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. 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 6 steps, while [[32/31]] is mapped to 5 steps.

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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== Notation ==

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

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

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):

=== 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):

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]]:

~~{{Sharpness-sharp10-qt1}}~~

== Approximation to JI ==

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</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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| 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. 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 6 steps, while [[32/31]] is mapped to 5 steps.
+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 19: / Line 21: @@
 == Notation ==
-=== Ups and downs notation ===
+[[Stein–Zimmermann–Gould notation]] uses sharps and flats combined with quartertone accidentals and arrows:
-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):
+{{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}}
-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]]:
-{{Sharpness-sharp10-qt1}}
 == Approximation to JI ==
@@ Line 208: / Line 210: @@
 </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 314: / Line 345: @@
 | 339.394
 | 128/105
-| [[Amity]] (99ef) / hitchcock (99)
+| [[Amity]] (99ef) / stalagmite (99ef) / hitchcock (99)
 |-
 | 1
@@ Line 362: / 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