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 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.
--tone homothetic just scale generated by 11-limit JI:
 {| 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
-|
 |-
-|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
-|
 |-
-|1981/1748
+| 1981/1748 || 216.628435 ||
-|216.628435
-|
 |-
-|97/84
+| 97/84 || 249.114503 || Homothetic semifourth
-|249.114503
-|semifourth
 |-
-|569/480
+| 569/480 || 294.473096 || Small homothetic supraminor third, quasi-Pythagorean minor third
-|294.473096
-|
 |-
-|1201/990
+| 1201/990 || 334.482865 || Large homothetic supraminor third
-|334.482865
-|
 |-
-|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
-|
 |-
-|511/376
+| 511/376 || 531.108755 || Homothetic acute fourth
-|531.108755
-|
 |-
-|1107/800
+| 1107/800 || 562.299980 || Homothetic augmented fourth
-|562.299980
-|
 |-
-|99/70
+| 99/70 || 600.088324 || Quasi-tempered tritone
-|600.088324
-|
 |-
-|159/110
+| 159/110 || 637.827890 || Homothetic diminished fifth
-|637.827890
-|
 |-
-|761/517
+| 761/517 || 669.278608 || Homothetic quasi-catafifth
-|669.278608
-|
 |-
-|6001/3933
+| 6001/3933 || 731.487292 || Homothetic super-fifth
-|731.487292
-|
 |-
-|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
-|
 |-
-|97/56
+| 97/56 || 951.069504 || Homothetic semitwelve
-|951.069504
-|semitwelve
 |-
-|3085/1748
+| 3085/1748 || 983.478365 ||
-|983.478365
-|
 |-
-|4429/2475
+| 4429/2475 || 1007.462966 || Quasi-meantone minor seventh
-|1007.462966
-|
 |-
-|2191/1210
+| 2191/1210 || 1027.898924 || Homothetic minor seventh
-|1027.898924
-|
 |-
-|241/132
+| 241/132 || 1042.194260 || Homothetic neutral seventh
-|1042.194260
-|
 |-
-|535/282
+| 535/282 || 1108.612475 || Homothetic major seventh
-|1108.612475
-|
 |-
-|[[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
+[[Category:Just intonation]]
+[[Category:Math]]
+[[Category:31-tone scales]]