50edo
| ← 49edo | 50edo | 51edo → |
50edo divides the octave into 50 equal parts of precisely 24 cents each.
Theory
In the 5-limit, 50edo tempers out 81/80, making it a meantone system, and in that capacity has historically has drawn some notice. In "Harmonics or the Philosophy of Musical Sounds" (1759) by Robert Smith, a musical temperament is described where the octave is divided into 50 equal parts – 50edo, in one word. Later, W.S.B. Woolhouse noted it was fairly close to the least squares tuning for 5-limit meantone. 50edo, however, is especially interesting from a higher limit point of view. While 31edo extends meantone with a 7/4 which is nearly pure, 50 has a flat 7/4 but both 11/8 and 13/8 are nearly pure. It is the highest edo which maps 9/8 and 10/9 to the same interval in a consistent manner, with two stacked fifths falling almost precisely in the middle of the two.
50edo tempers out 126/125, 225/224 and 3136/3125 in the 7-limit, indicating it supports septimal meantone; 245/242, 385/384 and 540/539 in the 11-limit and 105/104, 144/143 and 196/195 in the 13-limit, and can be used for even higher limits. Aside from meantone and its extension meanpop, it can be used to advantage for the coblack (15&50) temperament, and provides the optimal patent val for 11 and 13 limit bimeantone. It is also the unique equal temperament tempering out both 81/80 and the vishnuzma, [23 6 -14⟩;, so that in 50et seven chromatic semitones are a perfect fourth. In 12et by comparison this gives a fifth, in 31et a doubly diminished fifth, and in 19et a diminished fourth.
Relations
The 50edo system is related to 7edo, 12edo, 19edo, 31edo as the next approximation to the "Golden Tone System" (Das Goldene Tonsystem) of Thorvald Kornerup (and similarly as the next step from 31edo in Joseph Yasser's "A Theory of Evolving Tonality").
Intervals
| # | Cents | Ratios* | Ups and Downs Notation | Generator for* | |||
|---|---|---|---|---|---|---|---|
| 0 | 0 | 1/1 | Perfect 1sn | P1 | D | ||
| 1 | 24 | 45/44, 49/48, 56/55, 65/64, 66/65, 78/77, 91/90, 99/98, 100/99, 121/120, 169/168 | Up 1sn, Dim 2nd | ^1, d2 | ^D, Ebb | Sengagen | |
| 2 | 48 | 33/32, 36/35, 50/49, 55/54, 64/63 | Downaug 1sn, Updim 2nd | vA1, ^d2 | vD#, ^Ebb | ||
| 3 | 72 | 21/20, 25/24, 26/25, 27/26, 28/27 | Aug 1sn | A1 | D# | Vishnu (2/oct), Coblack (5/oct) | |
| 4 | 96 | 22/21 | Downminor 2nd | vm2 | vEb | Injera (50d val, 2/oct) | |
| 5 | 120 | 16/15, 15/14, 14/13 | Minor 2nd | m2 | Eb | ||
| 6 | 144 | 13/12, 12/11 | Upminor 2nd | ^m2 | ^Eb | ||
| 7 | 168 | 11/10 | Downmajor 2nd | vM2 | vE | ||
| 8 | 192 | 9/8, 10/9 | Major 2nd | M2 | E | ||
| 9 | 216 | 25/22 | Upmajor 2nd | ^M2 | ^E | Tremka, Machine (50b val) | |
| 10 | 240 | 8/7, 15/13 | Downaug 2nd, Dim 3rd | vA2, d3 | vE#, Fb | ||
| 11 | 264 | 7/6 | Updim 3rd, Aug 2nd | ^d3, A2 | ^Fb, E# | Septimin (13-limit) | |
| 12 | 288 | 13/11 | Downminor 3rd | vm3 | vF | ||
| 13 | 312 | 6/5 | Minor 3rd | m3 | F | Oolong | |
| 14 | 336 | 27/22, 39/32, 40/33, 49/40 | Upminor 3rd | ^m3 | ^F | ||
| 15 | 360 | 16/13, 11/9 | Downmajor 3rd | vM3 | vF# | ||
| 16 | 384 | 5/4 | Major 3rd | M3 | F# | Wizard (2/oct) | |
| 17 | 408 | 14/11 | Upmajor 3rd | ^M3 | ^F# | Ditonic | |
| 18 | 432 | 9/7 | Downaug 3rd, Dim 4th | vA3, d4 | vFx, Gb | Hedgehog (50cc val, 2/oct) | |
| 19 | 456 | 13/10 | Updim 4th, Aug 3rd | A3, ^d4 | ^Gb, Fx | Bisemidim (2/oct) | |
| 20 | 480 | 33/25, 55/42, 64/49 | Down 4th | v4 | vG | ||
| 21 | 504 | 4/3 | Perfect 4th | P4 | G | Meantone/Meanpop | |
| 22 | 528 | 15/11 | Up 4th | ^4 | ^G | ||
| 23 | 552 | 11/8, 18/13 | Downaug 4th | vA4 | vG# | Barton, Emka | |
| 24 | 576 | 7/5 | Aug 4th | A4 | G# | ||
| 25 | 600 | 63/44, 88/63, 78/55, 55/39 | Upaug 4th, Downdim 5th | ^A4, vd5 | ^G#, vAb | ||
| 26 | 624 | 10/7 | Dim 5th | d5 | Ab | ||
| 27 | 648 | 16/11, 13/9 | Updim 5th | ^d5 | ^Ab | ||
| 28 | 672 | 22/15 | Down 5th | v5 | vA | ||
| 29 | 696 | 3/2 | Perfect 5th | P5 | A | ||
| 30 | 720 | 50/33, 84/55, 49/32 | Up 5th | ^5 | ^A | ||
| 31 | 744 | 20/13 | Downaug 5th, Dim 6th | vA5, d6 | vA#, Bbb | ||
| 32 | 768 | 14/9 | Updim 6th, Aug 5th | ^d6, A5 | ^Bbb, A# | ||
| 33 | 792 | 11/7 | Downminor 6th | vm6 | vBb | ||
| 34 | 816 | 8/5 | Minor 6th | m6 | Bb | ||
| 35 | 840 | 13/8, 18/11 | Upminor 6th | ^m6 | ^Bb | ||
| 36 | 864 | 44/27, 64/39, 33/20, 80/49 | Downmajor 6th | vM6 | vB | ||
| 37 | 888 | 5/3 | Major 6th | M6 | B | ||
| 38 | 912 | 22/13 | Upmajor 6th | ^M6 | ^B | ||
| 39 | 936 | 12/7 | Downaug 6th, Dim 7th | vA6, d7 | vB#, Cb | ||
| 40 | 960 | 7/4 | Updim 7th, Aug 6th | ^d7, A6 | ^Cb, B# | ||
| 41 | 984 | 44/25 | Downminor 7th | vm7 | vC | ||
| 42 | 1008 | 16/9, 9/5 | Minor 7th | m7 | C | ||
| 43 | 1032 | 20/11 | Upminor 7th | ^m7 | ^C | ||
| 44 | 1056 | 24/13, 11/6 | Downmajor 7th | vM7 | vC# | ||
| 45 | 1080 | 15/8, 28/15, 13/7 | Major 7th | M7 | C# | ||
| 46 | 1104 | 21/11 | Upmajor 7th | ^M7 | ^C# | ||
| 47 | 1128 | 40/21, 48/25, 25/13, 52/27, 27/14 | Downaug 7th, Dim 8ve | vA7, d8 | vCx, Db | ||
| 48 | 1152 | 64/33, 35/18, 49/25, 108/55, 63/32 | Updim 8ve, Aug 7th | ^d8, A7 | ^Db, Cx | ||
| 49 | 1176 | 88/45, 96/49, 55/28, 128/65, 65/33, 77/39, 180/91, 196/99, 99/50, 240/121, 336/169 | Down 8ve | v8 | vD | ||
| 50 | 1200 | 2/1 | Perfect 8ve | P8 | D | ||
* Using the 13-limit patent val, except as noted.
Just approximation
Selected just intervals
Script error: No such module "primes_in_edo".
15-odd-limit mappings
The following table shows how 15-odd-limit intervals are represented in 50edo (ordered by absolute error). Prime harmonics are in bold; inconsistent intervals are in italic.
| Interval, complement | Error (abs, ¢) |
|---|---|
| 16/13, 13/8 | 0.528 |
| 15/14, 28/15 | 0.557 |
| 11/8, 16/11 | 0.682 |
| 13/11, 22/13 | 1.210 |
| 13/10, 20/13 | 1.786 |
| 5/4, 8/5 | 2.314 |
| 7/6, 12/7 | 2.871 |
| 11/10, 20/11 | 2.996 |
| 9/7, 14/9 | 3.084 |
| 6/5, 5/3 | 3.641 |
| 13/12, 24/13 | 5.427 |
| 4/3, 3/2 | 5.955 |
| 7/5, 10/7 | 6.512 |
| 12/11, 11/6 | 6.637 |
| 15/13, 26/15 | 7.741 |
| 16/15, 15/8 | 8.269 |
| 14/13, 13/7 | 8.298 |
| 8/7, 7/4 | 8.826 |
| 15/11, 22/15 | 8.951 |
| 14/11, 11/7 | 9.508 |
| 10/9, 9/5 | 9.596 |
| 18/13, 13/9 | 11.382 |
| 11/9, 18/11 | 11.408 |
| 9/8, 16/9 | 11.910 |
| Interval, complement | Error (abs, ¢) |
|---|---|
| 16/13, 13/8 | 0.528 |
| 15/14, 28/15 | 0.557 |
| 11/8, 16/11 | 0.682 |
| 13/11, 22/13 | 1.210 |
| 13/10, 20/13 | 1.786 |
| 5/4, 8/5 | 2.314 |
| 7/6, 12/7 | 2.871 |
| 11/10, 20/11 | 2.996 |
| 9/7, 14/9 | 3.084 |
| 6/5, 5/3 | 3.641 |
| 13/12, 24/13 | 5.427 |
| 4/3, 3/2 | 5.955 |
| 7/5, 10/7 | 6.512 |
| 12/11, 11/6 | 6.637 |
| 15/13, 26/15 | 7.741 |
| 16/15, 15/8 | 8.269 |
| 14/13, 13/7 | 8.298 |
| 8/7, 7/4 | 8.826 |
| 15/11, 22/15 | 8.951 |
| 14/11, 11/7 | 9.508 |
| 10/9, 9/5 | 9.596 |
| 18/13, 13/9 | 11.382 |
| 9/8, 16/9 | 11.910 |
| 11/9, 18/11 | 12.592 |
Temperament measures
The following table shows TE temperament measures (RMS normalized by the rank) of 50et.
| 3-limit | 5-limit | 7-limit | 11-limit | 13-limit | ||
|---|---|---|---|---|---|---|
| Octave stretch (¢) | +1.88 | +1.58 | +1.98 | +1.54 | +1.31 | |
| Error | absolute (¢) | 1.88 | 1.59 | 1.54 | 1.63 | 1.57 |
| relative (%) | 7.83 | 6.62 | 6.39 | 6.76 | 6.54 | |
Commas
50 EDO tempers out the following commas. (Note: This assumes the val ⟨50 79 116 140 173 185 204 212 226], comma values in cents rounded to 2 decimal places.) This list is not all-inclusive, and is based on the interval table from Scala version 2.2.
| Prime Limit |
Ratio[1] | Monzo | Cents | Name(s) |
|---|---|---|---|---|
| 5 | 81/80 | [-4 4 -1⟩ | 21.51 | Syntonic comma, Didymus comma |
| 5 | (20 digits) | [-27 -2 13⟩ | 18.17 | Ditonma |
| 5 | (20 digits) | [23 6 -14⟩ | 3.34 | Vishnuzma, Vishnu comma |
| 7 | 59049/57344 | [-13 10 0 -1⟩ | 50.72 | Harrison's comma |
| 7 | 126/125 | [1 2 -3 1⟩ | 13.79 | Starling comma, Small septimal comma |
| 7 | 225/224 | [-5 2 2 -1⟩ | 7.71 | Septimal kleisma, Marvel comma |
| 7 | 3136/3125 | [6 0 -5 2⟩ | 6.08 | Hemimean, Middle second comma |
| 7 | (24 digits) | [11 -10 -10 10⟩ | 5.57 | Linus |
| 7 | (12 digits) | [-11 2 7 -3⟩ | 1.63 | Meter |
| 7 | (12 digits) | [-6 -8 2 5⟩ | 1.12 | Wizma |
| 11 | 245/242 | [-1 0 1 2 -2⟩ | 21.33 | Cassacot |
| 11 | 385/384 | [-7 -1 1 1 1⟩ | 4.50 | Keenanisma, undecimal kleisma |
| 11 | 540/539 | [2 3 1 -2 -1⟩ | 3.21 | Swets' comma, Swetisma |
| 11 | 4000/3993 | [5 -1 3 0 -3⟩ | 3.03 | Wizardharry, undecimal schisma |
| 11 | 9801/9800 | [-3 4 -2 -2 2⟩ | 0.18 | Kalisma, Gauss' comma |
| 13 | 105/104 | [-3 1 1 1 0 -1⟩ | 16.57 | Animist comma, small tridecimal comma |
| 13 | 144/143 | [4 2 0 0 -1 -1⟩ | 12.06 | Grossma |
| 13 | 196/195 | [2 -1 -1 2 0 -1⟩ | 8.86 | Mynucuma |
| 13 | 1188/1183 | [2 3 0 -1 1 -2⟩ | 7.30 | Kestrel Comma |
| 13 | 364/363 | [2 -1 0 1 -2 1⟩ | 4.76 | Gentle comma |
| 13 | 2200/2197 | [3 0 2 0 1 -3⟩ | 2.36 | Petrma, Parizek comma |
| 17 | 170/169 | [1 0 1 0 0 -2 1⟩ | 10.21 | |
| 17 | 221/220 | [-2 0 -1 0 -1 1 1⟩ | 7.85 | |
| 17 | 289/288 | [-5 -2 0 0 0 0 2⟩ | 6.00 | minor seconds comma |
| 17 | 375/374 | [-1 1 3 0 -1 0 -1⟩ | 4.62 | |
| 19 | 153/152 | [-3 2 0 0 0 0 1 -1⟩ | 11.35 | ganassisma |
| 19 | 171/170 | [-1 2 -1 0 0 0 -1 1⟩ | 10.15 | |
| 19 | 210/209 | [1 1 1 1 -1 0 0 1⟩ | 8.26 | |
| 19 | 324/323 | [2 4 0 0 0 0 -1 -1⟩ | 5.35 | |
| 19 | 361/360 | [-3 -2 -1 0 0 0 0 2⟩ | 4.80 | |
| 19 | 495/494 | [-1 2 1 0 1 -1 0 -1⟩ | 3.50 | |
| 23 | 1288/1287 | [3 -2 0 1 -1 -1 0 0 1⟩ | 1.34 | Triaphonisma |
- ↑ Ratios longer than 10 digits are presented by placeholders with informative hints
Music
- Twinkle canon – 50 edo by Claudi Meneghin
- Fantasia Catalana by Claudi Meneghin
- Fugue on the Dragnet theme by Claudi Meneghin
- the late little xmas album by Cam Taylor
- Harpsichord meantone improvisation 1 in 50EDO by Cam Taylor
- Long improvisation 2 in 50EDO by Cam Taylor
- Chord sequence for Difference tones in 50EDO by Cam Taylor
- Enharmonic Modulations in 50EDO by Cam Taylor
- Harmonic Clusters on 50EDO Harpsichord by Cam Taylor
- Fragment in Fifty by Cam Taylor