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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]] '''x₀ = (r₂x₁ + r₁x₂) / (r₁ + r₂)'''. The tuning 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
! Frequency ratio
ratio
! Cents
! cents
! Names
value
! 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]]
Retrieved from "https://en.xen.wiki/w/Homothetic_just_intonation"