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List of superparticular intervals: Difference between revisions

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+23-limit monzos and improve some factorizations
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Line 33: Line 33:
| 3/2
| 3/2
| {{monzo|-1 1}}
| {{monzo|-1 1}}
| [[perfect fifth]], 3rd harmonic (octave reduced), diapente
| [[perfect fifth]], octave-reduced 3rd harmonic, diapente
|
|
|-
|-
Line 40: Line 40:
| 2<sup>2</sup>/3
| 2<sup>2</sup>/3
| {{monzo|2 -1}}
| {{monzo|2 -1}}
| perfect fourth, 3rd subharmonic (octave reduced), diatessaron
| perfect fourth, octave-reduced 3rd subharmonic, diatessaron
| S2
| S2
|-
|-
Line 47: Line 47:
| 3<sup>2</sup>/2<sup>3</sup>
| 3<sup>2</sup>/2<sup>3</sup>
| {{monzo|-3 2}}
| {{monzo|-3 2}}
| (Pythagorean) (whole) tone, Pythagorean major second, major whole tone, 9th harmonic or harmonic ninth (octave reduced)
| (Pythagorean) (whole) tone, Pythagorean major second, major whole tone, octave-reduced 9th harmonic or harmonic ninth
| S3
| S3
|-
|-
Line 56: Line 56:
| 5/2<sup>2</sup>
| 5/2<sup>2</sup>
| {{monzo|-2 0 1}}
| {{monzo|-2 0 1}}
| classic/just major third, 5th harmonic (octave reduced)
| classic(al)/just major third, octave-reduced 5th harmonic
|
|
|-
|-
Line 63: Line 63:
| (2*3)/5
| (2*3)/5
| {{monzo|1 1 -1}}
| {{monzo|1 1 -1}}
| classic/just minor third
| classic(al)/just minor third
|
|
|-
|-
Line 70: Line 70:
| (2*5)/3<sup>2</sup>
| (2*5)/3<sup>2</sup>
| {{monzo|1 -2 1}}
| {{monzo|1 -2 1}}
| classic (whole) tone, classic major second, minor whole tone
| classic(al) (whole) tone, classic major second, minor whole tone
|
|
|-
|-
Line 77: Line 77:
| 2<sup>4</sup>/(3*5)
| 2<sup>4</sup>/(3*5)
| {{monzo|4 -1 -1}}
| {{monzo|4 -1 -1}}
| classic/just diatonic semitone, 15th subharmonic
| classic(al)/just diatonic semitone, 15th subharmonic
| S4
| S4
|-
|-
Line 84: Line 84:
| 5<sup>2</sup>/(2<sup>3</sup>*3)
| 5<sup>2</sup>/(2<sup>3</sup>*3)
| {{monzo|-3 -1 2}}
| {{monzo|-3 -1 2}}
| classic/just chromatic semitone, chroma, Zarlinian semitone
| classic(al)/just chromatic semitone, chroma, Zarlinian semitone
| S5
| S5
|-
|-
Line 107: Line 107:
| 2<sup>3</sup>/7
| 2<sup>3</sup>/7
| {{monzo|3 0 0 -1}}
| {{monzo|3 0 0 -1}}
| (septimal) supermajor second, septimal whole tone, 7th subharmonic
| (septimal) supermajor second, septimal whole tone, octave-reduced 7th subharmonic
|
|
|-
|-
Line 214: Line 214:
| (3*11)/2<sup>5</sup>
| (3*11)/2<sup>5</sup>
| {{monzo|-5 1 0 0 1}}
| {{monzo|-5 1 0 0 1}}
| undecimal 1/4-tone, undecimal diesis, al-Farabi's 1/4-tone, 33rd harmonic (octave reduced)
| undecimal 1/4-tone, undecimal diesis, al-Farabi's 1/4-tone, octave-reduced 33rd harmonic
|
|
|-
|-
Line 228: Line 228:
| (5*11)/(2*3<sup>3</sup>)
| (5*11)/(2*3<sup>3</sup>)
| {{monzo|-1 -3 1 0 1}}
| {{monzo|-1 -3 1 0 1}}
| undecimal diasecundal comma, eleventyfive comma
| telepathma, eleventyfive comma, undecimal diasecundal comma
|
|
|-
|-
Line 370: Line 370:
| (7*13)/(2*3<sup>2</sup>*5)
| (7*13)/(2*3<sup>2</sup>*5)
| {{monzo|-1 -2 -1 1 0 1}}
| {{monzo|-1 -2 -1 1 0 1}}
| [[Biome comma]], superleap comma
| Biome comma, superleap comma
|
|
|-
|-
Line 398: Line 398:
| (2*7)<sup>2</sup>/(3*5*13)
| (2*7)<sup>2</sup>/(3*5*13)
| {{monzo|2 -1 -1 2 0 -1}}
| {{monzo|2 -1 -1 2 0 -1}}
| [[Mynucumic_chords|mynucuma]]
| mynucuma
| S14
| S14
|-
|-
Line 405: Line 405:
| (5<sup>2</sup>*13)/(2<sup>2</sup>*3<sup>4</sup>)
| (5<sup>2</sup>*13)/(2<sup>2</sup>*3<sup>4</sup>)
| {{monzo|-2 -4 2 0 0 1}}
| {{monzo|-2 -4 2 0 0 1}}
| [[Marveltwin|marveltwin comma]]
| marveltwin comma
|
|
|-
|-
Line 512: Line 512:
| 17/2<sup>4</sup>
| 17/2<sup>4</sup>
| {{monzo|-4 0 0 0 0 0 1}}
| {{monzo|-4 0 0 0 0 0 1}}
| large septendecimal semitone, 17th harmonic (octave reduced)
| large septendecimal semitone, octave-reduced 17th harmonic
|
|
|-
|-
Line 594: Line 594:
| [[256/255]]
| [[256/255]]
| 6.7759
| 6.7759
| (2<sup>8</sup>)/(3*5*17)
| 2<sup>8</sup>/(3*5*17)
| {{monzo|8 -1 -1 0 0 0 -1}}
| {{monzo|8 -1 -1 0 0 0 -1}}
| septendecimal kleisma, 255th subharmonic
| septendecimal kleisma, octave-reduced 255th subharmonic
| S16
| S16
|-
|-
Line 645: Line 645:
| (5*11*13)/(2*3*7*17)
| (5*11*13)/(2*3*7*17)
| {{monzo|-1 -1 1 -1 1 1 -1}}
| {{monzo|-1 -1 1 -1 1 1 -1}}
| September comma, septembrisma, septendecimal bridge comma
| september comma, septembrisma
|
|
|-
|-
Line 652: Line 652:
| (7<sup>2</sup>*17)/(2<sup>6</sup>*13)
| (7<sup>2</sup>*17)/(2<sup>6</sup>*13)
| {{monzo|-6 0 0 2 0 -1 1}}
| {{monzo|-6 0 0 2 0 -1 1}}
| horizon comma
| horizma, horizon comma
|
|
|-
|-
Line 664: Line 664:
| [[1089/1088]]
| [[1089/1088]]
| 1.5905
| 1.5905
| (3<sup>2</sup>*11<sup>2</sup>)/(2<sup>6</sup>*17)
| (3*11)<sup>2</sup>/(2<sup>6</sup>*17)
| {{monzo|-6 2 0 0 2 0 -1}}
| {{monzo|-6 2 0 0 2 0 -1}}
| twosquare comma
| twosquare comma
Line 671: Line 671:
| [[1156/1155]]
| [[1156/1155]]
| 1.4983
| 1.4983
| (2<sup>2</sup>*17<sup>2</sup>)/(3*5*7*11)
| (2*17)<sup>2</sup>/(3*5*7*11)
| {{monzo|2 -1 -1 -1 -1 0 2}}
| {{monzo|2 -1 -1 -1 -1 0 2}}
| septendecimal 1/4-tones comma
| septendecimal 1/4-tones comma
Line 678: Line 678:
| [[1225/1224]]
| [[1225/1224]]
| 1.4138
| 1.4138
| (5<sup>2</sup>*7<sup>2</sup>)/(2<sup>3</sup>*3<sup>2</sup>*17)
| (5*7)<sup>2</sup>/(2<sup>3</sup>*3<sup>2</sup>*17)
| {{monzo|-3 -2 2 2 0 0 -1}}
| {{monzo|-3 -2 2 2 0 0 -1}}
| noellisma
| noellisma
Line 694: Line 694:
| (3<sup>5</sup>*7)/[(2*5)<sup>2</sup>*17]
| (3<sup>5</sup>*7)/[(2*5)<sup>2</sup>*17]
| {{monzo|-2 5 -2 1 0 0 -1}}
| {{monzo|-2 5 -2 1 0 0 -1}}
| palingenetic comma, palingenesis comma, palingenesma
| palingenetic comma, palingenesis
|
|
|-
|-
Line 713: Line 713:
| [[2500/2499]]
| [[2500/2499]]
| 0.69263
| 0.69263
| (2<sup>2</sup>*5<sup>4</sup>)/(3*7<sup>2</sup>*17)
| (2*5<sup>2</sup>)<sup>2</sup>/(3*7<sup>2</sup>*17)
| {{monzo|2 -1 4 -2 0 0 -1}}
| {{monzo|2 -1 4 -2 0 0 -1}}
| sperasma
| sperasma
Line 720: Line 720:
| [[2601/2600]]
| [[2601/2600]]
| 0.66573
| 0.66573
| (3<sup>2</sup>*17<sup>2</sup>)/(2<sup>3</sup>*5<sup>2</sup>*13)
| (3*17)<sup>2</sup>/(2<sup>3</sup>*5<sup>2</sup>*13)
| {{monzo|-3 2 -2 0 0 -1 2}}
| {{monzo|-3 2 -2 0 0 -1 2}}
| septendecimal 1/6-tones comma
| septendecimal 1/6-tones comma
Line 727: Line 727:
| 4914/4913
| 4914/4913
| 0.35234
| 0.35234
| (2*3<sup>3</sup>*7*13)/(17<sup>3</sup>)
| (2*3<sup>3</sup>*7*13)/17<sup>3</sup>
| {{monzo|1 3 0 1 0 1 -3}}
| {{monzo|1 3 0 1 0 1 -3}}
|  
|  
Line 734: Line 734:
| [[5832/5831]]
| [[5832/5831]]
| 0.29688
| 0.29688
| (2<sup>3</sup>*3<sup>6</sup>)/(7<sup>3</sup>*17)
| (2*3<sup>2</sup>)<sup>3</sup>/(7<sup>3</sup>*17)
| {{monzo|3 6 0 -3 0 0 -1}}
| {{monzo|3 6 0 -3 0 0 -1}}
| chlorisma
| chlorisma
Line 748: Line 748:
| [[sparkisma|14400/14399]]
| [[sparkisma|14400/14399]]
| 0.12023
| 0.12023
| (2<sup>6</sup>*3<sup>2</sup>*5<sup>2</sup>)/(7*11<sup>2</sup>*17)
| (2<sup>3</sup>*3*5)<sup>2</sup>/(7*11<sup>2</sup>*17)
| {{monzo|6 2 2 -1 -2 0 -1}}
| {{monzo|6 2 2 -1 -2 0 -1}}
| sparkisma
| sparkisma
Line 755: Line 755:
| 28561/28560
| 28561/28560
| 0.060616
| 0.060616
| (13<sup>4</sup>)/(2<sup>4</sup>*3*5*7*17)
| 13<sup>4</sup>/(2<sup>4</sup>*3*5*7*17)
| {{monzo|-4 -1 -1 -1 0 4 -1}}
| {{monzo|-4 -1 -1 -1 0 4 -1}}
|  
|  
Line 776: Line 776:
| 194481/194480
| 194481/194480
| 0.008902
| 0.008902
| (3<sup>4</sup>*7<sup>4</sup>)/(2<sup>4</sup>*5*11*13*17)
| (3*7)<sup>4</sup>/(2<sup>4</sup>*5*11*13*17)
| {{monzo|-4 4 -1 4 -1 -1 -1}}
| {{monzo|-4 4 -1 4 -1 -1 -1}}
| scintillisma
| scintillisma
Line 890: Line 890:
| [[324/323]]
| [[324/323]]
| 5.3516
| 5.3516
| (2<sup>2</sup>*3<sup>4</sup>)/(17*19)
| (2*3<sup>2</sup>)<sup>2</sup>/(17*19)
| {{monzo|2 4 0 0 0 0 -1 -1}}
| {{monzo|2 4 0 0 0 0 -1 -1}}
| nusu comma
| nusu comma
Line 911: Line 911:
| 400/399
| 400/399
| 4.3335
| 4.3335
| (2<sup>4</sup>*5<sup>2</sup>)/(3*7*19)
| (2<sup>2</sup>*5)<sup>2</sup>/(3*7*19)
| {{monzo|4 -1 2 -1 0 0 0 -1}}
| {{monzo|4 -1 2 -1 0 0 0 -1}}
|  
|  
Line 988: Line 988:
| [[1729/1728]]
| [[1729/1728]]
| 1.0016
| 1.0016
| (7*13*19)/(2<sup>6</sup>*3<sup>3</sup>)
| (7*13*19)/(2<sup>2</sup>*3)<sup>3</sup>
| {{monzo|-6 -3 0 1 0 1 0 1}}
| {{monzo|-6 -3 0 1 0 1 0 1}}
| ramanujanisma
| ramanujanisma
Line 1,016: Line 1,016:
| 3136/3135
| 3136/3135
| 0.5521
| 0.5521
| (2<sup>6</sup>*7<sup>2</sup>)/(3*5*11*19)
| (2<sup>3</sup>*7)<sup>2</sup>/(3*5*11*19)
| {{monzo|6 -1 -1 2 -1 0 0 -1}}
| {{monzo|6 -1 -1 2 -1 0 0 -1}}
|  
|  
Line 1,023: Line 1,023:
| 3250/3249
| 3250/3249
| 0.5328
| 0.5328
| (2*5<sup>3</sup>*13)/(3<sup>2</sup>*19<sup>2</sup>)
| (2*5<sup>3</sup>*13)/(3*19)<sup>2</sup>
| {{monzo|1 -2 3 0 0 1 0 -2}}
| {{monzo|1 -2 3 0 0 1 0 -2}}
|  
|  
Line 1,037: Line 1,037:
| 5776/5775
| 5776/5775
| 0.2998
| 0.2998
| (2<sup>4</sup>*19<sup>2</sup>)/(3*5<sup>2</sup>*7*11)
| (2<sup>2</sup>*19)<sup>2</sup>/(3*5<sup>2</sup>*7*11)
| {{monzo|4 -1 -2 -1 -1 0 0 2}}
| {{monzo|4 -1 -2 -1 -1 0 0 2}}
|  
|  
Line 1,044: Line 1,044:
| 5929/5928
| 5929/5928
| 0.2920
| 0.2920
| (7<sup>2</sup>*11<sup>2</sup>)/(2<sup>3</sup>*3*13*19)
| (7*11)<sup>2</sup>/(2<sup>3</sup>*3*13*19)
| {{monzo|-3 -1 0 2 2 -1 0 -1}}
| {{monzo|-3 -1 0 2 2 -1 0 -1}}
|  
|  
Line 1,065: Line 1,065:
| 6860/6859
| 6860/6859
| 0.2524
| 0.2524
| (2<sup>2</sup>*5*7<sup>3</sup>)/(19<sup>3</sup>)
| (2<sup>2</sup>*5*7<sup>3</sup>)/19<sup>3</sup>
| {{monzo|2 0 1 3 0 0 0 -3}}
| {{monzo|2 0 1 3 0 0 0 -3}}
|  
|  
Line 1,114: Line 1,114:
| 23409/23408
| 23409/23408
| 0.07396
| 0.07396
| (3<sup>4</sup>*17<sup>2</sup>)/(2<sup>4</sup>*7*11*19)
| (3<sup>2</sup>*17)<sup>2</sup>/(2<sup>4</sup>*7*11*19)
| {{monzo|-4 4 0 -1 -1 0 1 -1}}
| {{monzo|-4 4 0 -1 -1 0 1 -1}}
|  
|  
Line 1,121: Line 1,121:
| 27456/27455
| 27456/27455
| 0.06306
| 0.06306
| (2<sup>6</sup>*3*11*17)/(5*17<sup>2</sup>*19)
| (2<sup>6</sup>*3*11)/(5*17<sup>2</sup>*19)
| {{monzo|6 1 -1 0 1 0 -2 -1}}
| {{monzo|6 1 -1 0 1 0 -2 -1}}
|  
|  
Line 1,128: Line 1,128:
| 28900/28899
| 28900/28899
| 0.05991
| 0.05991
| (2<sup>2</sup>*5<sup>2</sup>*17<sup>2</sup>)/(3<sup>2</sup>*13<sup>2</sup>*19)
| (2*5*17)<sup>2</sup>/(3<sup>2</sup>*13<sup>2</sup>*19)
| {{monzo|2 -2 2 0 0 -2 2 -1}}
| {{monzo|2 -2 2 0 0 -2 2 -1}}
|  
|  
Line 1,135: Line 1,135:
| 43681/43680
| 43681/43680
| 0.03963
| 0.03963
| (11<sup>2</sup>*19<sup>2</sup>)/(2<sup>5</sup>*3*5*7*13)
| (11*19)<sup>2</sup>/(2<sup>5</sup>*3*5*7*13)
| {{monzo|-5 -1 -1 -1 2 -1 0 2}}
| {{monzo|-5 -1 -1 -1 2 -1 0 2}}
|  
|  
Line 1,149: Line 1,149:
| 104976/104975
| 104976/104975
| 0.01649
| 0.01649
| (2<sup>4</sup>*3<sup>8</sup>)/(5<sup>2</sup>*13*17*19)
| (2*3<sup>2</sup>)<sup>4</sup>/(5<sup>2</sup>*13*17*19)
| {{monzo|4 8 -2 0 0 0 -1 -1 -1}}
| {{monzo|4 8 -2 0 0 0 -1 -1 -1}}
|  
|  
Line 1,191: Line 1,191:
| 5909761/5909760
| 5909761/5909760
| 0.0002929
| 0.0002929
| (11<sup>2</sup>*13<sup>2</sup>*17<sup>2</sup>)/(2<sup>8</sup>*3<sup>5</sup>*5*19)
| (11*13*17)<sup>2</sup>/(2<sup>8</sup>*3<sup>5</sup>*5*19)
| {{monzo|-8 -5 -1 0 2 2 2 -1}}
| {{monzo|-8 -5 -1 0 2 2 2 -1}}
|  
|  
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