished seventh [[32768/19683]] by 28672/28431.

	The [[secorian comma]] is made up of ([[896/891]] ~~× [[~~352/351]]) or in other words (352/351 × 364/363 × 352/351), and is thus a harmonisma smaller than (352/351)<sup>3</sup>. In 29-HTT, each 13/11 is thus a third of a harmonisma or 0.054 cents wider than just. Secor's HTT fifth of 703.579 cents, or precisely (504/13)<sup>1/9</sup> or wide by 1/9 of a secorian comma, would thus need to be 1/9 harmonisma larger at 703.597 cents to produce a just 13/11, a temperament of the fifth by (352/351)<sup>1/3</sup>.		The [[secorian comma]] is made up of ([[896/891]])⋅(352/351) or in other words (352/351)⋅(364/363)⋅(352/351), and is thus a harmonisma smaller than (352/351)<sup>3</sup>. In 29-HTT, each 13/11 is thus a third of a harmonisma or 0.054 cents wider than just. Secor's HTT fifth of 703.579 cents, or precisely (504/13)<sup>1/9</sup> or wide by 1/9 of a secorian comma, would thus need to be 1/9 harmonisma larger at 703.597 cents to produce a just 13/11, a temperament of the fifth by (352/351)<sup>1/3</sup>.

	Another manifestation of the harmonisma in 29-HTT is the tuning of 11/9 at 347.353 cents, a third of a harmonisma narrow. Here 7/4 is just, with the spacing between the relevant chains of fifths in a subset of 29-HTT which served as a prototype for parapyth temperament at 58.090 cents, as compared with the [[91/88]] spacing which would produce a just 7/4 if the regular major sixth were at a just 22/13. But it is a third of a harmonisma narrow, and the spacing consequently a third of a harmonisma greater.		Another manifestation of the harmonisma in 29-HTT is the tuning of 11/9 at 347.353 cents, a third of a harmonisma narrow. Here [[7/4]] is just, with the spacing between the relevant chains of fifths in a subset of 29-HTT which served as a prototype for parapyth temperament at 58.090 cents, as compared with the [[91/88]] spacing which would produce a just 7/4 if the regular major sixth were at a just [[22/13]]. But it is a third of a harmonisma narrow, and the spacing consequently a third of a harmonisma greater.

	As it happens, the difference between 11/9 and 13/11 is [[121/117]], a harmonisma greater than 91/88 (e.g. 22/13 vs. 7/4). Since the tempered 13/11 is a third of a harmonisma greater than just, and the spacing a third of a harmonisma greater than 91/88, this leaves 1/3 of the harmonisma difference between 91/88 and 121/117 unaccounted for, the amount by which 11/9 is narrow.		As it happens, the difference between 11/9 and 13/11 is [[121/117]], a harmonisma greater than 91/88 (e.g. 22/13 vs. 7/4). Since the tempered 13/11 is a third of a harmonisma greater than just, and the spacing a third of a harmonisma greater than 91/88, this leaves 1/3 of the harmonisma difference between 91/88 and 121/117 unaccounted for, the amount by which 11/9 is narrow.

	In parapyth, generally, the spacing can represent four ratios, whose differences show the four commas tempered out: 33/32 (53.~~273¢~~) and 91/88 (58.~~036¢~~) at 364/363 (4.~~763¢~~) apart ; 91/88 and 121/117 at 10648/10647 (0.~~163¢~~) apart; 121/117 and 28/27 (62.~~961¢~~) at 364/363 (4.~~763¢~~) apart; and the smallest and largest intervals among these represented by the parapyth spacing, 33/32 and 28/27 at 896/891 (9.~~688¢~~) apart. Thus 896/891 = (352/351 × 364/363), and also (364/363 × 10648/10647 × 364/363). A difference of 352/351 (4.~~925¢~~) or (364/363 x 10648/10647) occurs between 121/117 and 33/32, and between 28/27 and 91/88.		In parapyth, generally, the spacing can represent four ratios, whose differences show the four commas tempered out: 33/32 (53.273{{c}}) and 91/88 (58.036{{c}}) at 364/363 (4.763{{c}}) apart; 91/88 and 121/117 at 10648/10647 (0.163{{c}}) apart; 121/117 and 28/27 (62.961{{c}}) at 364/363 (4.763{{c}}) apart; and the smallest and largest intervals among these represented by the parapyth spacing, 33/32 and 28/27 at 896/891 (9.688{{c}}) apart. Thus 896/891 = (352/351)⋅(364/363), and also (364/363)⋅(10648/10647)⋅(364/363). A difference of 352/351 (4.925{{c}}) or (364/363)⋅(10648/10647) occurs between 121/117 and 33/32, and between 28/27 and 91/88.

	Here, for example 33/32 is 4/3 vs. 11/8; 91/88 is 22/13 vs. 7/4; 121/117 is 13/11 vs. 11/9; and 28/27 is 9/8 vs. 7/6.		Here, for example 33/32 is 4/3 vs. 11/8; 91/88 is 22/13 vs. 7/4; 121/117 is 13/11 vs. 11/9; and 28/27 is 9/8 vs. 7/6.

BudjarnLambeth at 03:35, 4 November 2024

2024-11-04T03:35:02Z

← Older revision		Revision as of 03:35, 4 November 2024
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	[[Category:Harmonismic]]		[[Category:Harmonismic]]
			[[Category:Commas named for other reasons]]

Tristanbay: /* Temperaments */added example of EDO for very accurate harmonismic temperament

2024-10-28T10:13:30Z

Temperaments: added example of EDO for very accurate harmonismic temperament

← Older revision		Revision as of 10:13, 28 October 2024
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	== Temperaments ==		== Temperaments ==
	Tempering out this comma in the full 13-limit gives the rank-5 '''harmonismic temperament'''. Equal temperaments where this comma is tempered with very high accuracy will have an interval corresponding to a "sharp fifth" of (ideally) 706.7 to 706.9 cents, corresponding to the range of fifths from 13/11 × 14/11 (→ [[182/121]]) on the lower end and 11/9 × 16/13 (→ [[176/117]]) on the higher end, and this interval is not mapped to 3/2. However, such temperaments are generally very precise, so [[224edo]], [[270edo]] and [[311edo]] offer slightly more manageable tunings. For less accurate temperaments still, 10648/10647 is notable as a comma of [[parapyth]].		Tempering out this comma in the full 13-limit gives the rank-5 '''harmonismic temperament'''. Equal temperaments where this comma is tempered with very high accuracy, such as [[764edo]], will have an interval corresponding to a "sharp fifth" of (ideally) 706.7 to 706.9 cents, corresponding to the range of fifths from 13/11 × 14/11 (→ [[182/121]]) on the lower end and 11/9 × 16/13 (→ [[176/117]]) on the higher end, and this interval is not mapped to 3/2. However, such temperaments are generally very precise, so [[224edo]], [[270edo]] and [[311edo]] offer slightly more manageable tunings. For less accurate temperaments still, 10648/10647 is notable as a comma of [[parapyth]].

	The harmonisma, 10648/10647, plays a striking role in [[Secor29htt\|George Secor's 29-tone high tolerance temperament]] of 1975, the first temperament in the High Tolerance Temperament family. In this tuning, the fifth at 703.579 cents produces an augmented second (+9 fifths) at a just [[63/52]] (9/8 × 14/13), or a diminished seventh (-9 fifths) at [[104/63]], which exceeds three 13/11 thirds by a harmonisma. The 63/52 exceeds the Pythagorean augmented second, 19683/16384 (a 32805/32768 schisma larger than 6/5), by the [[secorian comma]], 28672/28431. Likewise 104/63 is narrower than the Pythagorean 32768/19683 by 28672/28431.		The harmonisma, 10648/10647, plays a striking role in [[Secor29htt\|George Secor's 29-tone high tolerance temperament]] of 1975, the first temperament in the High Tolerance Temperament family. In this tuning, the fifth at 703.579 cents produces an augmented second (+9 fifths) at a just [[63/52]] (9/8 × 14/13), or a diminished seventh (-9 fifths) at [[104/63]], which exceeds three 13/11 thirds by a harmonisma. The 63/52 exceeds the Pythagorean augmented second, 19683/16384 (a 32805/32768 schisma larger than 6/5), by the [[secorian comma]], 28672/28431. Likewise 104/63 is narrower than the Pythagorean 32768/19683 by 28672/28431.

BudjarnLambeth: BudjarnLambeth moved page 10648/10647 to Harmonisma over redirect: Big number

2024-10-13T00:48:47Z

BudjarnLambeth moved page 10648/10647 to Harmonisma over redirect: Big number

← Older revision	Revision as of 00:48, 13 October 2024
(No difference)

BudjarnLambeth at 00:48, 13 October 2024

2024-10-13T00:48:29Z

← Older revision		Revision as of 00:48, 13 October 2024
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	{{Infobox Interval		{{Infobox Interval
			\| Ratio = 10648/10647
	\| Name = harmonisma		\| Name = harmonisma
	\| Color name = 3uu1o<sup>3</sup>r-2, Thuthutrilo-aru comma		\| Color name = 3uu1o<sup>3</sup>r-2, Thuthutrilo-aru comma

FloraC: /* Temperaments */ formatting

2023-07-19T12:31:37Z

Temperaments: formatting

← Older revision		Revision as of 12:31, 19 July 2023
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	[[Mapping]]: <br>		[[Mapping]]: <br>
	[~~{{val~~\| 1 0 0 0 1 3 ~~}}<br>~~		{\| class="right-all"
	~~{{val~~\| 0 1 0 0 0 -1 ~~}}<br>~~		\|-
	~~{{val~~\| 0 0 1 0 0 0 ~~}}<br>~~		\| [⟨ \|\| 1 \|\| 0 \|\| 0 \|\| 0 \|\| 1 \|\| 3 \|\| ],
	~~{{val~~\| 0 0 0 1 1 1 ~~}}<br>~~		\|-
	~~{{val~~\| 0 0 0 0 -2 -3 }}]		\| ⟨ \|\| 0 \|\| 1 \|\| 0 \|\| 0 \|\| 0 \|\| -1 \|\| ],
			\|-
			\| ⟨ \|\| 0 \|\| 0 \|\| 1 \|\| 0 \|\| 0 \|\| 0 \|\| ],
			\|-
			\| ⟨ \|\| 0 \|\| 0 \|\| 0 \|\| 1 \|\| 1 \|\| 1 \|\| ],
			\|-
			\| ⟨ \|\| 0 \|\| 0 \|\| 0 \|\| 0 \|\| -2 \|\| -3 \|\| ]]
			\|}

	~~Mapping~~ generators: ~2, ~3, ~5, ~7, ~44/39		: mapping generators: ~2, ~3, ~5, ~7, ~44/39

	{{Optimal ET sequence\|legend=1\| 41, 46, 58, 80, 87, 103, 121, 149, 161, 183, 190, 224, 270, 494, 684, 764, 954, 1178, 1236, 1448, 1506, 2190, 2684, 4190, 4771, 6691, 6961, 7455, 9645, 17100 }}		{{Optimal ET sequence\|legend=1\| 41, 46, 58, 80, 87, 103, 121, 149, 161, 183, 190, 224, 270, 494, 684, 764, 954, 1178, 1236, 1448, 1506, 2190, 2684, 4190, 4771, 6691, 6961, 7455, 9645, 17100 }}

Cmloegcmluin: "optimal GPV sequence" → "optimal ET sequence", per Talk:Optimal_ET_sequence

2023-05-07T18:57:02Z

"optimal GPV sequence" → "optimal ET sequence", per Talk:Optimal_ET_sequence

← Older revision		Revision as of 18:57, 7 May 2023
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	Mapping generators: ~2, ~3, ~5, ~7, ~44/39		Mapping generators: ~2, ~3, ~5, ~7, ~44/39

	{{~~Val list~~\|legend=1\| 41, 46, 58, 80, 87, 103, 121, 149, 161, 183, 190, 224, 270, 494, 684, 764, 954, 1178, 1236, 1448, 1506, 2190, 2684, 4190, 4771, 6691, 6961, 7455, 9645, 17100 }}		{{Optimal ET sequence\|legend=1\| 41, 46, 58, 80, 87, 103, 121, 149, 161, 183, 190, 224, 270, 494, 684, 764, 954, 1178, 1236, 1448, 1506, 2190, 2684, 4190, 4771, 6691, 6961, 7455, 9645, 17100 }}

	== Etymology ==		== Etymology ==

Cmloegcmluin: add link

2022-12-30T19:06:49Z

add link

← Older revision		Revision as of 19:06, 30 December 2022
Line 12:		Line 12:
	The harmonisma, 10648/10647, plays a striking role in [[Secor29htt\|George Secor's 29-tone high tolerance temperament]] of 1975, the first temperament in the High Tolerance Temperament family. In this tuning, the fifth at 703.579 cents produces an augmented second (+9 fifths) at a just [[63/52]] (9/8 × 14/13), or a diminished seventh (-9 fifths) at [[104/63]], which exceeds three 13/11 thirds by a harmonisma. The 63/52 exceeds the Pythagorean augmented second, 19683/16384 (a 32805/32768 schisma larger than 6/5), by the [[secorian comma]], 28672/28431. Likewise 104/63 is narrower than the Pythagorean 32768/19683 by 28672/28431.		The harmonisma, 10648/10647, plays a striking role in [[Secor29htt\|George Secor's 29-tone high tolerance temperament]] of 1975, the first temperament in the High Tolerance Temperament family. In this tuning, the fifth at 703.579 cents produces an augmented second (+9 fifths) at a just [[63/52]] (9/8 × 14/13), or a diminished seventh (-9 fifths) at [[104/63]], which exceeds three 13/11 thirds by a harmonisma. The 63/52 exceeds the Pythagorean augmented second, 19683/16384 (a 32805/32768 schisma larger than 6/5), by the [[secorian comma]], 28672/28431. Likewise 104/63 is narrower than the Pythagorean 32768/19683 by 28672/28431.

	The secorian comma is made up of ([[896/891]] × [[352/351]]) or in other words (352/351 × 364/363 × 352/351), and is thus a harmonisma smaller than (352/351)<sup>3</sup>. In 29-HTT, each 13/11 is thus a third of a harmonisma or 0.054 cents wider than just. Secor's HTT fifth of 703.579 cents, or precisely (504/13)<sup>1/9</sup> or wide by 1/9 of a secorian comma, would thus need to be 1/9 harmonisma larger at 703.597 cents to produce a just 13/11, a temperament of the fifth by (352/351)<sup>1/3</sup>.		The [[secorian comma]] is made up of ([[896/891]] × [[352/351]]) or in other words (352/351 × 364/363 × 352/351), and is thus a harmonisma smaller than (352/351)<sup>3</sup>. In 29-HTT, each 13/11 is thus a third of a harmonisma or 0.054 cents wider than just. Secor's HTT fifth of 703.579 cents, or precisely (504/13)<sup>1/9</sup> or wide by 1/9 of a secorian comma, would thus need to be 1/9 harmonisma larger at 703.597 cents to produce a just 13/11, a temperament of the fifth by (352/351)<sup>1/3</sup>.

	Another manifestation of the harmonisma in 29-HTT is the tuning of 11/9 at 347.353 cents, a third of a harmonisma narrow. Here 7/4 is just, with the spacing between the relevant chains of fifths in a subset of 29-HTT which served as a prototype for parapyth temperament at 58.090 cents, as compared with the [[91/88]] spacing which would produce a just 7/4 if the regular major sixth were at a just 22/13. But it is a third of a harmonisma narrow, and the spacing consequently a third of a harmonisma greater.		Another manifestation of the harmonisma in 29-HTT is the tuning of 11/9 at 347.353 cents, a third of a harmonisma narrow. Here 7/4 is just, with the spacing between the relevant chains of fifths in a subset of 29-HTT which served as a prototype for parapyth temperament at 58.090 cents, as compared with the [[91/88]] spacing which would produce a just 7/4 if the regular major sixth were at a just 22/13. But it is a third of a harmonisma narrow, and the spacing consequently a third of a harmonisma greater.

FloraC: +link

2022-12-09T11:15:18Z

+link

← Older revision		Revision as of 11:15, 9 December 2022
Line 10:		Line 10:
	Tempering out this comma in the full 13-limit gives the rank-5 '''harmonismic temperament'''. Equal temperaments where this comma is tempered with very high accuracy will have an interval corresponding to a "sharp fifth" of (ideally) 706.7 to 706.9 cents, corresponding to the range of fifths from 13/11 × 14/11 (→ [[182/121]]) on the lower end and 11/9 × 16/13 (→ [[176/117]]) on the higher end, and this interval is not mapped to 3/2. However, such temperaments are generally very precise, so [[224edo]], [[270edo]] and [[311edo]] offer slightly more manageable tunings. For less accurate temperaments still, 10648/10647 is notable as a comma of [[parapyth]].		Tempering out this comma in the full 13-limit gives the rank-5 '''harmonismic temperament'''. Equal temperaments where this comma is tempered with very high accuracy will have an interval corresponding to a "sharp fifth" of (ideally) 706.7 to 706.9 cents, corresponding to the range of fifths from 13/11 × 14/11 (→ [[182/121]]) on the lower end and 11/9 × 16/13 (→ [[176/117]]) on the higher end, and this interval is not mapped to 3/2. However, such temperaments are generally very precise, so [[224edo]], [[270edo]] and [[311edo]] offer slightly more manageable tunings. For less accurate temperaments still, 10648/10647 is notable as a comma of [[parapyth]].

	The harmonisma, 10648/10647, plays a striking role in [[George Secor]]'s 29-~~HTT~~ of 1975, the first temperament in the High Tolerance Temperament family. In this tuning, the fifth at 703.579 cents produces an augmented second (+9 fifths) at a just [[63/52]] (9/8 × 14/13), or a diminished seventh (-9 fifths) at [[104/63]], which exceeds three 13/11 thirds by a harmonisma. The 63/52 exceeds the Pythagorean augmented second, 19683/16384 (a 32805/32768 schisma larger than 6/5), by the [[secorian comma]], 28672/28431. Likewise 104/63 is narrower than the Pythagorean 32768/19683 by 28672/28431.		The harmonisma, 10648/10647, plays a striking role in [[Secor29htt\|George Secor's 29-tone high tolerance temperament]] of 1975, the first temperament in the High Tolerance Temperament family. In this tuning, the fifth at 703.579 cents produces an augmented second (+9 fifths) at a just [[63/52]] (9/8 × 14/13), or a diminished seventh (-9 fifths) at [[104/63]], which exceeds three 13/11 thirds by a harmonisma. The 63/52 exceeds the Pythagorean augmented second, 19683/16384 (a 32805/32768 schisma larger than 6/5), by the [[secorian comma]], 28672/28431. Likewise 104/63 is narrower than the Pythagorean 32768/19683 by 28672/28431.

	The secorian comma is made up of ([[896/891]] × [[352/351]]) or in other words (352/351 × 364/363 × 352/351), and is thus a harmonisma smaller than (352/351)<sup>3</sup>. In 29-HTT, each 13/11 is thus a third of a harmonisma or 0.054 cents wider than just. Secor's HTT fifth of 703.579 cents, or precisely (504/13)<sup>1/9</sup> or wide by 1/9 of a secorian comma, would thus need to be 1/9 harmonisma larger at 703.597 cents to produce a just 13/11, a temperament of the fifth by (352/351)<sup>1/3</sup>.		The secorian comma is made up of ([[896/891]] × [[352/351]]) or in other words (352/351 × 364/363 × 352/351), and is thus a harmonisma smaller than (352/351)<sup>3</sup>. In 29-HTT, each 13/11 is thus a third of a harmonisma or 0.054 cents wider than just. Secor's HTT fifth of 703.579 cents, or precisely (504/13)<sup>1/9</sup> or wide by 1/9 of a secorian comma, would thus need to be 1/9 harmonisma larger at 703.597 cents to produce a just 13/11, a temperament of the fifth by (352/351)<sup>1/3</sup>.