Homothetic just intonation: Difference between revisions

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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''0 = {{sfrac|''r''2''x''1 + ''r''1''x''2|''r''1 + ''r''2}} 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.

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| 535/282 || 1108.612475 || Homothetic major seventh

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| [[2/1]] || 1200 || [[Octave~~| octave~~]], ~~[https://en.wikipedia.org/wiki/Diapason~~ diapason]

| [[2/1]] || 1200 || [[Octave]], {{w|diapason}}

|}

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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>&nbsp;=&nbsp;{{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"
@@ Line 70: / Line 70: @@
 | 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}}
 |}