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 are never conflated or reversed) in the following 7-prime-limited odd-limits:

• 7-OL: 1/1 8/7 7/6 6/5 5/4 4/3 7/5 10/7 3/2 8/5 5/3 7/4
• 9-OL: add 10/9 9/8 9/7 14/9 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. 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