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]]. It aims at producing the pitches in 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 {{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.


Octave-equivalent 31-tone homothetic just scale generated by 11-limit JI:
{| class="wikitable sortable"
{| class="wikitable sortable"
|+ style="font-size: 105%;" | Octave-equivalent 31-tone homothetic just scale generated by 11-limit JI
|-
|-
! frequency ratio
! Frequency ratio
! cents value
! Cents
! names
! Names
|-
|-
|[[1/1]]
| [[1/1]] || 0 || unison
|0
|unison
|-
|-
|546/517
| 546/517 || 94.484004 || Large homothetic semitone
|94.484004
|large homothetic semitone
|-
|-
|241/220
| 241/220 || 156.835547 ||  
|156.835547
|
|-
|-
|243/220
| 243/220 || 172.143348 ||  
|172.143348
|
|-
|-
|2213/1980
| 2213/1980 || 192.603625 || Quasi-meantone
|192.603625
|quasi-meantone
|-
|-
|1981/1748
| 1981/1748 || 216.628435 ||  
|216.628435
|
|-
|-
|97/84
| 97/84 || 249.114503 || Homothetic semifourth
|249.114503
|homothetic semifourth
|-
|-
|569/480
| 569/480 || 294.473096 || Small homothetic supraminor third, quasi-Pythagorean minor third
|294.473096
|small homothetic supraminor third,
 
quasi-Pythagorean minor third
|-
|-
|1201/990
| 1201/990 || 334.482865 || Large homothetic supraminor third
|334.482865
|large homothetic supraminor third
|-
|-
|977/792
| 977/792 || 363.429758 ||  
|363.429758
|
|-
|-
|1223/968
| 1223/968 || 404.814542 ||  
|404.814542
|
|-
|-
|281/220
| 281/220 || 423.679928 ||  
|423.679928
|
|-
|-
|573/437
| 573/437 || 469.082231 || Homothetic sub-fourth
|469.082231
|homothetic sub-fourth
|-
|-
|511/376
| 511/376 || 531.108755 || Homothetic acute fourth
|531.108755
|homothetic acute fourth
|-
|-
|1107/800
| 1107/800 || 562.299980 || Homothetic augmented fourth
|562.299980
|homothetic augmented fourth
|-
|-
|99/70
| 99/70 || 600.088324 || Quasi-tempered tritone
|600.088324
|quasi-tempered tritone
|-
|-
|159/110
| 159/110 || 637.827890 || Homothetic diminished fifth
|637.827890
|homothetic diminished fifth
|-
|-
|761/517
| 761/517 || 669.278608 || Homothetic quasi-catafifth
|669.278608
|homothetic quasi-catafifth
|-
|-
|6001/3933
| 6001/3933 || 731.487292 || Homothetic super-fifth
|731.487292
|homothetic super-fifth
|-
|-
|1973/1260
| 1973/1260 || 776.360667 ||  
|776.360667
|
|-
|-
|1219/770
| 1219/770 || 795.321330 ||  
|795.321330
|
|-
|-
|981/605
| 981/605 || 836.781593 ||  
|836.781593
|
|-
|-
|399/242
| 399/242 || 865.658039 ||  
|865.658039
|
|-
|-
|[[27/16]]
| [[27/16]] || 905.865003 || Pythagorean major sixth
|905.865003
|Pythagorean major sixth
|-
|-
|97/56
| 97/56 || 951.069504 || Homothetic semitwelve
|951.069504
|homothetic semitwelve
|-
|-
|3085/1748
| 3085/1748 || 983.478365 ||  
|983.478365
|
|-
|-
|4429/2475
| 4429/2475 || 1007.462966 || Quasi-meantone minor seventh
|1007.462966
|quasi-meantone minor seventh
|-
|-
|2191/1210
| 2191/1210 || 1027.898924 || Homothetic minor seventh
|1027.898924
|homothetic minor seventh
|-
|-
|241/132
| 241/132 || 1042.194260 || Homothetic neutral seventh
|1042.194260
|homothetic neutral seventh
|-
|-
|535/282
| 535/282 || 1108.612475 || Homothetic major seventh
|1108.612475
|homothetic major seventh
|-
|-
|[[2/1]]
| [[2/1]] || 1200 || [[Octave]], {{w|diapason}}
|1200
|octave
|}
|}


=Links=
== Links ==
* [https://medium.com/@maksuihin/homothetic-just-intonation-b468777f724b Homothetic Just Intonation] by Sui-hin Mak
* [https://medium.com/@maksuihin/homothetic-just-intonation-b468777f724b Homothetic Just Intonation] by Sui-hin Mak
[[Category:Just intonation]]
[[Category:Math]]
[[Category:31-tone scales]]

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.

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

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