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Homothetic just intonation: Difference between revisions

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Homothetic just intonation is an operation for extended [[just intonation]] pitches conceived by Sui-hin Mak in July 2019. It aims at producing the pitches in between notes of an existing JI pitch collection.
{{Novelty}}
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.  


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


=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]]
Retrieved from "https://en.xen.wiki/w/Homothetic_just_intonation"