99edo: Difference between revisions

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== Theory ==

99edo is a very strong [[7-limit]] (and [[9-odd-limit]]) tuning, with a sound defined by the slight sharpness (1.1, 1.6, 0.9 cents) of its [[3/1|3]], [[5/1|5]], and [[7/1|7]]. As an equal temperament, it [[tempering out|tempers out]] 393216/390625 ([[würschmidt comma]]) and 1600000/1594323 ([[amity comma]]) in the [[5-limit]]; 2401/2400 ([[2401/2400|breedsma]]), 3136/3125 ([[hemimean comma]]), and 4375/4374 ([[4375/4374|ragisma]]) in the [[7-limit]], [[support]]ing [[hemififths]], [[amity]], [[parakleismic]], [[hemiwürschmidt]] and [[ennealimmal]] temperaments, and is pretty well a perfect tuning for [[hendecatonic]] temperament.

99edo is a very strong [[7-limit]] (and [[9-odd-limit]]) tuning, with a sound defined by the slight sharpness (1.1, 1.6, 0.9 cents) of its [[3/1|3]], [[5/1|5]], and [[7/1|7]]. As an equal temperament, it [[tempering out|tempers out]] 393216/390625 ([[würschmidt comma]]) and 1600000/1594323 ([[amity comma]]) in the [[5-limit]]; 5120/5103 ([[5120/5103|argent comma]]), 2401/2400 ([[2401/2400|breedsma]]), 3136/3125 ([[hemimean comma]]), and 4375/4374 ([[4375/4374|ragisma]]) in the [[7-limit]], [[support]]ing [[hemififths]], [[amity]], [[parakleismic]], [[hemiwürschmidt]] and [[ennealimmal]] temperaments, and is pretty well a perfect tuning for [[hendecatonic (temperament)|hendecatonic]] temperament.

Extending it to the [[11-limit]] requires choosing which mapping one wants to use, as both are nearly equally far off the mark. Using the {{val| 99 157 230 278 '''343''' }} (99e) val, it tempers out [[243/242]], [[441/440]], [[540/539]] and [[896/891]], and is an excellent tuning for the 11-limit version of hemififths temperament. Using the [[patent val]], 99edo is the [[optimal patent val]] for the rank-4 temperament tempering out [[121/120]]; zeus, the rank-3 temperament tempering out 121/120 and [[176/175]]; [[hemiwür]], one of the rank-2 11-limit extensions of hemiwürschmidt; and [[hitchcock]] (an 11-limit amity extension), the rank-2 temperament which also tempers out [[2200/2187]]. The same can be said of the mapping for [[13/1|13]], with the 99ef val tempering out [[144/143]], [[196/195]], 352/351 and [[364/363]], and its patent val tempering out [[169/168]], [[351/350]] and [[352/351]]. Hence 99edo, in spite of the fact that it tunes 11 and 13 relatively badly, is an important 13-limit tuning in more than one way.

Being a [[zeta peak edo]], 99edo is also a very strong no-11 no-13 system, where it is consistent to the [[29-odd-limit]] with a sharp tendency. This favors the sharp mapping of 11 and 13, and allows these relatively weak approximations to somewhat blend with the rest for a full [[29-limit]] (or [[31-limit]], using the sharp-tending 99efk val) temperament.

Being a [[zeta peak edo]], 99edo is also a very strong no-11 no-13 system, where it is consistent to the [[29-odd-limit]] with a sharp tendency. This favors the sharp mapping of 11 and 13, and allows these relatively weak approximations to somewhat blend with the rest for a full [[29-limit]] (or [[31-limit]], using the sharp-tending 99efk val) temperament. In fact, the 99efk val is the first to achieve [[diamond monotone]] in the [[31-odd-limit]], though it fails in the [[33-odd-limit]] due to mapping [[33/32]] to 5 steps, while [[32/31]] is mapped to 4 steps.

One step of 99edo is close to [[144/143]], the grossma. Unfortunately, neither 99ef nor the patent val map it consistently, though [[198edo]] does.

=== Prime harmonics ===

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=== Subsets and supersets ===

Since 99 factors into {{nowrap| 32 × 11 }}, 99edo has subset edos {{EDOs| 3, 9, 11, and 33 }}.

Since 99 factors into primes as {{nowrap| 32 × 11 }}, 99edo has subset edos {{EDOs| 3, 9, 11, and 33 }}. Splitting 99edo's step in half yields [[198edo]], correcting prime 11, slightly improving prime 13, and aligning both 11 and 13 with the sharp tunings of the lower odd primes. Because of this, 198edo can be seen as a complex yet notable true full 13-limit tuning.

== Intervals ==

== Notation ==

[[Stein–Zimmermann–Gould notation]] uses sharps and flats combined with quartertone accidentals and arrows:

=== Kite's ups and downs notation ===

99edo can also be notated with [[Kite's ups and downs notation|Kite's ups and downs]]. Note that quip (quintuple-up) is the same as quudsharp (quadruple-down sharp) and that quid (quintuple-down) is the same as quupflat (quadruple-up flat):

== Approximation to JI ==

=== 7-prime-limited odd-limit analysis ===

Unlike all previous edos, 99edo is ''distinctly'' [[consistent]] and monotone (i.e. when tempered using the patent val, the relative sizes of any two intervals are never conflated ''or'' reversed) up to the 7-prime-limited 45-odd-limit:

{{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>

}}

=== 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):

# ↔a = 245/243, the [[sensamagic]] comma

# ↔b = 225/224, the [[marvel]] comma

# ↔c = 126/125

# ↔d = 2048/2025, the [[Diaschismic|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):

# ↔e = 81/80

# ↔f = 64/63

# ↔g = 875/864, the keema

=== Interval mappings ===

{{Q-odd-limit intervals|99.1|apx=val|header=none|tag=none|title=15-odd-limit intervals by 99ef val mapping}}

== Regular temperament properties ==

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| 7.27

|}

* 99et is lower in relative error than any previous equal temperaments in the 7-limit. Not until 171 do we find a better equal temperament in terms of either absolute error or relative error.

* 99et is lower in relative error than any previous equal temperaments in the 7-limit. Not until [[171edo|171]] do we find a better equal temperament in terms of either absolute error or relative error.

=== Rank-2 temperaments ===

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Line 345:

| 339.394

| 128/105

| [[Amity]] (99ef) / hitchcock (99)

| [[Amity]] (99ef) / stalagmite (99ef) / hitchcock (99)

|-

| 1

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| 48.485

| 36/35

| [[Ennealimmal]] (~~99e~~) / ~~ennealimmia~~ (99) / ennealimnic (99ef) / ennealim (99e) / ennealiminal (99)

| [[Ennealimmal]] / enneabiotic (99ef) / ennealympic (99) / ennealimnic (99ef) / ennealim (99e) / ennealiminal (99)

|-

| 11

Line 177:

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| 496.970 (48.485)

| 4/3 (36/35)

| [[Hendecatonic]]

| [[Hendecatonic (temperament)|Hendecatonic]]

|}

<nowiki/>* [[Normal ~~lists~~|Octave-reduced form]], reduced to the first half-octave, and [[~~Normal lists~~|minimal form]] in parentheses if distinct

<nowiki/>* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct

== Octave stretch or compression ==

99edo's approximations of harmonics 3, 5, and 7 can all be improved if slightly [[stretched and compressed tuning|compressing the octave]] is acceptable, using tunings such as [[157edt]] or [[256ed6]]. 157edt is especially performant if the 13-limit of the 99ef val is intended, but the 7-limit part is overcompressed, for which the milder 256ed6 is a better choice~~. If the 13-limit patent val is intended, then little to no compression, or even stretch, might be serviceable~~.

99edo's approximations of harmonics 3, 5, and 7 can all be improved if slightly [[stretched and compressed tuning|compressing the octave]] is acceptable, using tunings such as [[157edt]] or [[256ed6]]. 157edt is especially performant if the 13-limit of the 99ef val is intended, but the 7-limit part is overcompressed, for which the milder 256ed6 is a better choice.

~~What follows~~ is ~~a comparison of stretched- and compressed-octave 99edo tunings~~.

If the 13-limit patent val is intended, then little to no compression, or even stretch, might be serviceable, such as in [[zpi|567zpi]].

~~; [[zpi|567zpi]]~~

== Scales ==

* Step size: 12.138{{c}}, octave size: 1201.66{{c}}

Stretching the octave of 99edo by around 1.5{{c}} results in improved primes 11, 13, 17, and 19, but worse primes 2, 3, 5 and 7. This approximates all harmonics up to 16 within 5.54{{c}}. The tuning 567zpi does this.

~~{{Harmonics in cet|12.138|intervals~~=~~integer|columns~~=~~11|collapsed~~=~~true|title~~=~~Approximation of harmonics in 567zpi}}~~

{{~~Harmonics in cet|12.138|intervals=integer|columns=12|start=12~~|~~collapsed=true|title=Approximation~~ of ~~harmonics~~ in ~~567zpi (continued)}}~~

~~; [[WE|99et, 13-limit WE tuning]]~~

* Step size: 12.123{{c}}, octave size: 1200.18{{c}}

~~Stretching the octave of~~ 99edo by around a fifth of a cent results in improved primes 11 and 13, but worse primes 2, 3, 5 and 7. This approximates all harmonics up to 16 within 5.25{{c}}. Its 13-limit WE tuning and 13-limit [[TE]] tuning both do this.

~~{{Harmonics in cet|12.123|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 99et, 13-limit WE tuning}}~~

~~{{Harmonics in cet|12.123|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 99et, 13-limit WE tuning (continued)}}~~

~~; 99edo~~

* Step size: 12.121{{c}}, octave size: 1200.00{{c}}

~~Pure-octaves 99edo approximates all harmonics up to 16 within 5.86{{c}}.~~

~~{{Harmonics in equal|99|2|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 99edo}}~~

~~{{Harmonics in equal|99|2|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 99edo (continued)}}~~

~~; [[WE|99et, 7-limit WE tuning]] / [[256ed6]]~~

* Step size: 12.117{{c}}, octave size: 1199.58{{c}}

Compressing the octave of 99edo by around 0.6{{c}} results in improved primes 3, 5, 7 and 11, but worse primes 2 and 13. This approximates all harmonics up to 16 within 5.71{{c}}. Its 7-limit WE tuning and 7-limit [[TE]] tuning both do this. So does the tuning 256ed6 whose octave is identical within a thousandth of a cent.

~~{{Harmonics in cet|12.117|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 99et, 7-limit WE tuning}}~~

~~{{Harmonics in cet|12.117|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 99et, 7-limit WE tuning (continued)~~}}

~~; [[zpi|568zpi]]~~

* Step size: 12.115{{c}}, octave size: 1199.39{{c}}

Compressing the octave of 99edo by around 0.4{{c}} results in improved primes 3, 5, 7 and 11, but worse primes 2 and 13. This approximates all harmonics up to 16 within 5.68{{c}}. The tuning 568zpi does this.

~~{{Harmonics in cet|12.115|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 568zpi}}~~

~~{{Harmonics in cet|12.115|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 568zpi (continued)}}~~

~~; [[157edt]] / [[ed5|230ed5]]~~

* Step size: 12.114{{c}}, octave size: 1199.32{{c}}

Compressing the octave of 99edo by around 0.3{{c}} results in improved primes 3, 5, 7 and 11, but worse primes 2 and 13. This approximates all harmonics up to 16 within 5.44{{c}}. The tuning 157edt does this. So does 230ed5 whose octave is identical within a hundredth of a cent.

~~{{Harmonics in equal|157|3|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 157edt}}~~

~~{{Harmonics in equal|157|3|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 157edt (continued)}}~~

~~; [[58edf]]~~

* Step size: 12.103{{c}}, octave size: 1199.16{{c}}

Compressing the octave of 99edo by around 1{{c}} results in improved primes 11, 13, 17 and 19 but worse primes 2, 3, 5 and 7. This approximates all harmonics up to 16 within 5.98{{c}}. The tuning 58edf does this.

~~{{Harmonics in equal|58|3|2|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 58edf}}~~

~~{{Harmonics in equal|58|3|2|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 58edf (continued)}}~~

~~== Scales ==~~

* [[Tutone6]]

* [[Tutone7]]

@@ Line 3: / Line 3: @@
 == Theory ==
-edo is a very strong [[7-limit]] (and [[9-odd-limit]]) tuning, with a sound defined by the slight sharpness (1.1, 1.6, 0.9 cents) of its [[3/1|3]], [[5/1|5]], and [[7/1|7]]. As an equal temperament, it [[tempering out|tempers out]] 393216/390625 ([[würschmidt comma]]) and 1600000/1594323 ([[amity comma]]) in the [[5-limit]]; 2401/2400 ([[2401/2400|breedsma]]), 3136/3125 ([[hemimean comma]]), and 4375/4374 ([[4375/4374|ragisma]]) in the [[7-limit]], [[support]]ing [[hemififths]], [[amity]], [[parakleismic]], [[hemiwürschmidt]] and [[ennealimmal]] temperaments, and is pretty well a perfect tuning for [[hendecatonic]] temperament.
+edo is a very strong [[7-limit]] (and [[9-odd-limit]]) tuning, with a sound defined by the slight sharpness (1.1, 1.6, 0.9 cents) of its [[3/1|3]], [[5/1|5]], and [[7/1|7]]. As an equal temperament, it [[tempering out|tempers out]] 393216/390625 ([[würschmidt comma]]) and 1600000/1594323 ([[amity comma]]) in the [[5-limit]]; 5120/5103 ([[5120/5103|argent comma]]), 2401/2400 ([[2401/2400|breedsma]]), 3136/3125 ([[hemimean comma]]), and 4375/4374 ([[4375/4374|ragisma]]) in the [[7-limit]], [[support]]ing [[hemififths]], [[amity]], [[parakleismic]], [[hemiwürschmidt]] and [[ennealimmal]] temperaments, and is pretty well a perfect tuning for [[hendecatonic (temperament)|hendecatonic]] temperament.
 Extending it to the [[11-limit]] requires choosing which mapping one wants to use, as both are nearly equally far off the mark. Using the {{val| 99 157 230 278 '''343''' }} (99e) val, it tempers out [[243/242]], [[441/440]], [[540/539]] and [[896/891]], and is an excellent tuning for the 11-limit version of hemififths temperament. Using the [[patent val]], 99edo is the [[optimal patent val]] for the rank-4 temperament tempering out [[121/120]]; zeus, the rank-3 temperament tempering out 121/120 and [[176/175]]; [[hemiwür]], one of the rank-2 11-limit extensions of hemiwürschmidt; and [[hitchcock]] (an 11-limit amity extension), the rank-2 temperament which also tempers out [[2200/2187]]. The same can be said of the mapping for [[13/1|13]], with the 99ef val tempering out [[144/143]], [[196/195]], 352/351 and [[364/363]], and its patent val tempering out [[169/168]], [[351/350]] and [[352/351]]. Hence 99edo, in spite of the fact that it tunes 11 and 13 relatively badly, is an important 13-limit tuning in more than one way.
-Being a [[zeta peak edo]], 99edo is also a very strong no-11 no-13 system, where it is consistent to the [[29-odd-limit]] with a sharp tendency. This favors the sharp mapping of 11 and 13, and allows these relatively weak approximations to somewhat blend with the rest for a full [[29-limit]] (or [[31-limit]], using the sharp-tending 99efk val) temperament.
+Being a [[zeta peak edo]], 99edo is also a very strong no-11 no-13 system, where it is consistent to the [[29-odd-limit]] with a sharp tendency. This favors the sharp mapping of 11 and 13, and allows these relatively weak approximations to somewhat blend with the rest for a full [[29-limit]] (or [[31-limit]], using the sharp-tending 99efk val) temperament. In fact, the 99efk val is the first to achieve [[diamond monotone]] in the [[31-odd-limit]], though it fails in the [[33-odd-limit]] due to mapping [[33/32]] to 5 steps, while [[32/31]] is mapped to 4 steps.
+One step of 99edo is close to [[144/143]], the grossma. Unfortunately, neither 99ef nor the patent val map it consistently, though [[198edo]] does.
 === Prime harmonics ===
@@ Line 13: / Line 15: @@
 === Subsets and supersets ===
-Since 99 factors into {{nowrap| 3<sup>2</sup> × 11 }}, 99edo has subset edos {{EDOs| 3, 9, 11, and 33 }}.
+Since 99 factors into primes as {{nowrap| 3<sup>2</sup> × 11 }}, 99edo has subset edos {{EDOs| 3, 9, 11, and 33 }}. Splitting 99edo's step in half yields [[198edo]], correcting prime 11, slightly improving prime 13, and aligning both 11 and 13 with the sharp tunings of the lower odd primes. Because of this, 198edo can be seen as a complex yet notable true full 13-limit tuning.
 == Intervals ==
 {{Main| Table of 99edo intervals }}
+== Notation ==
+[[Stein–Zimmermann–Gould notation]] uses sharps and flats combined with quartertone accidentals and arrows:
+{{Sharpness-sharp10-qt1-szg}}
+=== Kite's ups and downs notation ===
+edo can also be notated with [[Kite's ups and downs notation|Kite's ups and downs]]. Note that quip (quintuple-up) is the same as quudsharp (quadruple-down sharp) and that quid (quintuple-down) is the same as quupflat (quadruple-up flat):
+{{Ups and downs sharpness|99|true}}
+== Approximation to JI ==
+=== 7-prime-limited odd-limit analysis ===
+Unlike all previous edos, 99edo is ''distinctly'' [[consistent]] and monotone (i.e. when tempered using the patent val, the relative sizes of any two intervals are never conflated ''or'' reversed) up to the 7-prime-limited 45-odd-limit:
+{{Databox
+|collapse=true
+|title=The 7-prime-limited 45-odd-limit, by 99edo mapping (SW3 format)
+|text=
+<pre>
+(*
+-PL 45-OL odds:
+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>
+}}
+=== 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):
+# ↔<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):
+# ↔<sub>e</sub> = 81/80
+# ↔<sub>f</sub> = 64/63
+# ↔<sub>g</sub> = 875/864, the keema
+=== Interval mappings ===
+{{Q-odd-limit intervals|99}}
+{{Q-odd-limit intervals|99.1|apx=val|header=none|tag=none|title=15-odd-limit intervals by 99ef val mapping}}
 == Regular temperament properties ==
@@ Line 65: / Line 287: @@
 | 7.27
 |}
-* 99et is lower in relative error than any previous equal temperaments in the 7-limit. Not until 171 do we find a better equal temperament in terms of either absolute error or relative error.
+* 99et is lower in relative error than any previous equal temperaments in the 7-limit. Not until [[171edo|171]] do we find a better equal temperament in terms of either absolute error or relative error.
 === Rank-2 temperaments ===
@@ Line 123: / Line 345: @@
 | 339.394
 | 128/105
-| [[Amity]] (99ef) / hitchcock (99)
+| [[Amity]] (99ef) / stalagmite (99ef) / hitchcock (99)
 |-
 | 1
@@ Line 171: / Line 393: @@
 | 48.485
 | 36/35
-| [[Ennealimmal]] (99e) / ennealimmia (99) / <br />ennealimnic (99ef) / ennealim (99e) / ennealiminal (99)
+| [[Ennealimmal]] / enneabiotic (99ef) / ennealympic (99) / <br>ennealimnic (99ef) / ennealim (99e) / ennealiminal (99)
 |-
 | 11
@@ Line 177: / Line 399: @@
 | 496.970<br>(48.485)
 | 4/3<br>(36/35)
-| [[Hendecatonic]]
+| [[Hendecatonic (temperament)|Hendecatonic]]
 |}
-<nowiki/>* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
+<nowiki/>* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct
 == Octave stretch or compression ==
-edo's approximations of harmonics 3, 5, and 7 can all be improved if slightly [[stretched and compressed tuning|compressing the octave]] is acceptable, using tunings such as [[157edt]] or [[256ed6]]. 157edt is especially performant if the 13-limit of the 99ef val is intended, but the 7-limit part is overcompressed, for which the milder 256ed6 is a better choice. If the 13-limit patent val is intended, then little to no compression, or even stretch, might be serviceable.
+edo's approximations of harmonics 3, 5, and 7 can all be improved if slightly [[stretched and compressed tuning|compressing the octave]] is acceptable, using tunings such as [[157edt]] or [[256ed6]]. 157edt is especially performant if the 13-limit of the 99ef val is intended, but the 7-limit part is overcompressed, for which the milder 256ed6 is a better choice.
-What follows is a comparison of stretched- and compressed-octave 99edo tunings.
+If the 13-limit patent val is intended, then little to no compression, or even stretch, might be serviceable, such as in [[zpi|567zpi]].
-; [[zpi|567zpi]]
+== Scales ==
-* Step size: 12.138{{c}}, octave size: 1201.66{{c}}
+{{Main| List of MOS scales in 99edo }}
-Stretching the octave of 99edo by around 1.5{{c}} results in improved primes 11, 13, 17, and 19, but worse primes 2, 3, 5 and 7. This approximates all harmonics up to 16 within 5.54{{c}}. The tuning 567zpi does this.
-{{Harmonics in cet|12.138|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 567zpi}}
-{{Harmonics in cet|12.138|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 567zpi (continued)}}
-; [[WE|99et, 13-limit WE tuning]]
-* Step size: 12.123{{c}}, octave size: 1200.18{{c}}
-Stretching the octave of 99edo by around a fifth of a cent results in improved primes 11 and 13, but worse primes 2, 3, 5 and 7. This approximates all harmonics up to 16 within 5.25{{c}}. Its 13-limit WE tuning and 13-limit [[TE]] tuning both do this.
-{{Harmonics in cet|12.123|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 99et, 13-limit WE tuning}}
-{{Harmonics in cet|12.123|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 99et, 13-limit WE tuning (continued)}}
-; 99edo
-* Step size: 12.121{{c}}, octave size: 1200.00{{c}}
-Pure-octaves 99edo approximates all harmonics up to 16 within 5.86{{c}}.
-{{Harmonics in equal|99|2|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 99edo}}
-{{Harmonics in equal|99|2|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 99edo (continued)}}
-; [[WE|99et, 7-limit WE tuning]] / [[256ed6]]
-* Step size: 12.117{{c}}, octave size: 1199.58{{c}}
-Compressing the octave of 99edo by around 0.6{{c}} results in improved primes 3, 5, 7 and 11, but worse primes 2 and 13. This approximates all harmonics up to 16 within 5.71{{c}}. Its 7-limit WE tuning and 7-limit [[TE]] tuning both do this. So does the tuning 256ed6 whose octave is identical within a thousandth of a cent.
-{{Harmonics in cet|12.117|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 99et, 7-limit WE tuning}}
-{{Harmonics in cet|12.117|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 99et, 7-limit WE tuning (continued)}}
-; [[zpi|568zpi]]
-* Step size: 12.115{{c}}, octave size: 1199.39{{c}}
-Compressing the octave of 99edo by around 0.4{{c}} results in improved primes 3, 5, 7 and 11, but worse primes 2 and 13. This approximates all harmonics up to 16 within 5.68{{c}}. The tuning 568zpi does this.
-{{Harmonics in cet|12.115|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 568zpi}}
-{{Harmonics in cet|12.115|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 568zpi (continued)}}
-; [[157edt]] / [[ed5|230ed5]]
-* Step size: 12.114{{c}}, octave size: 1199.32{{c}}
-Compressing the octave of 99edo by around 0.3{{c}} results in improved primes 3, 5, 7 and 11, but worse primes 2 and 13. This approximates all harmonics up to 16 within 5.44{{c}}. The tuning 157edt does this. So does 230ed5 whose octave is identical within a hundredth of a cent.
-{{Harmonics in equal|157|3|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 157edt}}
-{{Harmonics in equal|157|3|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 157edt (continued)}}
-; [[58edf]]
-* Step size: 12.103{{c}}, octave size: 1199.16{{c}}
-Compressing the octave of 99edo by around 1{{c}} results in improved primes 11, 13, 17 and 19 but worse primes 2, 3, 5 and 7. This approximates all harmonics up to 16 within 5.98{{c}}. The tuning 58edf does this.
-{{Harmonics in equal|58|3|2|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 58edf}}
-{{Harmonics in equal|58|3|2|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 58edf (continued)}}
-== Scales ==
 * [[Tutone6]]
 * [[Tutone7]]