99edo: Difference between revisions

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'''99edo''' is an equal tuning with steps of size 12.12... cents. It is known for being very accurate in the 7-limit.
'''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 ==
== Theory ==
=== Prime approximations ===
=== 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.
{{Harmonics in ED|99}}
{{Harmonics in ED|99}}
=== 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.
{{Databox
|collapse=true
|title=The 7-prime-limited 45-odd-limit, by 99edo mapping (SW3 format)
|text=
<pre>
(*
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
</pre>
}}
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):
{{Databox
|collapse=true
|title=The 7-prime-limited 567-odd-limit, by 99edo mapping (SW3 format)
|text=
<pre>
(* 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;
</pre>
}}
=== Edostep interpretations ===
=== Edostep interpretations ===
1\99 represents the following 7-limit ratios:
1\99 = 12.1c, the "normal kleisma", represents the following 7-limit ratios:
* 126/125
* 126/125, the difference between 12/7 and (6/5)^3
* 225/224
* 225/224, the difference between 9/7 and 32/25
* 245/243
* 245/243, the difference between (9/7)^2 and 5/3
* 1029/1024
* 1029/1024, the difference between (8/7)^3 and 3/2
* 1728/1715
* 1728/1715, the difference between 8/5 and (7/6)^3
* 2048/2025
* 2048/2025, the difference between (16/15)^2 and 9/8, and the difference between (45/32)^2 and 2/1
* 4000/3969
* 4000/3969, the difference between a stack of three 10/9's and a stack of two 7/6's


2\99 represents the following 7-limit ratios:
2\99 = 24.2c, the "normal comma", represents the following 7-limit ratios:
* 64/63
* 64/63
* 81/80
* 81/80
3\99 = 36.3c, the "normal diesis", represents the following 7-limit ratios:
* 49/48
* 50/49
* 128/125


=== Temperaments ===
=== Temperaments ===
Line 24: Line 214:
* [[Ennealimmal]]
* [[Ennealimmal]]
* [[Wurschmidt]]
* [[Wurschmidt]]
=== Derivation ===
== 99edo on a Lumatone ==
Todo: Derive 99edo from
=== Diatonic ===
* Didacus
Due to the size of the edo, a standard diatonic mapping will miss a large fraction of the notes.
* Aberschismic
{{Lumatone edo mapping|n=99|start=6|xstep=17|ystep=-10}}
* Don't temper out 81/80, 64/63, or 128/125
 
* Ennealimmal
=== 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 ====
{{Lumatone edo mapping|n=99|start=79|xstep=8|ystep=1}}
==== 12L 3s ====
{{Lumatone edo mapping|n=99|start=12|xstep=8|ystep=-7}}
 
=== 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/Hemiwur ====
Division by 2 to get 16\99 yields [[Hemimean_clan#Hemiwürschmidt|Hemiwürschmidt/Würschmidt/Hemiwur]] with a slightly flat septimal) middle whole tone ~[[28/25]] for the divided generator, with a scale [[6L&nbsp;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}}
 
==== 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&nbsp;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).
{{Lumatone edo mapping|n=99|start=12|xstep=8|ystep=-7}}
 
=== 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.
{{Lumatone edo mapping|n=99|start=93|xstep=7|ystep=-2}}
 
=== Ennealimmal (currently untested) ===
This mapping uses [[Ennealimmal]] temperament; it maps one axis to 1\9 and one axis to ~21/20, an Ennealimmal generator.
{{Lumatone edo mapping|n=99|start=0|xstep=11|ystep=-7}}


{{Navbox EDO}}
{{Navbox EDO}}
{{Cat|edos}}
{{Cat|edos}}

Latest revision as of 01:52, 11 April 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.

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

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

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

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
13
30
47
64
81
0
17
34
51
68
85
3
20
37
54
71
88
6
23
7
24
41
58
75
92
10
27
44
61
78
95
13
30
47
64
81
96
14
31
48
65
82
0
17
34
51
68
85
3
20
37
54
71
88
6
23
4
21
38
55
72
89
7
24
41
58
75
92
10
27
44
61
78
95
13
30
47
64
81
93
11
28
45
62
79
96
14
31
48
65
82
0
17
34
51
68
85
3
20
37
54
71
88
6
23
18
35
52
69
86
4
21
38
55
72
89
7
24
41
58
75
92
10
27
44
61
78
95
13
30
47
64
81
59
76
93
11
28
45
62
79
96
14
31
48
65
82
0
17
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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

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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/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).

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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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View • Talk • EditEqual temperaments
EDOs
Macrotonal 5 • 7 • 8 • 9 • 10 • 11
12-23 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23
24-35 24 • 25 • 26 • 27 • 29 • 31 • 32 • 34 • 35
36-47 36 • 37 • 39 • 40 • 41 • 43 • 44 • 45 • 46 • 47
48-59 48 • 50 • 51 • 53 • 54 • 56 • 57 • 58
60-71 60 • 63 • 64 • 65 • 67 • 68 • 70
72-83 72 • 77 • 80 • 81
84-95 84 • 87 • 89 • 90 • 93 • 94
Large EDOs 99 • 104 • 106 • 111 • 118 • 130 • 140 • 152 • 159 • 171 • 217 • 224 • 239 • 270 • 306 • 311 • 612 • 665
Nonoctave equal temperaments
Tritave 4 • 9 • 13 • 17 • 26 • 39
Fifth 8 • 9 • 11 • 20
Other
Retrieved from "https://xenreference.com/wiki/index.php?title=99edo&oldid=6112"