69ed7 - Revision history

BudjarnLambeth at 00:11, 28 December 2025

2025-12-28T00:11:33Z

← Older revision		Revision as of 00:11, 28 December 2025
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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 ==		== Theory ==

BudjarnLambeth: Uncategorised page

2025-12-28T00:11:15Z

Uncategorised page

← Older revision		Revision as of 00:11, 28 December 2025
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	[[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]]		[[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]]

Contribution: /* Commas of the 3.5.7.11.13 subgroup tempered out in 69ed7 */

2025-10-27T19:38:51Z

Commas of the 3.5.7.11.13 subgroup tempered out in 69ed7

← Older revision		Revision as of 19:38, 27 October 2025
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	== Commas of the 3.5.7.11.13 subgroup tempered out in 69ed7 ==		== 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]]		[[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]]

Contribution at 19:37, 27 October 2025

2025-10-27T19:37:22Z

← Older revision		Revision as of 19:37, 27 October 2025
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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 ¢.		* 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 ==		== Intervals and approximation to JI ==

Contribution at 19:35, 27 October 2025

2025-10-27T19:35:24Z

← Older revision		Revision as of 19:35, 27 October 2025
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 ¢.		* 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 Triple Bohlen-Pierce just intonation, yet still remaining a simple division.

	== Intervals and approximation to JI ==		== Intervals and approximation to JI ==

Contribution: Created page with "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 $7^{\frac{1}{69}}$ , or the 69th root of 7. == Theory == 69ed7 is a nice and strong 3.5.7.11.13 system. All ratios in the 3.5.7.11.13 subgroup and 21-integer-limit are approximated in 69ed7 with less than 7.7 ¢ error. 69ed7 is close to the different methods fo..."

2025-10-27T19:34:18Z

Created page with "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 == 69ed7 is a nice and strong 3.5.7.11.13 system. All ratios in the 3.5.7.11.13 subgroup and 21-integer-limit are approximated in 69ed7 with less than 7.7 ¢ error. 69ed7 is close to the different methods fo..."

New page

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 ==

69ed7 is a nice and strong 3.5.7.11.13 system.

All ratios in the 3.5.7.11.13 subgroup and 21-integer-limit are approximated in 69ed7 with less than 7.7 ¢ error.

69ed7 is close to the different methods for approaching the 3.5.7.11.13 subgroup in the Triple Bohlen-Piece aera:
* The finite Euler product maxima of the 3.5.7.11.13 subgroup (sigma = 1 gives 48.82085 ¢, sigma = 1/2 gives 48.82100 ¢).
* 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.

== Intervals and approximation to JI ==

{| class="wikitable sortable center-all right-2 left-3"
|-
! 69ed7<br>degree
! Cents
! Ratios in the<br>3.5.7.11.13<br>subgroup
! Error (abs, [[Cent|¢]])
! Error (rel, [[Relative cent|%]])
|-
| 5
| 244.1
| [[15/13]]
| -3.623
| -7.4
|-
| 6
| 292.9
| [[13/11]]
| 3.732
| 7.6
|-
| 7
| 341.8
| [[11/9]]
| -5.643
| -11.6
|-
| 9
| 439.4
| [[9/7]]
| 4.328
| 8.9
|-
| 11
| 537.1
| [[15/11]]
| 0.108
| 0.2
|-
| 12
| 585.9
| [[7/5]]
| 3.371
| 6.9
|-
| 13
| 634.7
| [[13/9]]
| -1.911
| -3.9
|-
| 16
| 781.2
| [[11/7]]
| -1.315
| -2.7
|-
| 17
| 830.0
| [[21/13]]
| -0.253
| -0.5
|-
| 18
| 878.8
| [[5/3]]
| -5.535
| -11.3
|-
| 21
| 1025.3
| [[9/5]]
| 7.699
| 15.8
|-
| 22
| 1074.1
| [[13/7]]
| 2.417
| 4.9
|-
| 23
| 1122.9
| [[21/11]]
| 3.479
| 7.1
|-
| 27
| 1318.2
| [[15/7]]
| -1.207
| -2.5
|-
| 28
| 1367.1
| [[11/5]]
| 2.056
| 4.2
|-
| 30
| 1464.7
| [[7/3]]
| -2.164
| -4.4
|-
| 34
| 1660.0
| [[13/5]]
| 5.787
| 11.9
|-
| 39
| 1904.1
| [[3/1]]
| 2.164
| 4.4
|-
| 46
| 2245.9
| [[11/3]]
| -3.479
| -7.1
|-
| 51
| 2490.0
| [[21/5]]
| 5.535
| 11.3
|-
| 52
| 2538.8
| [[13/3]]
| 0.253
| 0.5
|-
| 57
| 2782.9
| [[5/1]]
| -3.371
| -6.9
|-
| 69
| 3368.8
| [[7/1]]
| 0.0
| 0.0
|-
| 78
| 3808.2
| [[9/1]]
| 4.328
| 8.9
|-
| 85
| 4150.0
| [[11/1]]
| -1.315
| -2.7
|-
| 91
| 4442.9
| [[13/1]]
| 2.417
| 4.9
|-
| 96
| 4687.1
| [[15/1]]
| -1.207
| -2.5
|-
| 108
| 5272.9
| [[21/1]]
| 2.164
| 4.4
|}

== Harmonics ==

{{Harmonics in equal|69|7|1|columns=6|intervals=prime}}

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

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]]