99edo

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

Theory

Prime approximations

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. relative sizes of any two intervals are never conflated or reversed) in the following 7-prime-limited odd-limits:

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

The 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 in these odd-limits: (todo)

Edostep interpretations

1\99 represents the following 7-limit ratios:

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

2\99 represents the following 7-limit ratios:

• 64/63
• 81/80

3\99 represents the following 7-limit ratios:

• 49/48
• 50/49
• 128/125

Temperaments

99edo notably supports

• Aberschismic
• Didacus
• Ennealimmal
• Wurschmidt

Derivation

Todo: Derive 99edo from

• Didacus
• Aberschismic
• Don't temper out 81/80, 64/63, or 128/125
• Ennealimmal