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<nowiki/>* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct
<nowiki/>* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct
=== Interval mappings ===
{{Q-odd-limit intervals|99|31}}
{{Q-odd-limit intervals|99.1|31|apx=val|header=none|tag=none|title=31-odd-limit intervals by 99efk val mapping}}


== Octave stretch or compression ==
== Octave stretch or compression ==

Revision as of 10:57, 17 April 2026

← 98edo 99edo 100edo →
Prime factorization 32 × 11
Step size 12.1212 ¢ 
Fifth 58\99 (703.03 ¢)
Semitones (A1:m2) 10:7 (121.2 ¢ : 84.85 ¢)
Consistency limit 9
Distinct consistency limit 9

99 equal divisions of the octave (abbreviated 99edo or 99ed2), also called 99-tone equal temperament (99tet) or 99 equal temperament (99et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 99 equal parts of about 12.1 ¢ each. Each step represents a frequency ratio of 21/99, or the 99th root of 2. The step size of this system is close to 144/143, the grossma.

Theory

99edo is a very strong 7-limit (and 9-odd-limit) tuning, with a sound defined by the slight sharpness (1.1, 1.6, 0.9 cents) of its 3, 5, and 7. As an equal temperament, it tempers out 393216/390625 (würschmidt comma) and 1600000/1594323 (amity comma) in the 5-limit; 5120/5103 (argent comma), 2401/2400 (breedsma), 3136/3125 (hemimean comma), and 4375/4374 (ragisma) in the 7-limit, supporting hemififths, amity, parakleismic, hemiwürschmidt and ennealimmal temperaments, and is pretty well a perfect tuning for hendecatonic temperament.

Extending it to the 11-limit requires choosing which mapping one wants to use, as both are nearly equally far off the mark. Using the ⟨99 157 230 278 343] (99e) val, it tempers out 243/242, 441/440, 540/539 and 896/891, and is an excellent tuning for the 11-limit version of hemififths temperament. Using the patent val, 99edo is the optimal patent val for the rank-4 temperament tempering out 121/120; zeus, the rank-3 temperament tempering out 121/120 and 176/175; hemiwür, one of the rank-2 11-limit extensions of hemiwürschmidt; and hitchcock (an 11-limit amity extension), the rank-2 temperament which also tempers out 2200/2187. The same can be said of the mapping for 13, with the 99ef val tempering out 144/143, 196/195, 352/351 and 364/363, and its patent val tempering out 169/168, 351/350 and 352/351. Hence 99edo, in spite of the fact that it tunes 11 and 13 relatively badly, is an important 13-limit tuning in more than one way.

Being a zeta peak edo, 99edo is also a very strong no-11 no-13 system, where it is consistent to the 29-odd-limit with a sharp tendency. This favors the sharp mapping of 11 and 13, and allows these relatively weak approximations to somewhat blend with the rest for a full 29-limit (or 31-limit, using the sharp-tending 99efk val) temperament. In fact, the 99efk val is the first to achieve diamond monotone in the 31-odd-limit, though it fails in the 33-odd-limit due to mapping 33/32 to 5 steps, while 32/31 is mapped to 4 steps.

Prime harmonics

Approximation of prime harmonics in 99edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.00 +1.08 +1.57 +0.87 -5.86 -4.16 +4.14 +5.52 +2.03 +0.73 -5.64
Relative (%) +0.0 +8.9 +12.9 +7.2 -48.4 -34.4 +34.1 +45.5 +16.7 +6.0 -46.5
Steps
(reduced) 		99
(0) 		157
(58) 		230
(32) 		278
(80) 		342
(45) 		366
(69) 		405
(9) 		421
(25) 		448
(52) 		481
(85) 		490
(94)

Subsets and supersets

Since 99 factors into primes as 32 × 11, 99edo has subset edos 3, 9, 11, and 33. Splitting 99edo's step in half yields 198edo, correcting prime 11, slightly improving prime 13, and aligning both 11 and 13 with the sharp tunings of the lower odd primes. Because of this, 198edo can be seen as a complex yet notable true full 13-limit tuning.

Intervals

Main article: Table of 99edo intervals

Notation

Ups and downs notation

99edo can be notated with Kite's ups and downs notation. Note that quip (quintuple-up) is the same as quudsharp (quadruple-down sharp) and that quid (quintuple-down) is the same as quupflat (quadruple-up flat):

Step offset 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21
Sharp symbol   
 
  
    
   
  
   
    
    
   
  
   
    
    
   
  
   
    
    
   
  
   
Flat symbol
 
  
    
   
  
   
    
    
   
  
   
    
    
   
  
   
    
    
   
  
   

Another notation uses alternative ups and downs. Here, this can be done using sharps and flats with arrows, borrowed from extended Helmholtz–Ellis notation:

Step offset 0 1 2 3 4 5 6 7 8 9 10 11 12
Sharp symbol
Flat symbol
Step offset 13 14 15 16 17 18 19 20 21 22
Sharp symbol
Flat symbol

Approximation to JI

7-prime-limited odd-limit analysis

Unlike all previous edos, 99edo is distinctly consistent and monotone (i.e. when tempered using the patent val, the relative sizes of any two intervals are never conflated or reversed) up to the 7-prime-limited 45-odd-limit:

The 7-prime-limited 45-odd-limit, by 99edo mapping (SW3 format)
(*
 7-PL 45-OL odds:
 1 3 5 7 9 15 21 25 27 35 45
 
Mapping   Ratio     Error  *)
(* 4\99*) 36/35 (* -0.286c *)
(* 5\99*) 28/27 (* -2.355c *)
(* 6\99*) 25/24 (* +2.055c *)
(* 7\99*) 21/20 (* +0.381c *)
(* 9\99*) 16/15 (* -2.640c *)
(*10\99*) 15/14 (* +1.769c *)
(*11\99*) 27/25 (* +0.096c *)
(*13\99*) 35/32 (* +2.436c *)
(*15\99*) 10/9  (* -0.586c *)
(*16\99*) 28/25 (* -2.259c *)
(*17\99*) 9/8   (* +2.151c *)
(*19\99*) 8/7   (* -0.871c *)
(*22\99*) 7/6   (* -0.204c *)
(*24\99*) 32/27 (* -3.226c *)
(*25\99*) 25/21 (* +1.184c *)
(*26\99*) 6/5   (* -0.490c *)
(*31\99*) 56/45 (* -2.845c *)
(*32\99*) 5/4   (* +1.565c *)
(*35\99*) 32/25 (* -3.130c *)
(*36\99*) 9/7   (* +1.280c *)
(*37\99*) 35/27 (* -0.790c *)
(*39\99*) 21/16 (* +1.946c *)
(*41\99*) 4/3   (* -1.075c *)
(*43\99*) 27/20 (* +1.661c *)
(*45\99*) 48/35 (* -1.361c *)
(*47\99*) 25/18 (* +0.980c *)
(*48\99*) 7/5   (* -0.694c *)
(*49\99*) 45/32 (* +3.716c *)
(*50\99*) 64/45
(*51\99*) 10/7
(*52\99*) 36/25
(*54\99*) 35/24
(*56\99*) 40/27
(*58\99*) 3/2
(*60\99*) 32/21
(*62\99*) 54/35
(*63\99*) 14/9
(*64\99*) 25/16
(*67\99*) 8/5
(*68\99*) 45/28
(*73\99*) 5/3
(*74\99*) 42/25
(*75\99*) 27/16
(*77\99*) 12/7
(*80\99*) 7/4
(*82\99*) 16/9
(*83\99*) 25/14
(*84\99*) 9/5
(*86\99*) 64/35
(*88\99*) 50/27
(*89\99*) 28/15
(*90\99*) 15/8
(*92\99*) 40/21
(*93\99*) 48/25
(*94\99*) 27/14
(*95\99*) 35/18
(*99\99*) 2/1

The 7-prime-limited 49-odd-limit is where non-distinctness first shows up: namely, ~49/48 = ~50/49 (this is characteristic of all ennealimmal tunings). However, 99edo remains monotone and consistent up to the 7-prime-limited 567-odd-limit (the next 7-limit odd, 625, is inconsistent):

The 7-prime-limited 567-odd-limit, by 99edo mapping (SW3 format)
(* 1*) 225/224; 126/125; 245/243;
(* 2*) 81/80; 64/63;
(* 3*) 50/49; 49/48; 128/125;
(* 4*) 525/512; 36/35; 250/243;
(* 5*) 405/392; 28/27;
(* 6*) 25/24; 256/245; 392/375;
(* 7*) 360/343; 21/20; 256/243;
(* 8*) 135/128; 200/189; 343/324;
(* 9*) 16/15;
(*10*) 15/14; 343/320;
(*11*) 27/25; 175/162;
(*12*) 243/224; 160/147; 49/45;
(*13*) 375/343; 35/32; 192/175;
(*14*) 54/49; 441/400; 448/405;
(*15*) 567/512; 10/9;
(*16*) 125/112; 384/343; 28/25;
(*17*) 9/8; 640/567;
(*18*) 500/441; 567/500; 245/216; 256/225;
(*19*) 8/7; 343/300;
(*20*) 225/196; 147/128; 144/125; 280/243;
(*21*) 81/70; 125/108; 512/441;
(*22*) 400/343; 7/6;
(*23*) 75/64; 288/245; 147/125;
(*24*) 405/343; 189/160; 32/27;
(*25*) 25/21; 343/288; 448/375;
(*26*) 6/5;
(*27*) 135/112; 98/81;
(*28*) 243/200; 175/144; 128/105;
(*29*) 60/49; 49/40;
(*30*) 315/256; 216/175; 100/81;
(*31*) 243/196; 56/45;
(*32*) 5/4;
(*33*) 432/343; 63/50; 512/405;
(*34*) 81/64; 80/63; 343/270;
(*35*) 125/98; 245/192; 32/25;
(*36*) 9/7;
(*37*) 162/125; 35/27;
(*38*) 125/96; 64/49; 98/75;
(*39*) 450/343; 21/16; 320/243;
(*40*) 324/245; 250/189;
(*41*) 4/3;
(*42*) 75/56; 343/256; 168/125;
(*43*) 27/20; 256/189;
(*44*) 200/147; 49/36; 512/375;
(*45*) 175/128; 48/35; 343/250;
(*46*) 135/98; 441/320; 112/81;
(*47*) 243/175; 25/18;
(*48*) 480/343; 7/5;
(*49*) 45/32; 800/567; 343/243;
(*50*) 486/343; 567/400; 64/45;
(*51*) 10/7; 343/240;
(*52*) 36/25; 350/243;
(*53*) 81/56; 640/441; 196/135;
(*54*) 500/343; 35/24; 256/175;
(*55*) 375/256; 72/49; 147/100;
(*56*) 189/128; 40/27;
(*57*) 125/84; 512/343; 112/75;
(*58*) 3/2;
(*59*) 189/125; 245/162;
(*60*) 243/160; 32/21; 343/225;
(*61*) 75/49; 49/32; 192/125;
(*62*) 54/35; 125/81;
(*63*) 14/9;
(*64*) 25/16; 384/245; 196/125;
(*65*) 540/343; 63/40; 128/81;
(*66*) 405/256; 100/63; 343/216;
(*67*) 8/5;
(*68*) 45/28; 392/243;
(*69*) 81/50; 175/108; 512/315;
(*70*) 80/49; 49/30;
(*71*) 105/64; 288/175; 400/243;
(*72*) 81/49; 224/135;
(*73*) 5/3;
(*74*) 375/224; 576/343; 42/25;
(*75*) 27/16; 320/189; 686/405;
(*76*) 250/147; 245/144; 128/75;
(*77*) 12/7; 343/200;
(*78*) 441/256; 216/125; 140/81
(*79*) 243/140; 125/72; 256/147; 392/225;
(*80*) 600/343; 7/4;
(*81*) 225/128; 432/245; 1000/567; 441/250;
(*82*) 567/320; 16/9;
(*83*) 25/14; 343/192; 224/125;
(*84*) 9/5; 1024/567;
(*85*) 405/224; 800/441; 49/27;
(*86*) 175/96; 64/35; 686/375;
(*87*) 90/49; 147/80; 448/243;
(*88*) 324/175; 50/27;
(*89*) 640/343; 28/15;
(*90*) 15/8;
(*91*) 648/343; 189/100; 256/135;
(*92*) 243/128; 40/21; 343/180;
(*93*) 375/196; 245/128; 48/25;
(*94*) 27/14; 784/405;
(*95*) 243/125; 35/18; 1024/525;
(*96*) 125/64; 96/49; 49/25;
(*97*) 63/32; 160/81;
(*98*) 486/245; 125/63; 448/225;
(*99*) 2/1;

Regular temperament properties

Subgroup Comma list Mapping Optimal
8ve stretch (¢) 		Tuning error
Absolute (¢) Relative (%)
2.3 [157 -99⟩ [⟨99 157]] −0.339 0.339 2.80
2.3.5 393216/390625, 1600000/1594323 [⟨99 157 230]] −0.451 0.319 2.63
2.3.5.7 2401/2400, 3136/3125, 4375/4374 [⟨99 157 230 278]] −0.416 0.283 2.33
2.3.5.7.11 243/242, 441/440, 896/891, 3136/3125 [⟨99 157 230 278 343]] (99e) −0.694 0.612 5.05
2.3.5.7.11 121/120, 176/175, 1375/1372, 2200/2187 [⟨99 157 230 278 342]] (99) +0.006 0.881 7.27
  • 99et is lower in relative error than any previous equal temperaments in the 7-limit. Not until 171 do we find a better equal temperament in terms of either absolute error or relative error.

Rank-2 temperaments

Table of rank-2 temperaments by generator
Periods
per 8ve 		Generator* Cents* Associated
ratio* 		Temperament
1 2\99 24.242 686/675, 99/98 Sengagen (99e) / sengage (99ef)
1 7\99 84.848 21/20 Amicable
1 16\99 193.939 28/25 Hemiwürschmidt (99e) / hemithir (99ef) / hemiwur (99f)
1 19\99 230.303 8/7 Gamera
1 20\99 242.424 147/128 Septiquarter
1 25\99 303.030 25/21 Quinmite
1 26\99 315.152 6/5 Parakleismic (99) / paralytic (99e) / parkleismic (99) / paradigmic (99e)
1 28\99 339.394 128/105 Amity (99ef) / hitchcock (99)
1 29\99 351.515 49/40 Hemififths (99ef)
1 32\99 387.879 5/4 Würschmidt / whirrschmidt
1 41\99 496.970 4/3 Undecental
1 37\99 448.485 35/27 Semidimfourth
3 5\99 60.606 28/27 Chromat
3 13\99 157.576 35/32 Nessafof
3 41\99
(8\99) 		496.970
(96.970) 		4/3
(18/17~19/18) 		Misty
9 4\99 48.485 36/35 Ennealimmal (99e) / ennealimmia (99) /
ennealimnic (99ef) / ennealim (99e) / ennealiminal (99)
11 41\99
(4\99) 		496.970
(48.485) 		4/3
(36/35) 		Hendecatonic

* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct

Interval mappings

The following tables show how 31-odd-limit intervals are represented in 99edo. Prime harmonics are in bold; inconsistent intervals are in italics.

31-odd-limit intervals in 99edo (direct approximation, even if inconsistent)
Interval and complement Error (abs, ¢) Error (rel, %)
1/1, 2/1 0.000 0.0
23/21, 42/23 0.082 0.7
27/25, 50/27 0.096 0.8
23/18, 36/23 0.122 1.0
29/28, 56/29 0.145 1.2
7/6, 12/7 0.204 1.7
31/22, 44/31 0.222 1.8
29/24, 48/29 0.349 2.9
21/20, 40/21 0.381 3.1
23/20, 40/23 0.464 3.8
5/3, 6/5 0.490 4.0
9/5, 10/9 0.586 4.8
23/15, 30/23 0.612 5.0
7/5, 10/7 0.694 5.7
29/16, 32/29 0.726 6.0
19/11, 22/19 0.741 6.1
29/20, 40/29 0.839 6.9
7/4, 8/7 0.871 7.2
27/17, 34/27 0.910 7.5
23/12, 24/23 0.953 7.9
31/19, 38/31 0.962 7.9
25/18, 36/25 0.980 8.1
25/17, 34/25 1.005 8.3
3/2, 4/3 1.075 8.9
25/23, 46/25 1.101 9.1
23/14, 28/23 1.158 9.6
25/21, 42/25 1.184 9.8
27/23, 46/27 1.197 9.9
29/21, 42/29 1.221 10.1
9/7, 14/9 1.280 10.6
29/23, 46/29 1.303 10.7
19/17, 34/19 1.382 11.4
29/18, 36/29 1.425 11.8
31/26, 52/31 1.478 12.2
17/15, 30/17 1.495 12.3
5/4, 8/5 1.565 12.9
27/20, 40/27 1.661 13.7
13/11, 22/13 1.699 14.0
15/14, 28/15 1.769 14.6
29/15, 30/29 1.915 15.8
21/16, 32/21 1.946 16.1
17/9, 18/17 1.985 16.4
23/16, 32/23 2.029 16.7
25/24, 48/25 2.055 17.0
23/17, 34/23 2.107 17.4
17/11, 22/17 2.122 17.5
9/8, 16/9 2.151 17.7
21/17, 34/21 2.189 18.1
25/14, 28/25 2.259 18.6
27/19, 38/27 2.291 18.9
31/17, 34/31 2.344 19.3
27/14, 28/27 2.355 19.4
25/19, 38/25 2.387 19.7
29/25, 50/29 2.404 19.8
19/13, 26/19 2.440 20.1
29/27, 54/29 2.500 20.6
17/10, 20/17 2.570 21.2
15/8, 16/15 2.640 21.8
19/15, 30/19 2.877 23.7
27/22, 44/27 3.032 25.0
17/12, 24/17 3.060 25.2
25/22, 44/25 3.128 25.8
25/16, 32/25 3.130 25.8
27/16, 32/27 3.226 26.6
31/27, 54/31 3.254 26.8
17/14, 28/17 3.264 26.9
31/25, 50/31 3.349 27.6
19/18, 36/19 3.367 27.8
29/17, 34/29 3.410 28.1
23/19, 38/23 3.489 28.8
21/19, 38/21 3.571 29.5
15/11, 22/15 3.617 29.8
17/13, 26/17 3.822 31.5
31/30, 60/31 3.839 31.7
19/10, 20/19 3.952 32.6
11/9, 18/11 4.107 33.9
17/16, 32/17 4.135 34.1
13/8, 16/13 4.164 34.4
23/22, 44/23 4.229 34.9
21/11, 22/21 4.311 35.6
31/18, 36/31 4.329 35.7
19/12, 24/19 4.442 36.6
31/23, 46/31 4.451 36.7
31/21, 42/31 4.533 37.4
19/14, 28/19 4.646 38.3
11/10, 20/11 4.693 38.7
27/26, 52/27 4.731 39.0
29/19, 38/29 4.791 39.5
25/13, 26/25 4.827 39.8
29/26, 52/29 4.890 40.3
31/20, 40/31 4.915 40.5
13/7, 14/13 5.035 41.5
11/6, 12/11 5.183 42.8
13/12, 24/13 5.239 43.2
15/13, 26/15 5.317 43.9
11/7, 14/11 5.387 44.4
31/24, 48/31 5.404 44.6
19/16, 32/19 5.517 45.5
29/22, 44/29 5.532 45.6
31/28, 56/31 5.609 46.3
31/16, 32/31 5.642 46.5
13/10, 20/13 5.729 47.3
31/29, 58/31 5.754 47.5
13/9, 18/13 5.807 47.9
11/8, 16/11 5.863 48.4
23/13, 26/23 5.929 48.9
21/13, 26/21 6.011 49.6
31-odd-limit intervals in 99edo (patent val mapping)
Interval and complement Error (abs, ¢) Error (rel, %)
1/1, 2/1 0.000 0.0
23/21, 42/23 0.082 0.7
27/25, 50/27 0.096 0.8
23/18, 36/23 0.122 1.0
29/28, 56/29 0.145 1.2
7/6, 12/7 0.204 1.7
31/22, 44/31 0.222 1.8
29/24, 48/29 0.349 2.9
21/20, 40/21 0.381 3.1
23/20, 40/23 0.464 3.8
5/3, 6/5 0.490 4.0
9/5, 10/9 0.586 4.8
23/15, 30/23 0.612 5.0
7/5, 10/7 0.694 5.7
29/16, 32/29 0.726 6.0
29/20, 40/29 0.839 6.9
7/4, 8/7 0.871 7.2
27/17, 34/27 0.910 7.5
23/12, 24/23 0.953 7.9
25/18, 36/25 0.980 8.1
25/17, 34/25 1.005 8.3
3/2, 4/3 1.075 8.9
25/23, 46/25 1.101 9.1
23/14, 28/23 1.158 9.6
25/21, 42/25 1.184 9.8
27/23, 46/27 1.197 9.9
29/21, 42/29 1.221 10.1
9/7, 14/9 1.280 10.6
29/23, 46/29 1.303 10.7
19/17, 34/19 1.382 11.4
29/18, 36/29 1.425 11.8
31/26, 52/31 1.478 12.2
17/15, 30/17 1.495 12.3
5/4, 8/5 1.565 12.9
27/20, 40/27 1.661 13.7
13/11, 22/13 1.699 14.0
15/14, 28/15 1.769 14.6
29/15, 30/29 1.915 15.8
21/16, 32/21 1.946 16.1
17/9, 18/17 1.985 16.4
23/16, 32/23 2.029 16.7
25/24, 48/25 2.055 17.0
23/17, 34/23 2.107 17.4
9/8, 16/9 2.151 17.7
21/17, 34/21 2.189 18.1
25/14, 28/25 2.259 18.6
27/19, 38/27 2.291 18.9
27/14, 28/27 2.355 19.4
25/19, 38/25 2.387 19.7
29/25, 50/29 2.404 19.8
29/27, 54/29 2.500 20.6
17/10, 20/17 2.570 21.2
15/8, 16/15 2.640 21.8
19/15, 30/19 2.877 23.7
17/12, 24/17 3.060 25.2
25/16, 32/25 3.130 25.8
27/16, 32/27 3.226 26.6
17/14, 28/17 3.264 26.9
19/18, 36/19 3.367 27.8
29/17, 34/29 3.410 28.1
23/19, 38/23 3.489 28.8
21/19, 38/21 3.571 29.5
19/10, 20/19 3.952 32.6
17/16, 32/17 4.135 34.1
13/8, 16/13 4.164 34.4
19/12, 24/19 4.442 36.6
19/14, 28/19 4.646 38.3
29/19, 38/29 4.791 39.5
29/26, 52/29 4.890 40.3
13/7, 14/13 5.035 41.5
13/12, 24/13 5.239 43.2
19/16, 32/19 5.517 45.5
31/16, 32/31 5.642 46.5
13/10, 20/13 5.729 47.3
11/8, 16/11 5.863 48.4
21/13, 26/21 6.110 50.4
23/13, 26/23 6.193 51.1
13/9, 18/13 6.315 52.1
31/29, 58/31 6.367 52.5
31/28, 56/31 6.513 53.7
29/22, 44/29 6.589 54.4
31/24, 48/31 6.717 55.4
11/7, 14/11 6.734 55.6
15/13, 26/15 6.804 56.1
11/6, 12/11 6.939 57.2
31/20, 40/31 7.207 59.5
25/13, 26/25 7.294 60.2
27/26, 52/27 7.390 61.0
11/10, 20/11 7.428 61.3
31/21, 42/31 7.588 62.6
31/23, 46/31 7.670 63.3
31/18, 36/31 7.792 64.3
21/11, 22/21 7.810 64.4
23/22, 44/23 7.892 65.1
11/9, 18/11 8.014 66.1
31/30, 60/31 8.282 68.3
17/13, 26/17 8.300 68.5
15/11, 22/15 8.504 70.2
31/25, 50/31 8.772 72.4
31/27, 54/31 8.868 73.2
25/22, 44/25 8.994 74.2
27/22, 44/27 9.089 75.0
19/13, 26/19 9.681 79.9
31/17, 34/31 9.777 80.7
17/11, 22/17 9.999 82.5
31/19, 38/31 11.159 92.1
19/11, 22/19 11.381 93.9
31-odd-limit intervals by 99efk val mapping
Interval and complement Error (abs, ¢) Error (rel, %)
1/1, 2/1 0.000 0.0
23/21, 42/23 0.082 0.7
27/25, 50/27 0.096 0.8
23/18, 36/23 0.122 1.0
29/28, 56/29 0.145 1.2
7/6, 12/7 0.204 1.7
31/22, 44/31 0.222 1.8
29/24, 48/29 0.349 2.9
21/20, 40/21 0.381 3.1
23/20, 40/23 0.464 3.8
5/3, 6/5 0.490 4.0
9/5, 10/9 0.586 4.8
23/15, 30/23 0.612 5.0
7/5, 10/7 0.694 5.7
29/16, 32/29 0.726 6.0
19/11, 22/19 0.741 6.1
29/20, 40/29 0.839 6.9
7/4, 8/7 0.871 7.2
27/17, 34/27 0.910 7.5
23/12, 24/23 0.953 7.9
31/19, 38/31 0.962 7.9
25/18, 36/25 0.980 8.1
25/17, 34/25 1.005 8.3
3/2, 4/3 1.075 8.9
25/23, 46/25 1.101 9.1
23/14, 28/23 1.158 9.6
25/21, 42/25 1.184 9.8
27/23, 46/27 1.197 9.9
29/21, 42/29 1.221 10.1
9/7, 14/9 1.280 10.6
29/23, 46/29 1.303 10.7
19/17, 34/19 1.382 11.4
29/18, 36/29 1.425 11.8
31/26, 52/31 1.478 12.2
17/15, 30/17 1.495 12.3
5/4, 8/5 1.565 12.9
27/20, 40/27 1.661 13.7
13/11, 22/13 1.699 14.0
15/14, 28/15 1.769 14.6
29/15, 30/29 1.915 15.8
21/16, 32/21 1.946 16.1
17/9, 18/17 1.985 16.4
23/16, 32/23 2.029 16.7
25/24, 48/25 2.055 17.0
23/17, 34/23 2.107 17.4
17/11, 22/17 2.122 17.5
9/8, 16/9 2.151 17.7
21/17, 34/21 2.189 18.1
25/14, 28/25 2.259 18.6
27/19, 38/27 2.291 18.9
31/17, 34/31 2.344 19.3
27/14, 28/27 2.355 19.4
25/19, 38/25 2.387 19.7
29/25, 50/29 2.404 19.8
19/13, 26/19 2.440 20.1
29/27, 54/29 2.500 20.6
17/10, 20/17 2.570 21.2
15/8, 16/15 2.640 21.8
19/15, 30/19 2.877 23.7
27/22, 44/27 3.032 25.0
17/12, 24/17 3.060 25.2
25/22, 44/25 3.128 25.8
25/16, 32/25 3.130 25.8
27/16, 32/27 3.226 26.6
31/27, 54/31 3.254 26.8
17/14, 28/17 3.264 26.9
31/25, 50/31 3.349 27.6
19/18, 36/19 3.367 27.8
29/17, 34/29 3.410 28.1
23/19, 38/23 3.489 28.8
21/19, 38/21 3.571 29.5
15/11, 22/15 3.617 29.8
17/13, 26/17 3.822 31.5
31/30, 60/31 3.839 31.7
19/10, 20/19 3.952 32.6
11/9, 18/11 4.107 33.9
17/16, 32/17 4.135 34.1
23/22, 44/23 4.229 34.9
21/11, 22/21 4.311 35.6
31/18, 36/31 4.329 35.7
19/12, 24/19 4.442 36.6
31/23, 46/31 4.451 36.7
31/21, 42/31 4.533 37.4
19/14, 28/19 4.646 38.3
11/10, 20/11 4.693 38.7
27/26, 52/27 4.731 39.0
29/19, 38/29 4.791 39.5
25/13, 26/25 4.827 39.8
31/20, 40/31 4.915 40.5
11/6, 12/11 5.183 42.8
15/13, 26/15 5.317 43.9
11/7, 14/11 5.387 44.4
31/24, 48/31 5.404 44.6
19/16, 32/19 5.517 45.5
29/22, 44/29 5.532 45.6
31/28, 56/31 5.609 46.3
31/29, 58/31 5.754 47.5
13/9, 18/13 5.807 47.9
23/13, 26/23 5.929 48.9
21/13, 26/21 6.011 49.6
11/8, 16/11 6.258 51.6
13/10, 20/13 6.392 52.7
31/16, 32/31 6.480 53.5
13/12, 24/13 6.882 56.8
13/7, 14/13 7.086 58.5
29/26, 52/29 7.231 59.7
13/8, 16/13 7.957 65.6

Octave stretch or compression

99edo's approximations of harmonics 3, 5, and 7 can all be improved if slightly compressing the octave is acceptable, using tunings such as 157edt or 256ed6. 157edt is especially performant if the 13-limit of the 99ef val is intended, but the 7-limit part is overcompressed, for which the milder 256ed6 is a better choice.

If the 13-limit patent val is intended, then little to no compression, or even stretch, might be serviceable, such as in 567zpi.

Scales

Main article: List of MOS scales in 99edo

Instruments

Skip fretting

Skip fretting system 99 6 11 is a skip fretting system for 99edo. The frets correspond to 16.5edo (33ed4). All intervals are for 7-string guitar.

Harmonics

1/1: string 2 open

2/1: string 5 fret 11

3/2: string 4 fret 6

5/4 is not easily accessible, but the next-best approximation is at string 5 open.

7/4: string 6 fret 6

11/8: string 5 fret 2

13/8: string 5 fret 6

Keyboards

Lumatone mappings for 99edo are now available.

Music

Bryan Deister
Mundoworld
Gene Ward Smith

See also

  • 58edf – relative edf
  • 157edt – relative edt
  • 87edo, 94edo, 111edo – similarly sized edos all with consistency in higher harmonics.
  • 198edo, the half-sized edo to reconcile the mappings of 11 and 13.
  • 105edo, a similarly sized edo that supports meantone, septimal meantone, undecimal meantone, and grosstone
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