45edo: Difference between revisions
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| Line 28: | Line 28: | ||
| 0.000 | | 0.000 | ||
| Perfect Unison | | Perfect Unison | ||
|P1 | | P1 | ||
| D | | D | ||
|- | |- | ||
| Line 37: | Line 37: | ||
| -0.174 | | -0.174 | ||
| Up unison | | Up unison | ||
|^1 | | ^1 | ||
| ^D | | ^D | ||
|- | |- | ||
| Line 46: | Line 46: | ||
| 0.060 | | 0.060 | ||
| Augmented Unison | | Augmented Unison | ||
|A1 | | A1 | ||
| D# | | D# | ||
|- | |- | ||
| Line 55: | Line 55: | ||
| -0.537 | | -0.537 | ||
| Diminished 2nd | | Diminished 2nd | ||
|d2 | | d2 | ||
| Ebb | | Ebb | ||
|- | |- | ||
| Line 73: | Line 73: | ||
| 0.095 | | 0.095 | ||
| Minor 2nd | | Minor 2nd | ||
|m2 | | m2 | ||
| Eb | | Eb | ||
|- | |- | ||
| Line 82: | Line 82: | ||
| -5.004 | | -5.004 | ||
| Mid 2nd | | Mid 2nd | ||
|~2 | | ~2 | ||
| vE | | vE | ||
|- | |- | ||
| Line 91: | Line 91: | ||
| 4.262 | | 4.262 | ||
| Major 2nd | | Major 2nd | ||
|M2 | | M2 | ||
| E | | E | ||
|- | |- | ||
| Line 100: | Line 100: | ||
| 9.423 | | 9.423 | ||
| Upmajor 2nd | | Upmajor 2nd | ||
|^M2 | | ^M2 | ||
| ^E | | ^E | ||
|- | |- | ||
| Line 109: | Line 109: | ||
| 8.826 | | 8.826 | ||
| Augmented 2nd | | Augmented 2nd | ||
|A2 | | A2 | ||
| E# | | E# | ||
|- | |- | ||
| Line 118: | Line 118: | ||
| -0.205 | | -0.205 | ||
| Diminished 3rd | | Diminished 3rd | ||
|d3 | | d3 | ||
| Fb | | Fb | ||
|- | |- | ||
| Line 127: | Line 127: | ||
| -0.802 | | -0.802 | ||
| Downminor 3rd | | Downminor 3rd | ||
|vm3 | | vm3 | ||
| vF | | vF | ||
|- | |- | ||
| Line 136: | Line 136: | ||
| 4.359 | | 4.359 | ||
| Minor 3rd | | Minor 3rd | ||
|m3 | | m3 | ||
| F | | F | ||
|- | |- | ||
| Line 145: | Line 145: | ||
| -0.741 | | -0.741 | ||
| Mid 3rd | | Mid 3rd | ||
|~3 | | ~3 | ||
| ^F | | ^F | ||
|- | |- | ||
| Line 154: | Line 154: | ||
| -12.980 | | -12.980 | ||
| Major 3rd | | Major 3rd | ||
|M3 | | M3 | ||
| F# | | F# | ||
|- | |- | ||
| Line 163: | Line 163: | ||
| -0.108 | | -0.108 | ||
| Upmajor 3rd | | Upmajor 3rd | ||
|^M3 | | ^M3 | ||
| ^F# | | ^F# | ||
|- | |- | ||
| Line 172: | Line 172: | ||
| -8.418 | | -8.418 | ||
| Augmented 3rd | | Augmented 3rd | ||
|A3 | | A3 | ||
| Fx | | Fx | ||
|- | |- | ||
| Line 181: | Line 181: | ||
| -0.961 | | -0.961 | ||
| Diminished 4th | | Diminished 4th | ||
|d4 | | d4 | ||
| Gb | | Gb | ||
|- | |- | ||
| Line 190: | Line 190: | ||
| 9.219 | | 9.219 | ||
| Down 4th | | Down 4th | ||
|v4 | | v4 | ||
| vG | | vG | ||
|- | |- | ||
| Line 199: | Line 199: | ||
| 8.622 | | 8.622 | ||
| Perfect 4th | | Perfect 4th | ||
|P4 | | P4 | ||
| G | | G | ||
|- | |- | ||
| Line 208: | Line 208: | ||
| -0.409 | | -0.409 | ||
| Up 4th or Mid 4th | | Up 4th or Mid 4th | ||
|^4, ~4 | | ^4, ~4 | ||
| ^G | | ^G | ||
|- | |- | ||
| Line 217: | Line 217: | ||
| -3.382 | | -3.382 | ||
| Augmented 4th | | Augmented 4th | ||
|A4 | | A4 | ||
| G# | | G# | ||
|- | |- | ||
| Line 226: | Line 226: | ||
| 4.155 | | 4.155 | ||
| Upaugmented 4th | | Upaugmented 4th | ||
|^A4 | | ^A4 | ||
| ^G# | | ^G# | ||
|- | |- | ||
| Line 235: | Line 235: | ||
| -4.155 | | -4.155 | ||
| Downdiminshed 5th | | Downdiminshed 5th | ||
|vd5 | | vd5 | ||
| vAb | | vAb | ||
|- | |- | ||
| Line 244: | Line 244: | ||
| 3.382 | | 3.382 | ||
| Diminished 5th | | Diminished 5th | ||
|d5 | | d5 | ||
| Ab | | Ab | ||
|- | |- | ||
| Line 253: | Line 253: | ||
| 0.409 | | 0.409 | ||
| Down 5th or Mid 5th | | Down 5th or Mid 5th | ||
|v5, ~5 | | v5, ~5 | ||
| vA | | vA | ||
|- | |- | ||
| Line 262: | Line 262: | ||
| -8.622 | | -8.622 | ||
| Perfect 5th | | Perfect 5th | ||
|P5 | | P5 | ||
| A | | A | ||
|- | |- | ||
| Line 271: | Line 271: | ||
| -9.219 | | -9.219 | ||
| Up 5th | | Up 5th | ||
|^5 | | ^5 | ||
| ^A | | ^A | ||
|- | |- | ||
| Line 280: | Line 280: | ||
| 0.961 | | 0.961 | ||
| Augmented 5th | | Augmented 5th | ||
|A5 | | A5 | ||
| A# | | A# | ||
|- | |- | ||
| Line 289: | Line 289: | ||
| 8.418 | | 8.418 | ||
| Diminished 6th | | Diminished 6th | ||
|d6 | | d6 | ||
| Bbb | | Bbb | ||
|- | |- | ||
| Line 298: | Line 298: | ||
| 0.108 | | 0.108 | ||
| Downminor 6th | | Downminor 6th | ||
|vm6 | | vm6 | ||
| vBb | | vBb | ||
|- | |- | ||
| Line 307: | Line 307: | ||
| 12.980 | | 12.980 | ||
| Minor 6th | | Minor 6th | ||
|m6 | | m6 | ||
| Bb | | Bb | ||
|- | |- | ||
| Line 316: | Line 316: | ||
| 0.741 | | 0.741 | ||
| Mid 6th | | Mid 6th | ||
|~6 | | ~6 | ||
| vB | | vB | ||
|- | |- | ||
| Line 325: | Line 325: | ||
| -4.359 | | -4.359 | ||
| Major 6th | | Major 6th | ||
|M6 | | M6 | ||
| B | | B | ||
|- | |- | ||
| Line 334: | Line 334: | ||
| 0.802 | | 0.802 | ||
| Upmajor 6th | | Upmajor 6th | ||
|^M6 | | ^M6 | ||
| ^B | | ^B | ||
|- | |- | ||
| Line 343: | Line 343: | ||
| 0.205 | | 0.205 | ||
| Augmented 6th | | Augmented 6th | ||
|A6 | | A6 | ||
| B# | | B# | ||
|- | |- | ||
| Line 352: | Line 352: | ||
| -8.826 | | -8.826 | ||
| Diminished 7th | | Diminished 7th | ||
|d7 | | d7 | ||
| Cb | | Cb | ||
|- | |- | ||
| Line 361: | Line 361: | ||
| -9.423 | | -9.423 | ||
| Downminor 7th | | Downminor 7th | ||
|vm7 | | vm7 | ||
| vC | | vC | ||
|- | |- | ||
| Line 370: | Line 370: | ||
| -4.262 | | -4.262 | ||
| Minor 7th | | Minor 7th | ||
|m7 | | m7 | ||
| C | | C | ||
|- | |- | ||
| Line 379: | Line 379: | ||
| 5.004 | | 5.004 | ||
| Mid 7th | | Mid 7th | ||
|~7 | | ~7 | ||
| ^C | | ^C | ||
|- | |- | ||
| Line 388: | Line 388: | ||
| -0.095 | | -0.095 | ||
| Major 7th | | Major 7th | ||
|M7 | | M7 | ||
| C# | | C# | ||
|- | |- | ||
| Line 397: | Line 397: | ||
| -1.711 | | -1.711 | ||
| Upmajor 7th | | Upmajor 7th | ||
|^M7 | | ^M7 | ||
| ^C# | | ^C# | ||
|- | |- | ||
| Line 406: | Line 406: | ||
| 0.537 | | 0.537 | ||
| Augmented 7th | | Augmented 7th | ||
|A7 | | A7 | ||
| Cx | | Cx | ||
|- | |- | ||
| Line 415: | Line 415: | ||
| -0.060 | | -0.060 | ||
| Diminished 8ve | | Diminished 8ve | ||
|d8 | | d8 | ||
| Db | | Db | ||
|- | |- | ||
| Line 424: | Line 424: | ||
| 0.174 | | 0.174 | ||
| Down 8ve | | Down 8ve | ||
|v8 | | v8 | ||
| vD | | vD | ||
|- | |- | ||
| Line 433: | Line 433: | ||
| 0.000 | | 0.000 | ||
| Perfect Octave | | Perfect Octave | ||
|P8 | | P8 | ||
| D | | D | ||
|} | |} | ||
Revision as of 17:05, 29 May 2024
| ← 44edo | 45edo | 46edo → |
Theory
45edo effectively 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.
45edo tempers out the quartisma and provides an excellent tuning for the 2.33/32.7/6 subgroup direct quartismic temperament, in which it approximates 33/32 quartertone with 2 steps and 7/6 with 10 steps. It is also the unique equal temperament tuning that tempers out both the syntonic comma and the ennealimma.
Odd harmonics
| 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 | P1 | D | |
| 1 | 26.666 | 65/64 | 26.841 | -0.174 | Up unison | ^1 | ^D | |
| 2 | 53.333 | 33/32 | 53.273 | 0.060 | Augmented Unison | A1 | D# | |
| 3 | 80.000 | 22/21 | 80.537 | -0.537 | Diminished 2nd | d2 | Ebb | |
| 4 | 106.666 | 17/16 | 104.955 | 1.711 | Downminor 2nd | vm2 | vEb | |
| 5 | 133.333 | 27/25 | 133.238 | 0.095 | Minor 2nd | m2 | Eb | |
| 6 | 160.000 | 11/10 | 165.004 | -5.004 | Mid 2nd | ~2 | vE | |
| 7 | 186.666 | 10/9 | 182.404 | 4.262 | Major 2nd | M2 | E | |
| 8 | 213.333 | 9/8 | 203.910 | 9.423 | Upmajor 2nd | ^M2 | ^E | |
| 9 | 240.000 | 8/7 | 231.174 | 8.826 | Augmented 2nd | A2 | E# | |
| 10 | 266.666 | 7/6 | 266.871 | -0.205 | Diminished 3rd | d3 | Fb | |
| 11 | 293.333 | 32/27 | 294.135 | -0.802 | Downminor 3rd | vm3 | vF | |
| 12 | 320.000 | 6/5 | 315.641 | 4.359 | Minor 3rd | m3 | F | |
| 13 | 346.666 | 11/9 | 347.408 | -0.741 | Mid 3rd | ~3 | ^F | |
| 14 | 373.333 | 5/4 | 386.314 | -12.980 | Major 3rd | M3 | F# | |
| 15 | 400.000 | 63/50 | 400.108 | -0.108 | Upmajor 3rd | ^M3 | ^F# | |
| 16 | 426.666 | 9/7 | 435.084 | -8.418 | Augmented 3rd | A3 | Fx | |
| 17 | 453.333 | 13/10 | 454.294 | -0.961 | Diminished 4th | d4 | Gb | |
| 18 | 480.000 | 21/16 | 470.781 | 9.219 | Down 4th | v4 | vG | |
| 19 | 506.666 | 4/3 | 498.045 | 8.622 | Perfect 4th | P4 | G | |
| 20 | 533.333 | 49/36 | 533.742 | -0.409 | Up 4th or Mid 4th | ^4, ~4 | ^G | |
| 21 | 560.000 | 18/13 | 563.382 | -3.382 | Augmented 4th | A4 | G# | |
| 22 | 586.666 | 7/5 | 582.512 | 4.155 | Upaugmented 4th | ^A4 | ^G# | |
| 23 | 613.333 | 10/7 | 617.488 | -4.155 | Downdiminshed 5th | vd5 | vAb | |
| 24 | 640.000 | 13/9 | 636.618 | 3.382 | Diminished 5th | d5 | Ab | |
| 25 | 666.666 | 72/49 | 666.258 | 0.409 | Down 5th or Mid 5th | v5, ~5 | vA | |
| 26 | 693.333 | 3/2 | 701.955 | -8.622 | Perfect 5th | P5 | A | |
| 27 | 720.000 | 32/21 | 729.219 | -9.219 | Up 5th | ^5 | ^A | |
| 28 | 746.666 | 20/13 | 745.786 | 0.961 | Augmented 5th | A5 | A# | |
| 29 | 773.333 | 14/9 | 764.916 | 8.418 | Diminished 6th | d6 | Bbb | |
| 30 | 800.000 | 100/63 | 799.892 | 0.108 | Downminor 6th | vm6 | vBb | |
| 31 | 826.666 | 8/5 | 813.686 | 12.980 | Minor 6th | m6 | Bb | |
| 32 | 853.333 | 18/11 | 852.592 | 0.741 | Mid 6th | ~6 | vB | |
| 33 | 880.000 | 5/3 | 884.359 | -4.359 | Major 6th | M6 | B | |
| 34 | 906.666 | 27/16 | 905.865 | 0.802 | Upmajor 6th | ^M6 | ^B | |
| 35 | 933.333 | 12/7 | 933.129 | 0.205 | Augmented 6th | A6 | B# | |
| 36 | 960.000 | 7/4 | 968.826 | -8.826 | Diminished 7th | d7 | Cb | |
| 37 | 986.666 | 16/9 | 996.089 | -9.423 | Downminor 7th | vm7 | vC | |
| 38 | 1013.333 | 9/5 | 1017.596 | -4.262 | Minor 7th | m7 | C | |
| 39 | 1040.000 | 20/11 | 1034.996 | 5.004 | Mid 7th | ~7 | ^C | |
| 40 | 1066.666 | 50/27 | 1066.762 | -0.095 | Major 7th | M7 | C# | |
| 41 | 1093.333 | 32/17 | 1095.044 | -1.711 | Upmajor 7th | ^M7 | ^C# | |
| 42 | 1120.000 | 21/11 | 1119.463 | 0.537 | Augmented 7th | A7 | Cx | |
| 43 | 1146.666 | 64/33 | 1146.727 | -0.060 | Diminished 8ve | d8 | Db | |
| 44 | 1173.333 | 128/65 | 1173.158 | 0.174 | Down 8ve | v8 | vD | |
| 45 | 1200.000 | 2/1 | 1200.000 | 0.000 | Perfect Octave | P8 | D | |
Regular temperament properties
Commas
This is a partial list of the commas that 45edo tempers out with its patent val, ⟨45 71 104 126 143 156 167].
| Prime Limit |
Ratio[1] | Monzo | Cents | Color name | Name(s) |
|---|---|---|---|---|---|
| 5 | 81/80 | [-4 4 -1⟩ | 21.51 | Gu | Syntonic comma, Didymus comma, meantone comma |
| 5 | (26 digits) | [1 -27 18⟩ | 0.86 | Satritribiyo | Ennealimma |
| 7 | 16807/16384 | [-14 0 0 5⟩ | 44.13 | Laquinzo | Cloudy comma |
| 7 | 525/512 | [-9 1 2 1⟩ | 43.41 | Lazoyoyo | Avicennma, Avicenna's enharmonic diesis |
| 7 | 875/864 | [-5 -3 3 1⟩ | 21.90 | Zotrigu | Keema |
| 7 | 3125/3087 | [0 -2 5 -3⟩ | 21.18 | Triru-aquinyo | Gariboh comma |
| 7 | (16 digits) | [-11 -9 0 9⟩ | 1.84 | Tritrizo | Septimal ennealimma |
| 7 | 4375/4374 | [-1 -7 4 1⟩ | 0.40 | Zoquadyo | Ragisma |
| 11 | 45/44 | [-2 2 1 0 -1⟩ | 38.91 | Luyo | Undecimal 1/5-tone |
| 11 | 385/384 | [-7 -1 1 1 1⟩ | 4.50 | Lozoyo | Keenanisma |
| 11 | (18 digits) | [24 -6 0 1 -5⟩ | 0.51 | Saquinlu-azo | Quartisma |
Instruments
Lumatone
See Lumatone mapping for 45edo
Music
- Fishbowl (2023)
- Archipelago Arpeggio (2024)
- ↑ Ratios longer than 10 digits are presented by placeholders with informative hints