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

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) 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):

The7-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, 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, the "normal comma", represents the following 7-limit ratios:

• 64/63
• 81/80

3\99, the "normal diesis", 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