99edo: Difference between revisions
From Xenharmonic Reference
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* 50/49 | * 50/49 | ||
* 128/125 | * 128/125 | ||
==== Intervals made equidistant by 99edo ==== | |||
Runs of 7-prime-limited 45-odd-limit intervals separated by 1\99: | |||
# 36/35 ↔<sub>a</sub> 28/27 ↔<sub>b</sub> 25/24 ↔<sub>c</sub> 21/20 | |||
# 16/15 ↔<sub>b</sub> 15/14 ↔<sub>c</sub> 27/25 | |||
# 10/9 ↔<sub>c</sub> 28/25 ↔<sub>b</sub> 9/8 | |||
# 32/27 ↔<sub>b</sub> 25/21 ↔<sub>c</sub> 6/5 | |||
# 32/25 ↔<sub>b</sub> 9/7 ↔<sub>a</sub> 35/27 | |||
# 25/18 ↔<sub>c</sub> 7/5 ↔<sub>b</sub> 45/32 ↔<sub>d</sub> 64/45 ↔<sub>b</sub> 10/7 ↔<sub>c</sub> 36/25 | |||
The separating intervals (all equated to 1\99): | |||
# ↔<sub>a</sub> = 245/243, the [[Sensamagic]] comma | |||
# ↔<sub>b</sub> = 225/224, the [[Marvel]] comma | |||
# ↔<sub>c</sub> = 126/125 | |||
# ↔<sub>d</sub> = 2048/2025, the [[Diaschismic|diaschisma]] | |||
Runs of intervals separated by 2\99: | |||
# 28/27 ↔<sub>e</sub> 21/20 ↔<sub>f</sub> 16/15 ↔<sub>e</sub> 27/25 ↔<sub>g</sub> 35/32 ↔<sub>f</sub> 10/9 ↔<sub>e</sub> 9/8 ↔<sub>f</sub> 8/7 | |||
# 7/6 ↔<sub>f</sub> 32/27 ↔<sub>e</sub> 6/5 | |||
# 32/25 ↔<sub>g</sub> 35/27 ↔<sub>e</sub> 21/16 ↔<sub>f</sub> 4/3 ↔<sub>e</sub> 27/20 ↔<sub>f</sub> 48/35 ↔<sub>g</sub> 25/18 ↔<sub>e</sub> 45/32 ↔<sub>f</sub> 10/7 | |||
The separating intervals (all equated to 2\99): | |||
# ↔<sub>e</sub> = 81/80 | |||
# ↔<sub>f</sub> = 64/63 | |||
# ↔<sub>g</sub> = 875/864 | |||
Runs of intervals separated by 3\99: | |||
# 36/35 ↔<sub>h</sub> 21/20 ↔<sub>i</sub> 15/14 ↔<sub>h</sub> 35/32 ↔<sub>j</sub> 28/25 ↔<sub>i</sub> 8/7 ↔<sub>h</sub> 7/6 ↔<sub>i</sub> 25/21 | |||
# 48/35 ↔<sub>h</sub> 7/5 ↔<sub>i</sub> 10/7 ↔<sub>h</sub> 35/24 | |||
The separating intervals (all equated to 3\99): | |||
# ↔<sub>h</sub> = 49/48 | |||
# ↔<sub>i</sub> = 50/49 | |||
# ↔<sub>j</sub> = 128/125 | |||
=== Temperaments === | === Temperaments === | ||
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* [[Wurschmidt]] | * [[Wurschmidt]] | ||
=== | ==== Tempering the 7-prime-limited 45-odd-limit ==== | ||
99edo can be derived from any three of the following non-JI identities between melodic steps in the 7-prime-limited 35-odd-limit, namely 9/8, 28/25, 10/9, 35/32, 27/25, 15/14, 16/15, 21/20, 25/24, 28/27, 36/35: | 99edo can be derived from any three of the following non-JI identities between melodic steps in the 7-prime-limited 35-odd-limit, namely 9/8, 28/25, 10/9, 35/32, 27/25, 15/14, 16/15, 21/20, 25/24, 28/27, 36/35: | ||
(~= means "is equated to", = means "equal in JI") | (~= means "is equated to", = means "equal in JI") | ||
# (36/35)^4 ~= 28/25 | # (36/35)^4 ~= 28/25 | ||
# | # the following equivalent relations: | ||
# 25/24 * 15/14 ~= 28/25 = 21/20 * 16/15 | #* 28/27 * 25/24 ~= 27/25 | ||
# (28/27)^3 ~= 10/9 | #* 27/25 * 36/35 ~= 10/9 | ||
# 16/15 * 36/35 ~= 35/32 = 25/24 * 21/20 | # the following equivalent relations: | ||
#* 25/24 * 15/14 ~= 28/25 = 21/20 * 16/15 | |||
#* 9/8 * 10/9 ~= (28/25)^2 | |||
#* 27/25 * 16/15 ~= (15/14)^2 | |||
#* 21/20 * 28/27 ~= (25/24)^2 | |||
# the following equivalent relations: | |||
#* 16/15 * 28/27 ~= (21/20)^2 | |||
#* 21/20 * 27/25 ~= (16/15)^2 | |||
# the following equivalent relations: | |||
#* (28/27)^3 ~= 10/9 | |||
#* (28/27)^2 ~= 36/35 * 25/24 = 15/14 | |||
# the following equivalent relations: | |||
#* 16/15 * 36/35 ~= 35/32 = 25/24 * 21/20 | |||
#* 35/32 * 25/24 ~= 16/15 * 15/14 | |||
The first two identities define Ennealimmal temperament; identity 3 defines Didacus; identity 4 defines Aberschismic. | |||
Note that the steps are related in JI via: | |||
* 9/8 = 36/35 * 35/32 | |||
* 28/25 = 27/25 * 28/27 = 21/20 * 16/15 | |||
* 35/32 = 21/20 * 25/24 | |||
* 27/25 = 36/35 * 21/20 | |||
* 15/14 = 36/35 * 25/24 | |||
* 16/15 = 36/35 * 28/27 | |||
=== Derivation === | |||
The following square-superparticulars: | |||
* S4 ('''16'''/15) | |||
* S5 (25/'''24''') | |||
* S6 ('''36'''/35) | |||
* S7 (49/'''48''') | |||
can be placed in the ratio of 1/16:1/24:1/36:1/48 - that is, 9:6:4:3, which specifies 99edo, much analogously to how [[34edo]] in the 5-limit is specified by S3, S4, and S5. | |||
== 99edo on a Lumatone == | == 99edo on a Lumatone == | ||
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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]]. | 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 | ==== Hemiwürschmidt/Würschmidt ==== | ||
Division by 2 to get 16\99 yields [[ | Division by 2 to get 16\99 yields [[Didacus|Hemiwürschmidt]] 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 [https://www.youtube.com/shorts/p9OUaFuTUek ''99edo waltz''] (2025). | ||
{{Lumatone edo mapping|n=99|start=40|xstep=16|ystep=-13}} | {{Lumatone edo mapping|n=99|start=40|xstep=16|ystep=-13}} | ||
Latest revision as of 03:15, 18 May 2026
99edo is an equal tuning with steps of size 12.12... cents. It is the edo below 100 that most faithfully models 7-limit just intonation.
Theory
Prime approximations
99edo's prime mappings usually tend sharp. The 3 is sharp for Ennealimmal temperament (in fact, only 126edo and 27edo are sharper Ennealimmal edo tunings) and shows the tendency of Aberschismic temperament to have sharp 3.
| 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
Unlike all previous edos, 99edo is distinctly consistent and monotone up to the 7-prime-limited 45-odd-limit, i.e.
- when tempered using the patent val, the relative sizes of any two intervals are never conflated or reversed
- the direct approximation is equal to the approximation given by stacking patent val prime approximations, thus every interval has absolute error < 6.06c. In fact, only 6 intervals on this list have more than 25% (+-3.03c) relative error.
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;
Edostep interpretations
1\99 = 12.1c, the "normal kleisma", represents the following 7-limit ratios:
- 126/125, the difference between 12/7 and (6/5)^3
- 225/224, the difference between 9/7 and 32/25
- 245/243, the difference between (9/7)^2 and 5/3
- 1029/1024, the difference between (8/7)^3 and 3/2
- 1728/1715, the difference between 8/5 and (7/6)^3
- 2048/2025, the difference between (16/15)^2 and 9/8, and the difference between (45/32)^2 and 2/1
- 4000/3969, the difference between a stack of three 10/9's and a stack of two 7/6's
- 32805/32768, the schisma, the difference between a stack of two 81/64's and 8/5
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
Intervals made equidistant by 99edo
Runs of 7-prime-limited 45-odd-limit intervals separated by 1\99:
- 36/35 ↔a 28/27 ↔b 25/24 ↔c 21/20
- 16/15 ↔b 15/14 ↔c 27/25
- 10/9 ↔c 28/25 ↔b 9/8
- 32/27 ↔b 25/21 ↔c 6/5
- 32/25 ↔b 9/7 ↔a 35/27
- 25/18 ↔c 7/5 ↔b 45/32 ↔d 64/45 ↔b 10/7 ↔c 36/25
The separating intervals (all equated to 1\99):
- ↔a = 245/243, the Sensamagic comma
- ↔b = 225/224, the Marvel comma
- ↔c = 126/125
- ↔d = 2048/2025, the diaschisma
Runs of intervals separated by 2\99:
- 28/27 ↔e 21/20 ↔f 16/15 ↔e 27/25 ↔g 35/32 ↔f 10/9 ↔e 9/8 ↔f 8/7
- 7/6 ↔f 32/27 ↔e 6/5
- 32/25 ↔g 35/27 ↔e 21/16 ↔f 4/3 ↔e 27/20 ↔f 48/35 ↔g 25/18 ↔e 45/32 ↔f 10/7
The separating intervals (all equated to 2\99):
- ↔e = 81/80
- ↔f = 64/63
- ↔g = 875/864
Runs of intervals separated by 3\99:
- 36/35 ↔h 21/20 ↔i 15/14 ↔h 35/32 ↔j 28/25 ↔i 8/7 ↔h 7/6 ↔i 25/21
- 48/35 ↔h 7/5 ↔i 10/7 ↔h 35/24
The separating intervals (all equated to 3\99):
- ↔h = 49/48
- ↔i = 50/49
- ↔j = 128/125
Temperaments
99edo notably supports
Tempering the 7-prime-limited 45-odd-limit
99edo can be derived from any three of the following non-JI identities between melodic steps in the 7-prime-limited 35-odd-limit, namely 9/8, 28/25, 10/9, 35/32, 27/25, 15/14, 16/15, 21/20, 25/24, 28/27, 36/35:
(~= means "is equated to", = means "equal in JI")
- (36/35)^4 ~= 28/25
- the following equivalent relations:
- 28/27 * 25/24 ~= 27/25
- 27/25 * 36/35 ~= 10/9
- the following equivalent relations:
- 25/24 * 15/14 ~= 28/25 = 21/20 * 16/15
- 9/8 * 10/9 ~= (28/25)^2
- 27/25 * 16/15 ~= (15/14)^2
- 21/20 * 28/27 ~= (25/24)^2
- the following equivalent relations:
- 16/15 * 28/27 ~= (21/20)^2
- 21/20 * 27/25 ~= (16/15)^2
- the following equivalent relations:
- (28/27)^3 ~= 10/9
- (28/27)^2 ~= 36/35 * 25/24 = 15/14
- the following equivalent relations:
- 16/15 * 36/35 ~= 35/32 = 25/24 * 21/20
- 35/32 * 25/24 ~= 16/15 * 15/14
The first two identities define Ennealimmal temperament; identity 3 defines Didacus; identity 4 defines Aberschismic.
Note that the steps are related in JI via:
- 9/8 = 36/35 * 35/32
- 28/25 = 27/25 * 28/27 = 21/20 * 16/15
- 35/32 = 21/20 * 25/24
- 27/25 = 36/35 * 21/20
- 15/14 = 36/35 * 25/24
- 16/15 = 36/35 * 28/27
Derivation
The following square-superparticulars:
- S4 (16/15)
- S5 (25/24)
- S6 (36/35)
- S7 (49/48)
can be placed in the ratio of 1/16:1/24:1/36:1/48 - that is, 9:6:4:3, which specifies 99edo, much analogously to how 34edo in the 5-limit is specified by S3, S4, and S5.
99edo on a Lumatone
Diatonic
Due to the size of the edo, a standard diatonic mapping will miss a large fraction of the notes.
6
23
13
30
47
64
81
3
20
37
54
71
88
6
23
10
27
44
61
78
95
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30
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81
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34
51
68
85
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20
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23
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24
41
58
75
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31
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85
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23
4
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76
Misty
Keeping the 3/2 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.
3L 9s
79
87
88
96
5
13
21
89
97
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98
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79
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9
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38
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98
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12L 3s
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Würschmidt
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
Division by 2 to get 16\99 yields Hemiwürschmidt 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).
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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 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).
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Amity (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, but with the octave split into three equal parts, giving a 12L3s 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 Amity 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.
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Ennealimmal (currently untested)
This mapping uses Ennealimmal temperament; it maps one axis to 1\9 and one axis to ~21/20, an Ennealimmal generator.
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