99edo: Difference between revisions

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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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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:

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}}

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):

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

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(*10*) 15/14; 343/320;

(*11*) 27/25; 175/162;

(*12*) 243/224; 160/147;

(*12*) 243/224; 160/147; 49/45;

(*13*) 375/343; 35/32; 192/175;

(*14*) 54/49; 441/400; 448/405;

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(*18*) 500/441; 567/500; 245/216; 256/225;

(*19*) 8/7; 343/300;

(*20*) 225/196; 147/128; 280/243;

(*20*) 225/196; 147/128; 144/125; 280/243;

(*21*) 125/108; 512/441;

(*21*) 81/70; 125/108; 512/441;

(*22*) 400/343; 7/6;

(*23*) 75/64; 288/245; 147/125;

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(*27*) 135/112; 98/81;

(*28*) 243/200; 175/144; 128/105;

(*29*) 49/40;

(*29*) 60/49; 49/40;

(*30*) 315/256; 216/175; 100/81;

(*31*) 243/196; 56/45;

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(*33*) 432/343; 63/50; 512/405;

(*34*) 81/64; 80/63; 343/270;

(*35*) 245/192; 32/25;

(*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;

(*39*) 450/343; 21/16; 320/243;

(*40*) 324/245; 250/189;

(*41*) 4/3;

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(*58*) 3/2;

(*59*) 189/125; 245/162;

(*60*) 32/21; 343/225;

(*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;

(*64*) 25/16; 384/245; 196/125;

(*65*) 540/343; 63/40; 128/81;

(*66*) 405/256; 100/63; 343/216;

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(*68*) 45/28; 392/243;

(*69*) 81/50; 175/108; 512/315;

(*70*) 80/49;

(*70*) 80/49; 49/30;

(*71*) 105/64; 288/175; 400/243;

(*72*) 81/49; 224/135;

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(*76*) 250/147; 245/144; 128/75;

(*77*) 12/7; 343/200;

(*78*) 441/256; 216/125;

(*78*) 441/256; 216/125; 140/81

(*79*) 243/140; 256/147; 392/225;

(*79*) 243/140; 125/72; 256/147; 392/225;

(*80*) 600/343; 7/4;

(*81*) 225/128; 432/245; 1000/567; 441/250;

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(*85*) 405/224; 800/441; 49/27;

(*86*) 175/96; 64/35; 686/375;

(*87*) 147/80; 448/243;

(*87*) 90/49; 147/80; 448/243;

(*88*) 324/175; 50/27;

(*89*) 640/343; 28/15;

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}}

== Intervals ==

=== Intervals made equidistant by 99edo ===

~~{{Main~~| ~~Table~~ of ~~99edo~~ intervals }}

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

== ~~Notation ==~~

The separating intervals (all equated):

=~~== Ups and downs notation ===~~

# ↔e = 81/80

~~99edo can be notated with [[Kite's ups and downs notation]]. 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):~~

# ↔f = 64/63

~~{{Ups and downs sharpness|99|true}}~~

# ↔g = 875/864, the keema

~~Another notation uses [[Alternative symbols for ups and downs notation#Sharp~~-5|~~alternative ups and downs]]. Here, this can be done using sharps and flats with arrows, borrowed from extended [[Helmholtz–Ellis notation]]:~~

=== 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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| 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 7: / Line 7: @@
 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 15: / Line 17: @@
 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:
@@ Line 88: / Line 102: @@
 }}
-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):
+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
@@ Line 106: / Line 120: @@
 (*10*) 15/14; 343/320;
 (*11*) 27/25; 175/162;
-(*12*) 243/224; 160/147;
+(*12*) 243/224; 160/147; 49/45;
 (*13*) 375/343; 35/32; 192/175;
 (*14*) 54/49; 441/400; 448/405;
@@ Line 114: / Line 128: @@
 (*18*) 500/441; 567/500; 245/216; 256/225;
 (*19*) 8/7; 343/300;
-(*20*) 225/196; 147/128; 280/243;
+(*20*) 225/196; 147/128; 144/125; 280/243;
-(*21*) 125/108; 512/441;
+(*21*) 81/70; 125/108; 512/441;
 (*22*) 400/343; 7/6;
 (*23*) 75/64; 288/245; 147/125;
@@ Line 123: / Line 137: @@
 (*27*) 135/112; 98/81;
 (*28*) 243/200; 175/144; 128/105;
-(*29*) 49/40;
+(*29*) 60/49; 49/40;
 (*30*) 315/256; 216/175; 100/81;
 (*31*) 243/196; 56/45;
@@ Line 129: / Line 143: @@
 (*33*) 432/343; 63/50; 512/405;
 (*34*) 81/64; 80/63; 343/270;
-(*35*) 245/192; 32/25;
+(*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;
+(*39*) 450/343; 21/16; 320/243;
 (*40*) 324/245; 250/189;
 (*41*) 4/3;
@@ Line 154: / Line 168: @@
 (*58*) 3/2;
 (*59*) 189/125; 245/162;
-(*60*) 32/21; 343/225;
+(*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;
+(*64*) 25/16; 384/245; 196/125;
 (*65*) 540/343; 63/40; 128/81;
 (*66*) 405/256; 100/63; 343/216;
@@ Line 164: / Line 178: @@
 (*68*) 45/28; 392/243;
 (*69*) 81/50; 175/108; 512/315;
-(*70*) 80/49;
+(*70*) 80/49; 49/30;
 (*71*) 105/64; 288/175; 400/243;
 (*72*) 81/49; 224/135;
@@ Line 172: / Line 186: @@
 (*76*) 250/147; 245/144; 128/75;
 (*77*) 12/7; 343/200;
-(*78*) 441/256; 216/125;
+(*78*) 441/256; 216/125; 140/81
-(*79*) 243/140; 256/147; 392/225;
+(*79*) 243/140; 125/72; 256/147; 392/225;
 (*80*) 600/343; 7/4;
 (*81*) 225/128; 432/245; 1000/567; 441/250;
@@ Line 181: / Line 195: @@
 (*85*) 405/224; 800/441; 49/27;
 (*86*) 175/96; 64/35; 686/375;
-(*87*) 147/80; 448/243;
+(*87*) 90/49; 147/80; 448/243;
 (*88*) 324/175; 50/27;
 (*89*) 640/343; 28/15;
@@ Line 197: / Line 211: @@
 }}
-== Intervals ==
+=== Intervals made equidistant by 99edo ===
-{{Main| Table of 99edo intervals }}
+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
-== Notation ==
+The separating intervals (all equated):
-=== Ups and downs notation ===
+# ↔<sub>e</sub> = 81/80
-edo can be notated with [[Kite's ups and downs notation]]. 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):
+# ↔<sub>f</sub> = 64/63
-{{Ups and downs sharpness|99|true}}
+# ↔<sub>g</sub> = 875/864, the keema
-Another notation uses [[Alternative symbols for ups and downs notation#Sharp-5|alternative ups and downs]]. Here, this can be done using sharps and flats with arrows, borrowed from extended [[Helmholtz–Ellis notation]]:
+=== Interval mappings ===
-{{Sharpness-sharp10-qt1}}
+{{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 313: / Line 345: @@
 | 339.394
 | 128/105
-| [[Amity]] (99ef) / hitchcock (99)
+| [[Amity]] (99ef) / stalagmite (99ef) / hitchcock (99)
 |-
 | 1
@@ Line 361: / 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