89edo: Difference between revisions

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~~'''89edo''' divides the octave into~~ 89 ~~steps of~~ size 13.~~48 [[cent]]s each, and [[tempering out~~|~~tempers out]] the commas [[126/125]], [[1728/1715]], [[32805/32768]], [[2401/2400]], [[176/175]], [[243/242]], [[441/440]] and [[540/539]]~~. ~~It is an especially good tuning for [[Starling temperaments~~|~~myna temperament]], both in the [[~~7~~-limit]], tempering out 126/125 and 1728/1715, and in the [[11-limit]], where 176/175 is tempered out also~~. ~~It is likewise a good tuning for the rank three temperament thrush, tempering out 126/125 and 176/175~~.

{{Infobox ET

| Prime factorization = 89 (prime)

| Step size = 13.4831¢

| Fifth = 52\89 (701.1¢)

| Semitones = 8:7 (107.9¢ : 94.4¢)

| Consistency = 11

}}

89 has a fifth less than a cent flat and a major third less than five cents sharp, with a 7 two cents sharp and an 11 1.5 cents sharp. It thus delivers reasonably good 11-limit harmony and very good no-fives harmony along with the very useful approximations represented by its commas. On a related note, a notable characteristic of this ~~EDO~~ is that it is the lowest in a series of four consecutive ~~EDOs~~ to temper out [[~~Quartisma|117440512/117406179~~]].

== Theory ==

89et [[tempering out|tempers out]] the commas [[126/125]], [[1728/1715]], [[32805/32768]], [[2401/2400]], [[176/175]], [[243/242]], [[441/440]] and [[540/539]]. It is an especially good tuning for the [[myna]] temperament, both in the [[7-limit]], tempering out 126/125 and 1728/1715, and in the [[11-limit]], where 176/175 is tempered out also. It is likewise a good tuning for the rank-3 temperament [[thrush]], tempering out 126/125 and 176/175.

89 has a fifth less than a cent flat and a major third less than five cents sharp, with a 7 two cents sharp and an 11 1.5 cents sharp. It thus delivers reasonably good 11-limit harmony and very good no-fives harmony along with the very useful approximations represented by its commas. On a related note, a notable characteristic of this edo is that it is the lowest in a series of four consecutive edos to temper out [[quartisma]].

89edo is the 24th [[prime edo]], and the 11th in the Fibonacci sequence, which means its 55th step approximates logarithmic φ (i.e. (φ - 1)×1200 cents) within a fraction of a cent.

=== Prime harmonics ===

~~89edo is the 24th [[prime_numbers|prime]] edo, and the 11th in the Fibonacci sequence, which means its 55th step approximates logarithmic phi (i.e. (phi-1)*1200 cents) within a fraction of a cent.~~

~~{{harmonics in equal|89}}~~

== Scales ==

* [[~~myna7~~]]

* [[Myna7]]

* [[~~myna11~~]]

* [[Myna11]]

* [[~~myna15~~]]

* [[Myna15]]

[[Category:89edo| ]]

[[Category:Equal divisions of the octave]]

[[Category:Prime EDO]]

[[Category:Myna]]

[[Category:~~89edo~~]]

[[Category:Thrush]]

[[Category:Quartismic]]

~~[[Category:Thrush]]~~

@@ Line 1: / Line 1: @@
-'''89edo''' divides the octave into 89 steps of size 13.48 [[cent]]s each, and [[tempering out|tempers out]] the commas [[126/125]], [[1728/1715]], [[32805/32768]], [[2401/2400]], [[176/175]], [[243/242]], [[441/440]] and [[540/539]]. It is an especially good tuning for [[Starling temperaments|myna temperament]], both in the [[7-limit]], tempering out 126/125 and 1728/1715, and in the [[11-limit]], where 176/175 is tempered out also. It is likewise a good tuning for the rank three temperament thrush, tempering out 126/125 and 176/175.
+{{Infobox ET
+| Prime factorization = 89 (prime)
+| Step size = 13.4831¢
+| Fifth = 52\89 (701.1¢)
+| Semitones = 8:7 (107.9¢ : 94.4¢)
+| Consistency = 11
+}}
+{{EDO intro|89}}
-has a fifth less than a cent flat and a major third less than five cents sharp, with a 7 two cents sharp and an 11 1.5 cents sharp. It thus delivers reasonably good 11-limit harmony and very good no-fives harmony along with the very useful approximations represented by its commas.  On a related note, a notable characteristic of this EDO is that it is the lowest in a series of four consecutive EDOs to temper out [[Quartisma|117440512/117406179]].
+== Theory ==
+et [[tempering out|tempers out]] the commas [[126/125]], [[1728/1715]], [[32805/32768]], [[2401/2400]], [[176/175]], [[243/242]], [[441/440]] and [[540/539]]. It is an especially good tuning for the [[myna]] temperament, both in the [[7-limit]], tempering out 126/125 and 1728/1715, and in the [[11-limit]], where 176/175 is tempered out also. It is likewise a good tuning for the rank-3 temperament [[thrush]], tempering out 126/125 and 176/175.
+has a fifth less than a cent flat and a major third less than five cents sharp, with a 7 two cents sharp and an 11 1.5 cents sharp. It thus delivers reasonably good 11-limit harmony and very good no-fives harmony along with the very useful approximations represented by its commas. On a related note, a notable characteristic of this edo is that it is the lowest in a series of four consecutive edos to temper out [[quartisma]].
+edo is the 24th [[prime edo]], and the 11th in the Fibonacci sequence, which means its 55th step approximates logarithmic φ (i.e. (φ - 1)×1200 cents) within a fraction of a cent.
+=== Prime harmonics ===
+{{Harmonics in equal|89}}
-edo is the 24th [[prime_numbers|prime]] edo, and the 11th in the Fibonacci sequence, which means its 55th step approximates logarithmic phi (i.e. (phi-1)*1200 cents) within a fraction of a cent.
-{{harmonics in equal|89}}
 == Scales ==
-* [[myna7]]
+* [[Myna7]]
-* [[myna11]]
+* [[Myna11]]
-* [[myna15]]
+* [[Myna15]]
+[[Category:89edo| ]] <!-- main article -->
 [[Category:Equal divisions of the octave]]
 [[Category:Prime EDO]]
 [[Category:Myna]]
-[[Category:89edo]]
+[[Category:Thrush]]
 [[Category:Quartismic]]
-[[Category:Thrush]]