69ed7: Difference between revisions

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69 equal divisions of the 7th harmonic (abbreviated 69ed7), is the tuning system that divides the 7th harmonic into 69 equal parts of about 48.82356 ¢ each. Each step represents a frequency ratio of <math>7^{\frac{1}{69}}</math>, or the 69th root of 7.

{{Infobox ET}} 69 equal divisions of the 7th harmonic (abbreviated 69ed7), is the tuning system that divides the 7th harmonic into 69 equal parts of about 48.82356 ¢ each. Each step represents a frequency ratio of <math>7^{\frac{1}{69}}</math>, or the 69th root of 7.

== Theory ==

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* The Tenney–Euclidean regular temperement in the 3.5.7.11.13 subgroup mapped with [⟨39 57 69 85 91]] gives 48.82201 ¢.

With a size of 48.82356 ¢, 69ed7 gives a better approximation than [[39edt]] to Triple Bohlen-Pierce just intonation, yet still remaining a simple division.

With a size of 48.82356 ¢, 69ed7 gives a better approximation than [[39edt]] to [[Bohlen-Pierce|Triple Bohlen-Pierce]] just intonation, yet still remaining a simple division.

== Intervals and approximation to JI ==

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== Commas of the 3.5.7.11.13 subgroup tempered out in 69ed7 ==

Commas with numerator < 1000000 :

Commas with numerator < 1000000:

[[245/243]], [[275/273]], [[847/845]], [[1331/1323]], [[1575/1573]], [[1625/1617]], [[1875/1859]], [[2197/2187]], [[2205/2197]], [[3125/3087]], [[4459/4455]], [[6655/6561]], [[6655/6591]], [[8125/8019]], [[9295/9261]], [[9375/9317]], [[9625/9477]], [[11011/10935]], [[12005/11979]], [[14641/14625]], [[15625/15309]], [[15625/15379]], [[16807/16731]], [[16875/16807]], [[26411/26325]], [[28875/28561]], [[29575/29403]], [[41503/41067]], [[42875/42471]], [[46475/45927]], [[60025/59049]], [[60025/59319]], [[75625/74529]], [[78125/77077]], [[91125/91091]], [[107811/107653]], [[109375/107811]], [[117975/117649]], [[153125/150579]], [[161051/159705]], [[196625/194481]], [[200475/199927]], [[218491/216513]], [[219615/218491]], [[366025/361179]], [[378125/369603]], [[373527/371293]], [[378125/371293]], [[390625/382239]], [[398125/395307]], [[408375/405769]], [[456533/455625]], [[538265/531441]], [[539539/531441]], [[546875/531441]], [[546875/533871]], [[717409/710775]], [[714025/713097]], [[717409/714025]], [[759375/753571]], [[823543/820125]], [[823875/823543]], [[831875/823543]], [[859375/842751]], [[983125/964467]]

[[Category:ed7]]

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-equal divisions of the 7th harmonic (abbreviated 69ed7), is the tuning system that divides the 7th harmonic into 69 equal parts of about 48.82356 ¢ each. Each step represents a frequency ratio of <math>7^{\frac{1}{69}}</math>, or the 69th root of 7.
+{{Infobox ET}} 69 equal divisions of the 7th harmonic (abbreviated 69ed7), is the tuning system that divides the 7th harmonic into 69 equal parts of about 48.82356 ¢ each. Each step represents a frequency ratio of <math>7^{\frac{1}{69}}</math>, or the 69th root of 7.
 == Theory ==
@@ Line 11: / Line 11: @@
 * The Tenney–Euclidean regular temperement in the 3.5.7.11.13 subgroup mapped with [⟨39 57 69 85 91]] gives 48.82201 ¢.
-With a size of 48.82356 ¢, 69ed7 gives a better approximation than [[39edt]] to Triple Bohlen-Pierce just intonation, yet still remaining a simple division.
+With a size of 48.82356 ¢, 69ed7 gives a better approximation than [[39edt]] to [[Bohlen-Pierce|Triple Bohlen-Pierce]] just intonation, yet still remaining a simple division.
 == Intervals and approximation to JI ==
@@ Line 199: / Line 199: @@
 == Commas of the 3.5.7.11.13 subgroup tempered out in 69ed7 ==
-Commas with numerator < 1000000 :
+Commas with numerator < 1000000:
 [[245/243]], [[275/273]], [[847/845]], [[1331/1323]], [[1575/1573]], [[1625/1617]], [[1875/1859]], [[2197/2187]], [[2205/2197]], [[3125/3087]], [[4459/4455]], [[6655/6561]], [[6655/6591]], [[8125/8019]], [[9295/9261]], [[9375/9317]], [[9625/9477]], [[11011/10935]], [[12005/11979]], [[14641/14625]], [[15625/15309]], [[15625/15379]], [[16807/16731]], [[16875/16807]], [[26411/26325]], [[28875/28561]], [[29575/29403]], [[41503/41067]], [[42875/42471]], [[46475/45927]], [[60025/59049]], [[60025/59319]], [[75625/74529]], [[78125/77077]], [[91125/91091]], [[107811/107653]], [[109375/107811]], [[117975/117649]], [[153125/150579]], [[161051/159705]], [[196625/194481]], [[200475/199927]], [[218491/216513]], [[219615/218491]], [[366025/361179]], [[378125/369603]], [[373527/371293]], [[378125/371293]], [[390625/382239]], [[398125/395307]], [[408375/405769]], [[456533/455625]], [[538265/531441]], [[539539/531441]], [[546875/531441]], [[546875/533871]], [[717409/710775]], [[714025/713097]], [[717409/714025]], [[759375/753571]], [[823543/820125]], [[823875/823543]], [[831875/823543]], [[859375/842751]], [[983125/964467]]
+[[Category:ed7]]