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 | 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 < | 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" | ||
| Line 70: | Line 70: | ||
| 535/282 || 1108.612475 || Homothetic major seventh | | 535/282 || 1108.612475 || Homothetic major seventh | ||
|- | |- | ||
| [[2/1]] || 1200 || [[Octave | | [[2/1]] || 1200 || [[Octave]], {{w|diapason}} | ||
|} | |} | ||
Latest revision as of 16:38, 10 January 2025
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 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.
| Frequency ratio | Cents | 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