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 [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.

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

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