EDO

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An equal division of the octave (EDO or edo) is a tuning obtained by dividing the octave in a certain number of equal steps. This means that the interval between any two consecutive pitches is identical.

A tuning with n equal divisions of the octave is usually called "n-edo" ("n-EDO"). For instance, the predominant tuning system in the world today is 12edo (12-EDO).

An equal (pitch) division of the octave is the most common type of equal (pitch) division, which is a kind of equal-step tuning. Therefore, it is also a kind of arithmetic and harmonotonic tuning.

History

Tuning theorists first used the term "equal temperament" for edos designed to approximate low-complexity just intervals. The same term is still used today for all rank-1 temperaments. For example, 15edo can be referred to as 15-tone equal temperament (15-TET, 15-tET, 15tet, etc.), or more simply 15 equal temperament (15-ET, 15et, etc.).

The acronym "EDO" (EE-dee-oh) was coined by Daniel Anthony Stearns in 1999, originally standing for "equidistant divisions of the octave"[1][2]. More recently, the anacronym "edo" (EE-doh), spelled in lowercase, has also become common.

With the development of equal divisions of non-octave intervals (edonoi), some people started writing "ed2" ("ED2"), especially when naming a specific tuning.

Several alternate notations have been devised, including "edd" ("EDD"; equal division of the ditave), "DIV," and "EQ". [citation needed]

Formula

To find the step size of n-edo in terms of cents, divide 1200 by n. The size s of k steps of n-edo (k\n) is

[math]\displaystyle s = 1200 \cdot k/n[/math]

To find the step size of n-edo in terms of frequency ratio, take the n-th root of 2. For example, the step of 12edo is 21/12 (≈ 1.059). So the ratio c of the k steps of n-edo is

[math]\displaystyle c = 2^{k/n}[/math]

In particular, when k is 0, c is simply 1, because any number to the 0th power is 1. And when k = n, c is simply 2, because any number to the 1st power is itself.

EDO FAQ

What are EDO scales like?

Very straightforward to work with, the step size being so even and all. Some find the monotony bland, others find it a safe stable footing for musicmaking. The only property shared by all of them is the equality of their step-sizes; otherwise, their individual properties are as different as can be. The lower-numbered EDOs, especially 5 to 24, possess very strong and unique "characters", which some composers have found to be inspiring in their own right.

Why would I want to use an EDO?

If you are a guitarist (or a player of some other fretted string instrument, like a bass guitar, Appalachian dulcimer, ukulele, banjo, mandolin, sitar, saz, pipa, or zhong ruan), an EDO will provide you with the simplest possible fretboard layout, as all of the frets will go straight across the fretboard, regardless of how you want to tune the open strings. Speaking of string instruments fretted for EDOs, since ascending through the EDOs will crowd a fretboard relatively quickly, especially as one approaches the 30-something EDOs, equal divisions of the double octave (or higher multiple of the octave) are a relatively tidy compromise solution to the problem of laying out high-EDO fretboards.

More generally, EDOs allow for modulation to every single key in the tuning, without any alteration in harmonic properties, thus making transposition totally seamless. This also makes them somewhat easier to learn, as you do not have to memorize the harmonic and melodic variations that appear in various keys (which you would have to learn in JI, an unequal regular temperament, or a well-temperament, especially with smaller numbers of tones). For those accustomed to the "equality" of 12-TET, the equality of the alternative EDOs can be reassuringly familiar.

How do I explore so many?

It depends entirely on your desires as a musician!

If you are interested in exploring the unique merits and challenges of each EDO, irrespective of any desire to approximate Just intonation (or any other a priori musical goal), starting at the bottom and working your way up can be a most illuminating exercise.

If you're a classically-trained musician and you'd like to start with some EDOs that have some relationship to common-practice tonal music, starting with reasonably-low EDOs that give a good approximation to 3/2 (the perfect 5th) can be rewarding. These include 12, 17, 19, 22, 26, 27, 29, 31, 39, 41, 43, 45, 46, 49, 50, and 53. All of these can be notated with some variant on the A-G "circle of fifths" notation, while other EDOs, including 24, 34, 36, 38, 44, 48, or 51 involve more than one such circle.

Some EDOs, such as 26, 27, 32, 33, or 37 have fifths which are reasonably good but quite audibly not just. Other EDOs, such as 11, 13, 14, 15, 16, 18, 20, 21, 23, or 25, are of interest to the avid seeker of totally unusual sounds that have next-to-no connection with the common practice.

If your interest lies in the nuanced approximation of just intonation through EDOs, then delving into EDOs characterized by a strong local zeta peak could be especially captivating. Such EDOs, including 12, 19, 22, 27, 31, 34, 41, 46, 53, 58, 60, 65, 68, 72, 77, 80, 84, 87, 94, and 99, offer rich avenues for exploration in the quest for harmonic purity and transparent temperaments.

EDOs with a less pronounced, yet still noteworthy local zeta peak—specifically 10, 14, 15, 16, 17, 21, 24, 26, 29, 32, 36, 37, 38, 39, 43, 45, 48, 49, 50, 56, 62, 63, 82, 89, and 96 EDOs— present a unique palette for harmony explorers. Although these systems may lack the harmonic precision found in EDOs with more prominent zeta peaks, they strike an intriguing balance between consonance and more distant harmonic textures.

EDOs can be further subdivided and classified according to the size of the fifth, such as with Margo Schulter's gentle region or the distinction between negative, positive, doubly negative and doubly positive of R. H. M. Bosanquet. Kite Giedraitis has proposed these six categories, based on the size of the fifth. From narrowest to widest:

  • superflat EDOs (9, 11, 13b, 16, 18b & 23) have a fifth narrower than four-sevenths of an octave = 4\7 = 686¢
  • perfect EDOs (7, 14, 21, 28 & 35) have a fifth of 4\7 = 686¢
  • diatonic EDOs (12, 17, 19, 22, 24, etc.) have a fifth between 686¢ and 720¢
  • pentatonic EDOs (5, 10, 15, 20, 25 & 30) have a fifth of three-fifths of an octave = 3\5 = 720¢
  • supersharp EDOs (8, 13 & 18) have a fifth wider than 3\5 = 720¢
  • trivial EDOs (1, 2, 3, 4 and 6) have a fifth about 100¢ from just, and are contained in 12edo

Non-tuning properties

You will quickly find that the factorization of the total number of notes in each EDO has consequences for its structure and the way it relates to other EDOs. For example, 6 = 2 x 3, so 6edo contains all of the intervals in both 2edo and 3edo. On the other hand, 7 is a prime number, so no 7edo intervals are redundant with those of smaller EDOs. See Prime EDO and Highly composite EDO for more details.

The MOS paradigm is a fascinating way of thinking about building sub-scales of EDOs and relating them to non-EDO scales, as well as finding common melodic patterns between multiple EDOs.

Adding EDOs

Interesting phenomena may be observed when adding the cardinality of one equal division to that of another (octave or not). This really amounts to the consideration of adding the associated vals, which are the mappings to primes larger than 2. An EDO is defined by a certain number of steps equating to 2; if we have more steps equating to 3, we get a 3-limit val, and so forth. So, for example, 12edo can be written 12], saying that twelve steps maps to 2, but the 3-limit val for 12 is 12 19], telling us that 19 steps maps to 3, and the 5-limit val is 12 19 28], telling us that 28 steps maps to 5.

If we add 12 and 19 we get another good division, 12 + 19 = 31. We can understand why this works if we look at it as adding vals; 12 19 28] + 19 30 44] = 31 49 72]. The relative error in terms of relative cents is additive, and so sharpness and flatness cancel out, as they do for example with the approximation to 5 when adding 12 and 19. In terms of relative cents, the error of 12edo for the primes 3 and 5 is [-1.955 13.686] (the same as absolute cents) and the error of 19edo is [-11.429 -11.663], and this sums to [-13.384 2.023]. In relative cents the error of the fifth for 31edo is not much increased from 19edo, and on converting to absolute cents we find it is even better, and the error of the major third is much smaller due to the cancellation. When the errors are very sharp in one direction and very flat in another, as for instance with 15edo and 16edo, the sum (again 31edo) can have a much smaller error due to the cancellation. 24edo's flat fifth and 29edo's sharp fifth can be added to form 53edo.

We may also look at addition of EDOs in terms of MOS; if a\n is a generator for an n-edo MOS, and b\m for an m-edo MOS, where both of these are generators for the same linear temperament, then the mediant, (a + b)\(n + m), will be a generator for a MOS for the same temperament, this time in (n + m)-edo. A visual way of putting this is that through this addition of n and m, one becomes the accidentals or black keys, and the other the naturals or white keys. The choice of accidental/natural or black keys/white keys is a question of emphasis on the part of the composer or designer. Furthermore, one may add more than two numbers, hierarchically expanding the possibilities to double flats and sharps and beyond. This can be useful in designing keyboards and systems of notation.

Size of an EDO

When an edo divides the octave into fewer than 12 divisions (so that each step exceeds 100 cents), you might call it a macrotonal EDO. Of these, 1, 2, 3, 4 and 6 divide 12 and so are already available to anyone wishing to explore them. 5, 7 and 9 have arguably been used in various kinds of musical traditions in different parts of the world. The 13 EDOs of Xmas by Scott Thompson is a humorous demonstration of EDOs 1–13.

On the other hand, if you use the edo to tune a scale or regular temperament, the size of the edo does not matter so much (at least conceptually), as you don't need to use all of it. Some of the EDOs which can be used to tune various temperaments are listed on the optimal patent val page. Tuning a scale in just intonation by one of these EDOs can be regarded as automatically tempering it to the corresponding regular temperament.

To practically tune large edos through software tuning, one may take advantage of MIDI channels. See Tuning per channel.

All of these tools are also applicable to equal divisions of other (nonoctave) intervals as well.

What's the difference between EDOs and Equal Temperaments?

See EDO vs ET.

Individual pages for EDOs

0…999

0…99
0 1 2 3 4 5 6 7 8 9
10 11 12 13 14 15 16 17 18 19
20 21 22 23 24 25 26 27 28 29
30 31 32 33 34 35 36 37 38 39
40 41 42 43 44 45 46 47 48 49
50 51 52 53 54 55 56 57 58 59
60 61 62 63 64 65 66 67 68 69
70 71 72 73 74 75 76 77 78 79
80 81 82 83 84 85 86 87 88 89
90 91 92 93 94 95 96 97 98 99
100…199
100 101 102 103 104 105 106 107 108 109
110 111 112 113 114 115 116 117 118 119
120 121 122 123 124 125 126 127 128 129
130 131 132 133 134 135 136 137 138 139
140 141 142 143 144 145 146 147 148 149
150 151 152 153 154 155 156 157 158 159
160 161 162 163 164 165 166 167 168 169
170 171 172 173 174 175 176 177 178 179
180 181 182 183 184 185 186 187 188 189
190 191 192 193 194 195 196 197 198 199
200…299
200 201 202 203 204 205 206 207 208 209
210 211 212 213 214 215 216 217 218 219
220 221 222 223 224 225 226 227 228 229
230 231 232 233 234 235 236 237 238 239
240 241 242 243 244 245 246 247 248 249
250 251 252 253 254 255 256 257 258 259
260 261 262 263 264 265 266 267 268 269
270 271 272 273 274 275 276 277 278 279
280 281 282 283 284 285 286 287 288 289
290 291 292 293 294 295 296 297 298 299
300…399
300 301 302 303 304 305 306 307 308 309
310 311 312 313 314 315 316 317 318 319
320 321 322 323 324 325 326 327 328 329
330 331 332 333 334 335 336 337 338 339
340 341 342 343 344 345 346 347 348 349
350 351 352 353 354 355 356 357 358 359
360 361 362 363 364 365 366 367 368 369
370 371 372 373 374 375 376 377 378 379
380 381 382 383 384 385 386 387 388 389
390 391 392 393 394 395 396 397 398 399
400…499
400 401 402 403 404 405 406 407 408 409
410 411 412 413 414 415 416 417 418 419
420 421 422 423 424 425 426 427 428 429
430 431 432 433 434 435 436 437 438 439
440 441 442 443 444 445 446 447 448 449
450 451 452 453 454 455 456 457 458 459
460 461 462 463 464 465 466 467 468 469
470 471 472 473 474 475 476 477 478 479
480 481 482 483 484 485 486 487 488 489
490 491 492 493 494 495 496 497 498 499
500…599
500 501 502 503 504 505 506 507 508 509
510 511 512 513 514 515 516 517 518 519
520 521 522 523 524 525 526 527 528 529
530 531 532 533 534 535 536 537 538 539
540 541 542 543 544 545 546 547 548 549
550 551 552 553 554 555 556 557 558 559
560 561 562 563 564 565 566 567 568 569
570 571 572 573 574 575 576 577 578 579
580 581 582 583 584 585 586 587 588 589
590 591 592 593 594 595 596 597 598 599
600…699
600 601 602 603 604 605 606 607 608 609
610 611 612 613 614 615 616 617 618 619
620 621 622 623 624 625 626 627 628 629
630 631 632 633 634 635 636 637 638 639
640 641 642 643 644 645 646 647 648 649
650 651 652 653 654 655 656 657 658 659
660 661 662 663 664 665 666 667 668 669
670 671 672 673 674 675 676 677 678 679
680 681 682 683 684 685 686 687 688 689
690 691 692 693 694 695 696 697 698 699
700…799
700 701 702 703 704 705 706 707 708 709
710 711 712 713 714 715 716 717 718 719
720 721 722 723 724 725 726 727 728 729
730 731 732 733 734 735 736 737 738 739
740 741 742 743 744 745 746 747 748 749
750 751 752 753 754 755 756 757 758 759
760 761 762 763 764 765 766 767 768 769
770 771 772 773 774 775 776 777 778 779
780 781 782 783 784 785 786 787 788 789
790 791 792 793 794 795 796 797 798 799
800…899
800 801 802 803 804 805 806 807 808 809
810 811 812 813 814 815 816 817 818 819
820 821 822 823 824 825 826 827 828 829
830 831 832 833 834 835 836 837 838 839
840 841 842 843 844 845 846 847 848 849
850 851 852 853 854 855 856 857 858 859
860 861 862 863 864 865 866 867 868 869
870 871 872 873 874 875 876 877 878 879
880 881 882 883 884 885 886 887 888 889
890 891 892 893 894 895 896 897 898 899
900…999
900 901 902 903 904 905 906 907 908 909
910 911 912 913 914 915 916 917 918 919
920 921 922 923 924 925 926 927 928 929
930 931 932 933 934 935 936 937 938 939
940 941 942 943 944 945 946 947 948 949
950 951 952 953 954 955 956 957 958 959
960 961 962 963 964 965 966 967 968 969
970 971 972 973 974 975 976 977 978 979
980 981 982 983 984 985 986 987 988 989
990 991 992 993 994 995 996 997 998 999

1000…1999

1000, 1001, 1012, 1015, 1051, 1053, 1059, 1063, 1065, 1080, 1092, 1106, 1125, 1131, 1152, 1166, 1171, 1178, 1193, 1200, 1210, 1224, 1225, 1230, 1236, 1240, 1244, 1260, 1277, 1289, 1308, 1323, 1330, 1337, 1342, 1395, 1407, 1419, 1429, 1440, 1448, 1506, 1517, 1520, 1525, 1536, 1547, 1553, 1554, 1559, 1578, 1583, 1590, 1600, 1612, 1619, 1637, 1641, 1643, 1650, 1665, 1672, 1700, 1759, 1776, 1778, 1783, 1789, 1793, 1794, 1802, 1803, 1817, 1848, 1861, 1880, 1889, 1911, 1920, 1944, 1955, 1957, 1984, 1995

2000…9999

2000, 2016, 2019, 2022, 2023, 2024, 2048, 2072, 2100, 2113, 2118, 2129, 2190, 2200, 2320, 2444, 2460, 2513, 2520, 2554, 2619, 2660, 2684, 2711, 2730, 2777, 2809, 2814, 2897, 2901, 2912, 2960, 2964, 3072, 3079, 3125, 3178, 3325, 3395, 3422, 3476, 3498, 3558, 3566, 3578, 3600, 3684, 3696, 3776, 3920, 3990, 4004, 4096, 4172, 4190, 4231, 4296, 4320, 4327, 4349, 4380, 4501, 4650, 4655, 4973, 5040, 5280, 5320, 5544, 5585, 5902, 5941, 5985, 6079, 6380, 6650, 6664, 6691, 7033, 7315, 7980, 8103, 8192, 8269, 8539, 8736, 8745

10000 and up

10600, 11664, 12000, 12276, 12348, 12500, 13382, 14124, 14348, 14618, 14842, 15601, 15900, 16384, 16625, 16808, 17461, 18355, 20203, 20567, 21489, 25281, 28000, 30103, 30631, 31867, 31920, 32768, 34691, 46032, 50562, 58973, 65536, 73709, 78005, 79335, 80000, 86400, 98304, 99693, 102557, 103169, 111202, 120000, 148418, 190537, 196608, 241200, 253389, 258008, 571611, 762148, 1200000, 1714833, 1905370, 2547047, 2667518, 2901533, 4191814, 6000000, 11358058, 402653184, 5407372813

The largest physically possible EDO in a frequency range can be found through molecular physics such as the mean free path combined with the speed of sound in a given substance.

Non-integer EDO

A non-integer EDO can be defined as using a non-integer divisor to divide the octave. Typically, non-integer EDOs are understood as not containing the exact octave, so that they remain equal tunings. If the exact octave is retained and if the generator resets itself at each period, then this results in a MOS scale with only 1 small step.

All fractional EDOs are integer equal divisions of another integer interval. For example, (25/2)edo is equivalent to 25ed4. In general:

[math]\displaystyle (p/q) \text{edo} = p \text{-ed} 2^q[/math]

for integers p and q. Many irrational EDOs cannot be converted to integer equal divisions of another integer interval, so they are things of their own.

Non-integer EDOs can be written in decimal form, such as 12.1edo. This is often meant to be approximate, used in the context of octave stretch of an integer EDO, rather than as a fractional EDO.

Scale tree

The scale tree, or Stern-Brocot tree, provides a visual map of the world of EDOs, based on fifth size.

The Scale Tree.png

The regular EDOs, up to 72edo:

Scale Tree close-up.png

Pergens

Pergens provide a JI-agnostic way to name the rank-2 scales of an EDO. This table lists every possible period/generator combination for EDOs 5-24, and for each coprime combination, the simplest pergen that it can represent. Non-coprime combinations such as P = 6\12, G = 4\12 are marked as "-".

EDO Period Generator in EDO steps
in EDO steps 1 2 3 4 5 6 7 8 9 10 11
5 5 = P8 P4/2 P5
6 6 = P8 P4/2 -
3 = P8/2 P5
7 7 = P8 P4/3 P5/2 P5
8 8 = P8 P4/3 - P5
4 = P8/2 P5 -
9 9 = P8 P4/4 P4/2 - P5
3 = P8/3 P5
10 10 = P8 P4/4 - P5/2 -
5 = P8/2 P5 P4/2
11 11 = P8 P4/5 P5/3 P5/2 P11/4 P5
12 12 = P8 P4/5 - - - P5
6 = P8/2 P5 - -
4 = P8/3 P5 -
3 = P8/4 P5
13b 13 = P8 P4/6 P4/3 P4/2 P12/5 P12/4 P5
14 14 = P8 P4/6 - P4/2 - P11/4 -
7 = P8/2 P5 P4/3 P4/2
15 15 = P8 P4/6 P4/3 - P12/6 - - P11/3
5 = P8/3 P5 P4/2
3 = P8/5 P4/3
16 16 = P8 P4/7 - P5/3 - P12/5 - P5
8 = P8/2 P5 - P5/3 -
4 = P8/4 P5 -
17 17 = P8 P4/7 P5/5 P11/8 P11/6 P5/2 P11/4 P5 P11/3
18b 18 = P8 P4/8 - - - P5/2 - P12/4 -
9 = P8/2 P5 P4/4 - P4/2
6 = P8/3 P5/2 - -
3 = P8/6 P5
19 19 = P8 P4/8 P4/4 P11/9 P4/2 P12/6 P12/5 ccP5/7 P5 P11/3
20 20 = P8 P4/8 - P5/4 - - - P11/4 - c3P5/8
10 = P8/2 M2/4 - P5/4 - -
5 = P8/4 P4/2 P5
4 = P8/5 P5/4 -
21 21 = P8 P4/9 P5/6 - P5/3 P11/6 - - c3P4/9 - P11/3
7 = P8/3 P5/2 P5 P4/3
3 = P8/7 P5/3
22 22 = P8 P4/9 - P4/3 - P12/7 - P12/5 - P5 -
11 = P8/2 M2/4 P5 P4/3 P12/5 P12/7
23 23 = P8 P4/10 P4/5 P11/11 P12/9 P4/2 P12/6 ccP4/8 ccP4/7 P12/4 P5 P11/3
24 24 = P8 P4/10 - - - P4/2 - P5/2 - - - c4P5/10
12 = P8/2 M2/4 - - - P4/2
8 = P8/3 P5/2 - P4/2
6 = P8/4 P4/2 - -
4 = P8/6 P4/2 -
3 = P8/8 P5
1 2 3 4 5 6 7 8 9 10 11

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Notes