99edo

99edo is an equal tuning with steps of size 12.12... cents. It is arguably the edo below 100 that most faithfully models 7-limit just intonation.

Theory

Prime approximations

99edo's prime mappings usually tend sharp. The sharp 3, in particular, is a tendency of Aberschismic temperament.

Approximation of prime harmonics in 99edo

Harmonic		2	3	5	7	11	13	17	19	23	29	31
Error	Absolute (¢)	0.0	+1.1	+1.6	+0.9	-5.9	-4.2	+4.1	+5.5	+2.0	+0.7	-5.6
	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)

7-prime-limited odd-limit analysis

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, unlike all previous edos:

36/35; 28/27; 25/24; 21/20; 16/15; 15/14; 27/25; 35/32; 10/9; 28/25; 9/8; 8/7; 7/6; 32/27; 25/21; 6/5; 56/45; 5/4; 32/25; 9/7; 35/27; 21/16; 4/3; 27/20; 48/35; 7/5; 45/32; 64/45; 10/7; 35/24; 40/27; 3/2; 32/21; 54/35; 14/9; 25/16; 8/5; 45/28; 5/3; 42/25; 27/16; 12/7; 7/4; 16/9; 25/14; 9/5; 64/35; 50/27; 28/15; 15/8; 40/21; 48/25; 27/14; 35/18; 2/1

The 7-prime-limited 49-odd-limit is where non-distinctness first shows up: namely, T49/48 = T50/49 (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

225/224; 126/125; 245/243; 81/80; 64/63; 50/49; 49/48; 128/125; 525/512; 36/35; 250/243; 405/392; 28/27; 25/24; 256/245; 392/375; 360/343; 21/20; 256/243; 135/128; 200/189; 343/324; 16/15; 15/14; 343/320; 27/25; 175/162; 243/224; 160/147; 375/343; 35/32; 192/175; 54/49; 441/400; 448/405; 567/512; 10/9; 125/112; 384/343; 28/25; 9/8; 640/567; 500/441; 567/500; 245/216; 256/225; 8/7; 343/300; 225/196; 147/128; 280/243; 125/108; 512/441; 400/343; 7/6; 75/64; 288/245; 147/125; 405/343; 189/160; 32/27; 25/21; 343/288; 448/375; 6/5; 135/112; 98/81; 243/200; 175/144; 128/105; 49/40; 315/256; 216/175; 100/81; 243/196; 56/45; 5/4; 432/343; 63/50; 512/405; 81/64; 80/63; 343/270; 245/192; 32/25; 9/7; 162/125; 35/27; 125/96; 64/49; 98/75; 450/343; 21/16; 324/245; 250/189; 4/3; 75/56; 343/256; 168/125; 27/20; 256/189; 200/147; 49/36; 512/375; 175/128; 48/35; 343/250; 135/98; 441/320; 112/81; 243/175; 480/343; 7/5; 45/32; 800/567; 343/243; 486/343; 567/400; 64/45; 10/7; 343/240; 350/243; 81/56; 640/441; 196/135; 500/343; 35/24; 256/175; 375/256; 72/49; 147/100; 189/128; 40/27; 125/84; 512/343; 112/75; 3/2; 189/125; 245/162; 32/21; 343/225; 75/49; 49/32; 192/125; 54/35; 125/81; 14/9; 25/16; 384/245; 540/343; 63/40; 128/81; 405/256; 100/63; 343/216; 8/5; 45/28; 392/243; 81/50; 175/108; 512/315; 80/49; 105/64; 288/175; 400/243; 81/49; 224/135; 5/3; 375/224; 576/343; 42/25; 27/16; 320/189; 686/405; 250/147; 245/144; 128/75; 12/7; 343/200; 441/256; 216/125; 243/140; 256/147; 392/225; 600/343; 7/4; 225/128; 432/245; 1000/567; 441/250; 567/320; 16/9; 25/14; 343/192; 224/125; 9/5; 1024/567; 405/224; 800/441; 49/27; 175/96; 64/35; 686/375; 147/80; 448/243; 324/175; 50/27; 640/343; 28/15; 15/8; 648/343; 189/100; 256/135; 243/128; 40/21; 343/180; 375/196; 245/128; 48/25; 27/14; 784/405; 243/125; 35/18; 1024/525; 125/64; 96/49; 49/25; 63/32; 160/81; 486/245; 125/63; 448/225; 2/1

Edostep interpretations

1\99 = 12.1c, the "normal kleisma", represents the following 7-limit ratios:

• 126/125
• 225/224
• 245/243
• 1029/1024
• 1728/1715
• 2048/2025
• 4000/3969

2\99 = 24.2c, the "normal comma", represents the following 7-limit ratios:

• 64/63
• 81/80

3\99 = 36.3c, the "normal diesis", represents the following 7-limit ratios:

• 49/48
• 50/49
• 128/125

Temperaments

99edo notably supports

• Aberschismic
• Didacus
• Ennealimmal
• Wurschmidt

Derivation

Main article: 99edo/Derivation

Diatonic

Due to the size of the edo, a standard diatonic mapping will miss a large fraction of the notes. Template:Lumatone EDO mapping

Misty

Keeping the same generator but dividing the period in three gives you Misty. The 3L 9s mapping covers nearly all the notes with the occasional skip, while the 12L 3s one does cover the whole gamut, but has a smaller range and a very lopsided step size.

Würschmidt and extensions thereof

The Würschmidt generator, which is the classic major third ~5/4 (near-just), is 32\99 in 99edo, so it is divisible by 2 or 4 but not by 3 (seen with 65edo.

Hemiwürschmidt/Würschmidt/Hemiwur

Division by 2 to get 16\99 yields Hemiwürschmidt/Würschmidt/Hemiwur with a slightly flat septimal) middle whole tone ~28/25 for the divided generator, with a scale 6L 1s (16:3 step ratio). This mapping only splits the Würschmidt in half to get greater range (over four octaves) than when splitting it in quarters, but at the cost of missing many notes in each octave. Despite the missing notes, Bryan Deister has demonstrated this mapping in 99edo waltz (2025). Template:Lumatone EDO mapping

Würschmidt unnamed extension with generator divided by 4

Division of the generator by 2 again (for 4 overall) yields a further extension that uses this mapping's rightward generator 8\99 as a slightly sharp ptolemaic chromatic semitone (major limma) ~135/128, with a scale 12L 3s (8:1 step ratio), implying that the octave is also divided into three equal parts. As befits Würschmidt, eight classic major thirds (32\65) make a near-just 6th harmonic ~6/1. The range is just over two octaves, and the octaves slant up mildly, now with no missing notes and some repeated notes to ease vertical wraparound. Compared to the Amity (Amicable) mapping with split period, this mapping is more lopsided with the hard scale step ratio, but on the other hand gets some consonant ratios with only a few generator steps. Bryan Deister has experimented with this mapping, but no demonstration video is available yet (as of 2025-07-24). Template:Lumatone EDO mapping

Amity (Amicable) (currently untested, and shown for comparison)

Since 99edo falls on the Amity temperament line, it is tempting to use the generator 7\99 functioning as a near-just ~21/20 as in the Amicable extension, but with the octave split into three equal parts, giving a 12L 3s scale with 7:5 step ratio. The range is a bit over two octaves, slanting up mildly, with no missed notes and a few repeated notes to assist with vertical wraparounds. Relative to the mappings for Würschmidt and its extensions, the Amicable mapping has the advantage that the layout is less lopsided, but the disadvantage that stacking generators does not hit good ratios at low numbers of generators. Template:Lumatone EDO mapping

Ennealimmal (currently untested)

This mapping uses ennealimmal temperament; it maps one axis to 1\9 and one axis to ~21/20, an ennealimmal generator. Template:Lumatone EDO mapping