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]] '''x₀ = (r₂x₁ + r₁x₂) / (r₁ + r₂)'''. The | 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. | |||
{| class="wikitable sortable" | {| class="wikitable sortable" | ||
|+Octave-equivalent 31-tone homothetic just scale generated by 11-limit JI | |||
|- | |- | ||
! frequency | ! frequency | ||
Revision as of 14:26, 14 July 2019
Homothetic just intonation is a kind of extended just intonation conceived by Sui-hin Mak. The term 'homothetic' refers to the 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.
| frequency
ratio |
cents
value |
names |
|---|---|---|
| 1/1 | 0 | unison |
| 546/517 | 94.484004 | large homothetic semitone |
| 241/220 | 156.835547 | |
| 243/220 | 172.143348 | |
| 2213/1980 | 192.603625 | quasi-meantone |
| 1981/1748 | 216.628435 | |
| 97/84 | 249.114503 | homothetic semifourth |
| 569/480 | 294.473096 | small homothetic supraminor third, quasi-Pythagorean minor third |
| 1201/990 | 334.482865 | large homothetic supraminor third |
| 977/792 | 363.429758 | |
| 1223/968 | 404.814542 | |
| 281/220 | 423.679928 | |
| 573/437 | 469.082231 | homothetic sub-fourth |
| 511/376 | 531.108755 | homothetic acute fourth |
| 1107/800 | 562.299980 | homothetic augmented fourth |
| 99/70 | 600.088324 | quasi-tempered tritone |
| 159/110 | 637.827890 | homothetic diminished fifth |
| 761/517 | 669.278608 | homothetic quasi-catafifth |
| 6001/3933 | 731.487292 | homothetic super-fifth |
| 1973/1260 | 776.360667 | |
| 1219/770 | 795.321330 | |
| 981/605 | 836.781593 | |
| 399/242 | 865.658039 | |
| 27/16 | 905.865003 | Pythagorean major sixth |
| 97/56 | 951.069504 | homothetic semitwelve |
| 3085/1748 | 983.478365 | |
| 4429/2475 | 1007.462966 | quasi-meantone minor seventh |
| 2191/1210 | 1027.898924 | homothetic minor seventh |
| 241/132 | 1042.194260 | homothetic neutral seventh |
| 535/282 | 1108.612475 | homothetic major seventh |
| 2/1 | 1200 | octave, diapason |
Links
- Homothetic Just Intonation by Sui-hin Mak