# Homothetic just intonation

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 $x_0 = \frac{r_2 x_1 + r_1 x_2}{r_1 + r_2}$ 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.

Octave-equivalent 31-tone homothetic just scale generated by 11-limit JI
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