Homothetic just intonation - Revision history

Overthink: Novelty

2026-02-05T04:44:48Z

Novelty

← Older revision		Revision as of 04:44, 5 February 2026
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			{{Novelty}}
	Homothetic just intonation is a kind of extended [[just intonation]] conceived by [[Sui-hin Mak]]. The term 'homothetic' refers to the {{w\|Homothetic center#Computing homothetic centers\|homothetic formula}} for circles. The tuning aims at producing the pitches between notes of an existing prime limit JI pitch collection.		Homothetic just intonation is a kind of extended [[just intonation]] conceived by [[Sui-hin Mak]]. The term 'homothetic' refers to the {{w\|Homothetic center#Computing homothetic centers\|homothetic formula}} for circles. The tuning aims at producing the pitches between notes of an existing prime limit JI pitch collection.

ArrowHead294 at 16:38, 10 January 2025

2025-01-10T16:38:14Z

← Older revision		Revision as of 16:38, 10 January 2025
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	Homothetic just intonation is a kind of extended [[just intonation]] conceived by [[Sui-hin Mak]]. The term 'homothetic' refers to the {{w\|Homothetic center#Computing homothetic centers\|homothetic formula}} for circles. The tuning aims at producing the pitches between notes of an existing prime limit JI pitch collection.		Homothetic just intonation is a kind of extended [[just intonation]] conceived by [[Sui-hin Mak]]. The term 'homothetic' refers to the {{w\|Homothetic center#Computing homothetic centers\|homothetic formula}} for circles. The tuning aims at producing the pitches between notes of an existing prime limit JI pitch collection.

	Circles are drawn on an axis with the existing pitches as their centres, and with their sizes determined by its prime factors. The homothetic formula ''x''<sub>0</sub>~~ ~~=~~ ~~{{sfrac\|''r''<sub>2</sub>''x''<sub>1</sub> + ''r''<sub>1</sub>''x''<sub>2</sub>\|''r''<sub>1</sub> + ''r''<sub>2</sub>}} is used to locate the intersection of common tangents of two given circles. The new pitch between two successive existing pitches is determined by the homothetic centre of the two circles.		Circles are drawn on an axis with the existing pitches as their centres, and with their sizes determined by its prime factors. The homothetic formula {{nowrap\|''x''<sub>0</sub> {{=}} {{sfrac\|''r''<sub>2</sub>''x''<sub>1</sub> + ''r''<sub>1</sub>''x''<sub>2</sub>\|''r''<sub>1</sub> + ''r''<sub>2</sub>}}}} is used to locate the intersection of common tangents of two given circles. The new pitch between two successive existing pitches is determined by the homothetic centre of the two circles.

	{\| class="wikitable sortable"		{\| class="wikitable sortable"

ArrowHead294 at 16:35, 10 January 2025

2025-01-10T16:35:18Z

← Older revision		Revision as of 16:35, 10 January 2025
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	Homothetic just intonation is a kind of extended [[just intonation]] conceived by [[Sui-hin Mak]]. The term 'homothetic' refers to the ~~[[wikipedia:Homothetic_center~~#~~Computing_homothetic_centers~~\|homothetic formula]] for circles. The tuning aims at producing the pitches between notes of an existing prime limit JI pitch collection.		Homothetic just intonation is a kind of extended [[just intonation]] conceived by [[Sui-hin Mak]]. The term 'homothetic' refers to the {{w\|Homothetic center#Computing homothetic centers\|homothetic formula}} for circles. The tuning aims at producing the pitches between notes of an existing prime limit JI pitch collection.

	Circles are drawn on an axis with the existing pitches as their centres, and with their sizes determined by its prime factors. The homothetic formula <~~math~~>~~x_0~~ = ~~\frac~~{~~r_2 x_1~~ + ~~r_1 x_2}{r_1~~ + ~~r_2}~~</~~math~~> is used to locate the intersection of common tangents of two given circles. The new pitch between two successive existing pitches is determined by the homothetic centre of the two circles.		Circles are drawn on an axis with the existing pitches as their centres, and with their sizes determined by its prime factors. The homothetic formula ''x''<sub>0</sub> = {{sfrac\|''r''<sub>2</sub>''x''<sub>1</sub> + ''r''<sub>1</sub>''x''<sub>2</sub>\|''r''<sub>1</sub> + ''r''<sub>2</sub>}} is used to locate the intersection of common tangents of two given circles. The new pitch between two successive existing pitches is determined by the homothetic centre of the two circles.

	{\| class="wikitable sortable"		{\| class="wikitable sortable"
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	\| 535/282 \|\| 1108.612475 \|\| Homothetic major seventh		\| 535/282 \|\| 1108.612475 \|\| Homothetic major seventh
	\|-		\|-
	\| [[2/1]] \|\| 1200 \|\| [[Octave~~\| octave~~]], ~~[https://en.wikipedia.org/wiki/Diapason~~ diapason]		\| [[2/1]] \|\| 1200 \|\| [[Octave]], {{w\|diapason}}
	\|}		\|}

BudjarnLambeth: Add link to name

2024-09-18T23:44:16Z

Add link to name

← Older revision		Revision as of 23:44, 18 September 2024
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	Homothetic just intonation is a kind of extended [[just intonation]] conceived by Sui-hin Mak. The term 'homothetic' refers to the [[wikipedia:Homothetic_center#Computing_homothetic_centers\|homothetic formula]] for circles. The tuning aims at producing the pitches between notes of an existing prime limit JI pitch collection.		Homothetic just intonation is a kind of extended [[just intonation]] conceived by [[Sui-hin Mak]]. The term 'homothetic' refers to the [[wikipedia:Homothetic_center#Computing_homothetic_centers\|homothetic formula]] for circles. The tuning aims at producing the pitches between notes of an existing prime limit JI pitch collection.

	Circles are drawn on an axis with the existing pitches as their centres, and with their sizes determined by its prime factors. The homothetic formula <math>x_0 = \frac{r_2 x_1 + r_1 x_2}{r_1 + r_2}</math> is used to locate the intersection of common tangents of two given circles. The new pitch between two successive existing pitches is determined by the homothetic centre of the two circles.		Circles are drawn on an axis with the existing pitches as their centres, and with their sizes determined by its prime factors. The homothetic formula <math>x_0 = \frac{r_2 x_1 + r_1 x_2}{r_1 + r_2}</math> is used to locate the intersection of common tangents of two given circles. The new pitch between two successive existing pitches is determined by the homothetic centre of the two circles.

ArrowHead294 at 15:47, 9 September 2024

2024-09-09T15:47:42Z

Show changes

BudjarnLambeth at 11:22, 25 April 2023

2023-04-25T11:22:34Z

← Older revision		Revision as of 11:22, 25 April 2023
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	[[Category:Just intonation]]		[[Category:Just intonation]]
	[[Category:Math]]		[[Category:Math]]
			[[Category:31-tone scales]]

BudjarnLambeth: Categorised this uncategorised page

2023-04-25T11:20:17Z

Categorised this uncategorised page

← Older revision		Revision as of 11:20, 25 April 2023
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	\|[[2/1]]		\|[[2/1]]
	\|1200		\|1200
	\|[[Octave\|octave]], [~~http~~://en.wikipedia.org/wiki/Diapason diapason]		\|[[Octave\|octave]], [https://en.wikipedia.org/wiki/Diapason diapason]
	\|}		\|}

	=Links=		=Links=
	* [https://medium.com/@maksuihin/homothetic-just-intonation-b468777f724b Homothetic Just Intonation] by Sui-hin Mak		* [https://medium.com/@maksuihin/homothetic-just-intonation-b468777f724b Homothetic Just Intonation] by Sui-hin Mak
			[[Category:Just intonation]]
			[[Category:Math]]

Maksuihin at 14:44, 14 July 2019

2019-07-14T14:44:05Z

← Older revision		Revision as of 14:44, 14 July 2019
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	Homothetic just intonation is a kind of extended [[just intonation]] conceived by Sui-hin Mak. The term 'homothetic' refers to the [[wikipedia:Homothetic_center#Computing_homothetic_centers\|homothetic formula]] for circles. The tuning aims at producing the pitches between notes of an existing prime limit JI pitch collection.		Homothetic just intonation is a kind of extended [[just intonation]] conceived by Sui-hin Mak. The term 'homothetic' refers to the [[wikipedia:Homothetic_center#Computing_homothetic_centers\|homothetic formula]] for circles. The tuning aims at producing the pitches between notes of an existing prime limit JI pitch collection.

	Circles are drawn on an axis with the existing pitches as their centres, and with their sizes determined by its prime factors. The homothetic formula ~~'''x₀~~ = ~~(r₂x₁~~ + ~~r₁x₂)~~ / ~~(r₁ + r₂)'''~~ is used to locate the intersection of common tangents of two given circles. The new pitch between two successive existing pitches is determined by the homothetic centre of the two circles.		Circles are drawn on an axis with the existing pitches as their centres, and with their sizes determined by its prime factors. The homothetic formula <math>x_0 = \frac{r_2 x_1 + r_1 x_2}{r_1 + r_2}</math> is used to locate the intersection of common tangents of two given circles. The new pitch between two successive existing pitches is determined by the homothetic centre of the two circles.

	{\| class="wikitable sortable"		{\| class="wikitable sortable"

Maksuihin at 14:31, 14 July 2019

2019-07-14T14:31:25Z

← Older revision		Revision as of 14:31, 14 July 2019
Line 1:		Line 1:
	Homothetic just intonation is a kind of extended [[just intonation]] conceived by Sui-hin Mak. The term 'homothetic' refers to the [[wikipedia:Homothetic_center#Computing_homothetic_centers\|homothetic formula]] for circles. The tuning aims at producing the pitches between notes of an existing prime limit JI pitch collection.		Homothetic just intonation is a kind of extended [[just intonation]] conceived by Sui-hin Mak. The term 'homothetic' refers to the [[wikipedia:Homothetic_center#Computing_homothetic_centers\|homothetic formula]] for circles. The tuning aims at producing the pitches between notes of an existing prime limit JI pitch collection.

	Circles are drawn ~~at all points of~~ the existing pitches as their centres, with ~~its size~~ determined by its prime factors. The homothetic formula '''x₀ = (r₂x₁ + r₁x₂) / (r₁ + r₂)''' is used to ~~find~~ the intersection of common tangents of two given circles. The new pitch between two successive existing pitches is determined by the homothetic centre of the two circles.		Circles are drawn on an axis with the existing pitches as their centres, and with their sizes determined by its prime factors. The homothetic formula '''x₀ = (r₂x₁ + r₁x₂) / (r₁ + r₂)''' is used to locate the intersection of common tangents of two given circles. The new pitch between two successive existing pitches is determined by the homothetic centre of the two circles.

	{\| class="wikitable sortable"		{\| class="wikitable sortable"

Maksuihin at 14:26, 14 July 2019

2019-07-14T14:26:22Z

← Older revision		Revision as of 14:26, 14 July 2019
Line 1:		Line 1:
	Homothetic just intonation is a kind of extended [[just intonation]] conceived by Sui-hin Mak. The term 'homothetic' refers to the [[wikipedia:Homothetic_center#Computing_homothetic_centers\|homothetic formula]] '''x₀ = (r₂x₁ + r₁x₂) / (r₁ + r₂)'''. The ~~tuning aims at producing~~ the ~~pitches in between notes~~ of ~~an existing prime limit JI pitch collection~~.		Homothetic just intonation is a kind of extended [[just intonation]] conceived by Sui-hin Mak. The term 'homothetic' refers to the [[wikipedia:Homothetic_center#Computing_homothetic_centers\|homothetic formula]] for circles. The tuning aims at producing the pitches between notes of an existing prime limit JI pitch collection.

			Circles are drawn at all points of the existing pitches as their centres, with its size determined by its prime factors. The homothetic formula '''x₀ = (r₂x₁ + r₁x₂) / (r₁ + r₂)''' is used to find the intersection of common tangents of two given circles. The new pitch between two successive existing pitches is determined by the homothetic centre of the two circles.

	~~Octave-equivalent 31-tone homothetic just scale generated by 11-limit JI:~~
	{\| class="wikitable sortable"		{\| class="wikitable sortable"
			\|+Octave-equivalent 31-tone homothetic just scale generated by 11-limit JI
	\|-		\|-
	! frequency		! frequency

← Older revision		Revision as of 16:38, 10 January 2025
Line 1:		Line 1:
	Homothetic just intonation is a kind of extended [[just intonation]] conceived by [[Sui-hin Mak]]. The term 'homothetic' refers to the {{w\|Homothetic center#Computing homothetic centers\|homothetic formula}} for circles. The tuning aims at producing the pitches between notes of an existing prime limit JI pitch collection.		Homothetic just intonation is a kind of extended [[just intonation]] conceived by [[Sui-hin Mak]]. The term 'homothetic' refers to the {{w\|Homothetic center#Computing homothetic centers\|homothetic formula}} for circles. The tuning aims at producing the pitches between notes of an existing prime limit JI pitch collection.

	Circles are drawn on an axis with the existing pitches as their centres, and with their sizes determined by its prime factors. The homothetic formula ''x''<sub>0</sub>~~ ~~=~~ ~~{{sfrac\|''r''<sub>2</sub>''x''<sub>1</sub> + ''r''<sub>1</sub>''x''<sub>2</sub>\|''r''<sub>1</sub> + ''r''<sub>2</sub>}} is used to locate the intersection of common tangents of two given circles. The new pitch between two successive existing pitches is determined by the homothetic centre of the two circles.		Circles are drawn on an axis with the existing pitches as their centres, and with their sizes determined by its prime factors. The homothetic formula {{nowrap\|''x''<sub>0</sub> {{=}} {{sfrac\|''r''<sub>2</sub>''x''<sub>1</sub> + ''r''<sub>1</sub>''x''<sub>2</sub>\|''r''<sub>1</sub> + ''r''<sub>2</sub>}}}} is used to locate the intersection of common tangents of two given circles. The new pitch between two successive existing pitches is determined by the homothetic centre of the two circles.

	{\| class="wikitable sortable"		{\| class="wikitable sortable"