45edo: Difference between revisions
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m Removing from Category:Theory using Cat-a-lot |
→Intervals: Changed the gamut to be consistent: dim vmin min mid maj ^maj aug. Moved the up or down to before the note name, vC not Cv. |
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| Line 37: | Line 37: | ||
| -0.174 | | -0.174 | ||
| Up unison | | Up unison | ||
| D | | ^D | ||
|- | |- | ||
| 2 | | 2 | ||
| Line 52: | Line 52: | ||
| 80.537 | | 80.537 | ||
| -0.537 | | -0.537 | ||
| | | Diminished 2nd | ||
| | | Ebb | ||
|- | |- | ||
| 4 | | 4 | ||
| Line 61: | Line 61: | ||
| 1.711 | | 1.711 | ||
| Downminor 2nd | | Downminor 2nd | ||
| | | vEb | ||
|- | |- | ||
| 5 | | 5 | ||
| Line 76: | Line 76: | ||
| 165.004 | | 165.004 | ||
| -5.004 | | -5.004 | ||
| | | Mid 2nd | ||
| | | vE | ||
|- | |- | ||
| 7 | | 7 | ||
| Line 93: | Line 93: | ||
| 9.423 | | 9.423 | ||
| Upmajor 2nd | | Upmajor 2nd | ||
| E | | ^E | ||
|- | |- | ||
| 9 | | 9 | ||
| Line 117: | Line 117: | ||
| -0.802 | | -0.802 | ||
| Downminor 3rd | | Downminor 3rd | ||
| | | vF | ||
|- | |- | ||
| 12 | | 12 | ||
| Line 133: | Line 133: | ||
| -0.741 | | -0.741 | ||
| Mid 3rd | | Mid 3rd | ||
| F | | ^F | ||
|- | |- | ||
| 14 | | 14 | ||
| Line 149: | Line 149: | ||
| 13.686 | | 13.686 | ||
| Upmajor 3rd | | Upmajor 3rd | ||
| F# | | ^F# | ||
|- | |- | ||
| 16 | | 16 | ||
| Line 156: | Line 156: | ||
| 435.084 | | 435.084 | ||
| -8.418 | | -8.418 | ||
| | | Augmented 3rd | ||
| | | Fx | ||
|- | |- | ||
| 17 | | 17 | ||
| Line 173: | Line 173: | ||
| 9.219 | | 9.219 | ||
| Down 4th | | Down 4th | ||
| | | vG | ||
|- | |- | ||
| 19 | | 19 | ||
| Line 188: | Line 188: | ||
| 533.742 | | 533.742 | ||
| -0.409 | | -0.409 | ||
| Up 4th | | Up 4th or Mid 4th | ||
| G | | ^G | ||
|- | |- | ||
| 21 | | 21 | ||
| Line 205: | Line 205: | ||
| 4.155 | | 4.155 | ||
| Upaugmented 4th | | Upaugmented 4th | ||
| G# | | ^G# | ||
|- | |- | ||
| 23 | | 23 | ||
| Line 213: | Line 213: | ||
| -4.155 | | -4.155 | ||
| Downdiminshed 5th | | Downdiminshed 5th | ||
| | | vAb | ||
|- | |- | ||
| 24 | | 24 | ||
| Line 228: | Line 228: | ||
| 666.258 | | 666.258 | ||
| 0.409 | | 0.409 | ||
| Down 5th | | Down 5th or Mid 5th | ||
| | | vA | ||
|- | |- | ||
| 26 | | 26 | ||
| Line 245: | Line 245: | ||
| -9.219 | | -9.219 | ||
| Up 5th | | Up 5th | ||
| A | | ^A | ||
|- | |- | ||
| 28 | | 28 | ||
| Line 260: | Line 260: | ||
| 764.916 | | 764.916 | ||
| 8.418 | | 8.418 | ||
| | | Diminished 6th | ||
| | | Bbb | ||
|- | |- | ||
| 30 | | 30 | ||
| Line 269: | Line 269: | ||
| -13.686 | | -13.686 | ||
| Downminor 6th | | Downminor 6th | ||
| | | vBb | ||
|- | |- | ||
| 31 | | 31 | ||
| Line 285: | Line 285: | ||
| 0.741 | | 0.741 | ||
| Mid 6th | | Mid 6th | ||
| | | vB | ||
|- | |- | ||
| 33 | | 33 | ||
| Line 301: | Line 301: | ||
| 0.802 | | 0.802 | ||
| Upmajor 6th | | Upmajor 6th | ||
| B | | ^B | ||
|- | |- | ||
| 35 | | 35 | ||
| Line 325: | Line 325: | ||
| -9.423 | | -9.423 | ||
| Downminor 7th | | Downminor 7th | ||
| | | vC | ||
|- | |- | ||
| 38 | | 38 | ||
| Line 340: | Line 340: | ||
| 1034.996 | | 1034.996 | ||
| 5.004 | | 5.004 | ||
| | | Mid 7th | ||
| C | | ^C | ||
|- | |- | ||
| 40 | | 40 | ||
| Line 357: | Line 357: | ||
| -1.711 | | -1.711 | ||
| Upmajor 7th | | Upmajor 7th | ||
| C# | | ^C# | ||
|- | |- | ||
| 42 | | 42 | ||
| Line 364: | Line 364: | ||
| 1119.463 | | 1119.463 | ||
| 0.537 | | 0.537 | ||
| | | Augmented 7th | ||
| | | Cx | ||
|- | |- | ||
| 43 | | 43 | ||
| Line 381: | Line 381: | ||
| 0.174 | | 0.174 | ||
| Down 8ve | | Down 8ve | ||
| | | vD | ||
|- | |- | ||
| 45 | | 45 | ||
Revision as of 08:05, 22 June 2022
| ← 44edo | 45edo | 46edo → |
45edo divides the octave into 45 equal parts of 26.667 cents. It has two major thirds, each almost equally far from Just, but as the flat one is slightly closer, it qualifies as a meantone temperament, forming a good approximation to 2/5 comma meantone. It is the optimal patent val for flattone temperament, the 7-limit 525/512 planar avicennmic temperament, the 11-limit calliope temperament tempering out 45/44 and 81/80, and the rank four temperament tempering out 45/44. It tempers out 81/80, 3125/3087, 525/512, 875/864 and 45/44. It is a flat-tending system in the 7-limit, with 3, 5 and 7 all flat, but the 11 is sharp. Also supports a very inaccurate version of ennealimmal, if you want to view it that way. Since 45 is a multiple of 5 and 9, it can be used to model Indonesian music in both Slendro (~ 5edo) and Pelog (~ modes of 9edo) tunings.
| Harmonic | 3 | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | -8.6 | -13.0 | -8.8 | +9.4 | +8.7 | +12.8 | +5.1 | +1.7 | -4.2 | +9.2 | +11.7 |
| Relative (%) | -32.3 | -48.7 | -33.1 | +35.3 | +32.6 | +48.0 | +19.0 | +6.4 | -15.7 | +34.6 | +44.0 | |
| Steps (reduced) |
71 (26) |
104 (14) |
126 (36) |
143 (8) |
156 (21) |
167 (32) |
176 (41) |
184 (4) |
191 (11) |
198 (18) |
204 (24) | |
Intervals
| Step # | ET | Just (JI) | Error (ET−JI) |
Ups and Downs Notation | |||
|---|---|---|---|---|---|---|---|
| Cents | Interval | Cents | |||||
| 0 | 0.000 | 1/1 | 0.000 | 0.000 | Perfect Unison | D | |
| 1 | 26.666 | 65/64 | 26.841 | -0.174 | Up unison | ^D | |
| 2 | 53.333 | 33/32 | 53.273 | 0.060 | Augmented Unison | D# | |
| 3 | 80.000 | 22/21 | 80.537 | -0.537 | Diminished 2nd | Ebb | |
| 4 | 106.666 | 17/16 | 104.955 | 1.711 | Downminor 2nd | vEb | |
| 5 | 133.333 | 27/25 | 133.238 | 0.095 | Minor 2nd | Eb | |
| 6 | 160.000 | 11/10 | 165.004 | -5.004 | Mid 2nd | vE | |
| 7 | 186.666 | 10/9 | 182.404 | 4.262 | Major 2nd | E | |
| 8 | 213.333 | 9/8 | 203.910 | 9.423 | Upmajor 2nd | ^E | |
| 9 | 240.000 | 8/7 | 231.174 | 8.826 | Augmented 2nd | E# | |
| 10 | 266.666 | 7/6 | 266.871 | -0.205 | Diminished 3rd | Fb | |
| 11 | 293.333 | 32/27 | 294.135 | -0.802 | Downminor 3rd | vF | |
| 12 | 320.000 | 6/5 | 315.641 | 4.359 | Minor 3rd | F | |
| 13 | 346.666 | 11/9 | 347.408 | -0.741 | Mid 3rd | ^F | |
| 14 | 373.333 | 5/4- | 386.314 | -12.980 | Major 3rd | F# | |
| 15 | 400.000 | 5/4+ | 386.314 | 13.686 | Upmajor 3rd | ^F# | |
| 16 | 426.666 | 9/7 | 435.084 | -8.418 | Augmented 3rd | Fx | |
| 17 | 453.333 | 13/10 | 454.294 | -0.961 | Diminished 4th | Gb | |
| 18 | 480.000 | 21/16 | 470.781 | 9.219 | Down 4th | vG | |
| 19 | 506.666 | 4/3 | 498.045 | 8.622 | Perfect 4th | G | |
| 20 | 533.333 | 49/36 | 533.742 | -0.409 | Up 4th or Mid 4th | ^G | |
| 21 | 560.000 | 18/13 | 563.382 | -3.382 | Augmented 4th | G# | |
| 22 | 586.666 | 7/5 | 582.512 | 4.155 | Upaugmented 4th | ^G# | |
| 23 | 613.333 | 10/7 | 617.488 | -4.155 | Downdiminshed 5th | vAb | |
| 24 | 640.000 | 13/9 | 636.618 | 3.382 | Diminished 5th | Ab | |
| 25 | 666.666 | 72/49 | 666.258 | 0.409 | Down 5th or Mid 5th | vA | |
| 26 | 693.333 | 3/2 | 701.955 | -8.622 | Perfect 5th | A | |
| 27 | 720.000 | 32/21 | 729.219 | -9.219 | Up 5th | ^A | |
| 28 | 746.666 | 20/13 | 745.786 | 0.961 | Augmented 5th | A# | |
| 29 | 773.333 | 14/9 | 764.916 | 8.418 | Diminished 6th | Bbb | |
| 30 | 800.000 | 8/5- | 813.686 | -13.686 | Downminor 6th | vBb | |
| 31 | 826.666 | 8/5+ | 813.686 | 12.980 | Minor 6th | Bb | |
| 32 | 853.333 | 18/11 | 852.592 | 0.741 | Mid 6th | vB | |
| 33 | 880.000 | 5/3 | 884.359 | -4.359 | Major 6th | B | |
| 34 | 906.666 | 27/16 | 905.865 | 0.802 | Upmajor 6th | ^B | |
| 35 | 933.333 | 12/7 | 933.129 | 0.205 | Augmented 6th | B# | |
| 36 | 960.000 | 7/4 | 968.826 | -8.826 | Diminished 7th | Cb | |
| 37 | 986.666 | 16/9 | 996.089 | -9.423 | Downminor 7th | vC | |
| 38 | 1013.333 | 9/5 | 1017.596 | -4.262 | Minor 7th | C | |
| 39 | 1040.000 | 20/11 | 1034.996 | 5.004 | Mid 7th | ^C | |
| 40 | 1066.666 | 50/27 | 1066.762 | -0.095 | Major 7th | C# | |
| 41 | 1093.333 | 32/17 | 1095.044 | -1.711 | Upmajor 7th | ^C# | |
| 42 | 1120.000 | 21/11 | 1119.463 | 0.537 | Augmented 7th | Cx | |
| 43 | 1146.666 | 64/33 | 1146.727 | -0.060 | Diminished 8ve | Db | |
| 44 | 1173.333 | 128/65 | 1173.158 | 0.174 | Down 8ve | vD | |
| 45 | 1200.000 | 2/1 | 1200.000 | 0.000 | Perfect Octave | D | |