680edo - Revision history

Lériendil: fixed headings

2025-04-15T03:39:09Z

fixed headings

← Older revision		Revision as of 03:39, 15 April 2025
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	{{ED intro}}		{{ED intro}}

	=== Odd harmonics ===		== Odd harmonics ==
	680edo retains a reasonable 3rd and 5th [[harmonic]], though nowhere near the accuracy of the prior multiple [[612edo]]; as a multiple of [[34edo]], 680edo borrows that edo's accurate representation of the interval [[25/24]], implying that the error on [[prime]] 3 is approximately twice that on prime 5. However, 680edo is most notable for its approximation of the 7th harmonic, 680 being the denominator of a semiconvergent to log<sub>2</sub>([[7/4]]).		680edo retains a reasonable 3rd and 5th [[harmonic]], though nowhere near the accuracy of the prior multiple [[612edo]]; as a multiple of [[34edo]], 680edo borrows that edo's accurate representation of the interval [[25/24]], implying that the error on [[prime]] 3 is approximately twice that on prime 5. However, 680edo is most notable for its approximation of the 7th harmonic, 680 being the denominator of a semiconvergent to log<sub>2</sub>([[7/4]]).

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	{{Harmonics in equal\|680\|columns=11\|start=12\|collapsed=true\|title=Approximation of prime harmonics in 680edo (continued)}}		{{Harmonics in equal\|680\|columns=11\|start=12\|collapsed=true\|title=Approximation of prime harmonics in 680edo (continued)}}

	=== Subsets and supersets ===		== Subsets and supersets ==
	Since 680 factors into {{factorization\|680}}, 680edo has subset edos {{EDOs\| 2, 4, 5, 8, 10, 17, 20, 34, 40, 68, 85, 136, 170, and 340 }}.		Since 680 factors into {{factorization\|680}}, 680edo has subset edos {{EDOs\| 2, 4, 5, 8, 10, 17, 20, 34, 40, 68, 85, 136, 170, and 340 }}.

	{{todo\|expand}}		{{todo\|expand}}

Francium: changed EDO intro to ED intro

2025-02-20T17:10:09Z

changed EDO intro to ED intro

← Older revision		Revision as of 17:10, 20 February 2025
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	{{Infobox ET}}		{{Infobox ET}}
	{{~~EDO~~ intro~~\|680~~}}		{{ED intro}}

	=== Odd harmonics ===		=== Odd harmonics ===

BudjarnLambeth at 19:23, 25 December 2024

2024-12-25T19:23:17Z

← Older revision		Revision as of 19:23, 25 December 2024
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	Since 680 factors into {{factorization\|680}}, 680edo has subset edos {{EDOs\| 2, 4, 5, 8, 10, 17, 20, 34, 40, 68, 85, 136, 170, and 340 }}.		Since 680 factors into {{factorization\|680}}, 680edo has subset edos {{EDOs\| 2, 4, 5, 8, 10, 17, 20, 34, 40, 68, 85, 136, 170, and 340 }}.

	{{todo:expand}}		{{todo\|expand}}

BudjarnLambeth: Change stub to todo:expand, add links

2024-12-25T19:23:10Z

Change stub to todo:expand, add links

← Older revision		Revision as of 19:23, 25 December 2024
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	=== Odd harmonics ===		=== Odd harmonics ===
	680edo retains a reasonable 3rd and 5th harmonic, though nowhere near the accuracy of the prior multiple [[612edo]]; as a multiple of [[34edo]], 680edo borrows that edo's accurate representation of the interval [[25/24]], implying that the error on prime 3 is approximately twice that on prime 5. However, 680edo is most notable for its approximation of the 7th harmonic, 680 being the denominator of a semiconvergent to log<sub>2</sub>([[7/4]]).		680edo retains a reasonable 3rd and 5th [[harmonic]], though nowhere near the accuracy of the prior multiple [[612edo]]; as a multiple of [[34edo]], 680edo borrows that edo's accurate representation of the interval [[25/24]], implying that the error on [[prime]] 3 is approximately twice that on prime 5. However, 680edo is most notable for its approximation of the 7th harmonic, 680 being the denominator of a semiconvergent to log<sub>2</sub>([[7/4]]).

	Its primes 11, 13, 17, and 19 are all approximated rather badly, but 680edo actually shines in very high prime ~~limits~~, with great representation of prime 23 (inherited from [[170edo]]) and accurate representation of prime 31 as well as the entire stretch of primes from 41 to 73; even the remaining primes are often off by similar enough margins in the same direction that there are many instances of intervals between them that are approximated quite precisely, such as 37/29, of which 680edo is a weak circle.		Its primes 11, 13, 17, and 19 are all approximated rather badly, but 680edo actually shines in very high [[prime limit]]s, with great representation of prime 23 (inherited from [[170edo]]) and accurate representation of prime 31 as well as the entire stretch of primes from 41 to 73; even the remaining primes are often off by similar enough margins in the same direction that there are many instances of intervals between them that are approximated quite precisely, such as [[37/29]], of which 680edo is a weak [[circle]].
	{{Harmonics in equal\|680\|columns=11}}		{{Harmonics in equal\|680\|columns=11}}
	{{Harmonics in equal\|680\|columns=11\|start=12\|collapsed=true\|title=Approximation of prime harmonics in 680edo (continued)}}		{{Harmonics in equal\|680\|columns=11\|start=12\|collapsed=true\|title=Approximation of prime harmonics in 680edo (continued)}}
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	Since 680 factors into {{factorization\|680}}, 680edo has subset edos {{EDOs\| 2, 4, 5, 8, 10, 17, 20, 34, 40, 68, 85, 136, 170, and 340 }}.		Since 680 factors into {{factorization\|680}}, 680edo has subset edos {{EDOs\| 2, 4, 5, 8, 10, 17, 20, 34, 40, 68, 85, 136, 170, and 340 }}.

	{{~~Stub~~}}		{{todo:expand}}

Lériendil at 22:55, 14 December 2024

2024-12-14T22:55:36Z

← Older revision		Revision as of 22:55, 14 December 2024
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	Its primes 11, 13, 17, and 19 are all approximated rather badly, but 680edo actually shines in very high prime limits, with great representation of prime 23 (inherited from [[170edo]]) and accurate representation of prime 31 as well as the entire stretch of primes from 41 to 73; even the remaining primes are often off by similar enough margins in the same direction that there are many instances of intervals between them that are approximated quite precisely, such as 37/29, of which 680edo is a weak circle.		Its primes 11, 13, 17, and 19 are all approximated rather badly, but 680edo actually shines in very high prime limits, with great representation of prime 23 (inherited from [[170edo]]) and accurate representation of prime 31 as well as the entire stretch of primes from 41 to 73; even the remaining primes are often off by similar enough margins in the same direction that there are many instances of intervals between them that are approximated quite precisely, such as 37/29, of which 680edo is a weak circle.
	{{Harmonics in equal\|680\|columns=11}}		{{Harmonics in equal\|680\|columns=11}}
	{{Harmonics in equal\|680\|columns=10\|start=12\|collapsed=true\|title=Approximation of prime harmonics in 680edo (continued)}}		{{Harmonics in equal\|680\|columns=11\|start=12\|collapsed=true\|title=Approximation of prime harmonics in 680edo (continued)}}

	=== Subsets and supersets ===		=== Subsets and supersets ===

Lériendil: elaborated on the below

2024-12-14T00:59:25Z

elaborated on the below

← Older revision		Revision as of 00:59, 14 December 2024
Line 3:		Line 3:

	=== Odd harmonics ===		=== Odd harmonics ===
	680edo retains a reasonable 3rd and 5th harmonic, though nowhere near the accuracy of the prior multiple [[612edo]]; as a multiple of [[34edo]], 680edo borrows that edo's accurate representation of the interval [[25/24]], implying that the error on prime 3 is approximately twice that on prime 5. However, 680edo is most notable for its approximation of the 7th harmonic, 680 being the denominator of a semiconvergent to log<sub>2</sub>([[7/4]]). ~~It also has a~~ very ~~accurate 23rd harmonic~~, inherited from [[170edo]].		680edo retains a reasonable 3rd and 5th harmonic, though nowhere near the accuracy of the prior multiple [[612edo]]; as a multiple of [[34edo]], 680edo borrows that edo's accurate representation of the interval [[25/24]], implying that the error on prime 3 is approximately twice that on prime 5. However, 680edo is most notable for its approximation of the 7th harmonic, 680 being the denominator of a semiconvergent to log<sub>2</sub>([[7/4]]).

			Its primes 11, 13, 17, and 19 are all approximated rather badly, but 680edo actually shines in very high prime limits, with great representation of prime 23 (inherited from [[170edo]]) and accurate representation of prime 31 as well as the entire stretch of primes from 41 to 73; even the remaining primes are often off by similar enough margins in the same direction that there are many instances of intervals between them that are approximated quite precisely, such as 37/29, of which 680edo is a weak circle.
	{{Harmonics in equal\|680\|columns=11}}		{{Harmonics in equal\|680\|columns=11}}
	{{Harmonics in equal\|680\|columns=10\|start=12\|collapsed=true\|title=Approximation of prime harmonics in 680edo (continued)}}		{{Harmonics in equal\|680\|columns=10\|start=12\|collapsed=true\|title=Approximation of prime harmonics in 680edo (continued)}}

Lériendil at 00:52, 14 December 2024

2024-12-14T00:52:38Z

← Older revision		Revision as of 00:52, 14 December 2024
Line 5:		Line 5:
	680edo retains a reasonable 3rd and 5th harmonic, though nowhere near the accuracy of the prior multiple [[612edo]]; as a multiple of [[34edo]], 680edo borrows that edo's accurate representation of the interval [[25/24]], implying that the error on prime 3 is approximately twice that on prime 5. However, 680edo is most notable for its approximation of the 7th harmonic, 680 being the denominator of a semiconvergent to log<sub>2</sub>([[7/4]]). It also has a very accurate 23rd harmonic, inherited from [[170edo]].		680edo retains a reasonable 3rd and 5th harmonic, though nowhere near the accuracy of the prior multiple [[612edo]]; as a multiple of [[34edo]], 680edo borrows that edo's accurate representation of the interval [[25/24]], implying that the error on prime 3 is approximately twice that on prime 5. However, 680edo is most notable for its approximation of the 7th harmonic, 680 being the denominator of a semiconvergent to log<sub>2</sub>([[7/4]]). It also has a very accurate 23rd harmonic, inherited from [[170edo]].
	{{Harmonics in equal\|680\|columns=11}}		{{Harmonics in equal\|680\|columns=11}}
	{{Harmonics in equal\|680\|columns=11\|start=12\|collapsed=true\|title=Approximation of prime harmonics in 680edo (continued)}}		{{Harmonics in equal\|680\|columns=10\|start=12\|collapsed=true\|title=Approximation of prime harmonics in 680edo (continued)}}

	=== Subsets and supersets ===		=== Subsets and supersets ===

Lériendil at 00:52, 14 December 2024

2024-12-14T00:52:25Z

← Older revision		Revision as of 00:52, 14 December 2024
Line 4:		Line 4:
	=== Odd harmonics ===		=== Odd harmonics ===
	680edo retains a reasonable 3rd and 5th harmonic, though nowhere near the accuracy of the prior multiple [[612edo]]; as a multiple of [[34edo]], 680edo borrows that edo's accurate representation of the interval [[25/24]], implying that the error on prime 3 is approximately twice that on prime 5. However, 680edo is most notable for its approximation of the 7th harmonic, 680 being the denominator of a semiconvergent to log<sub>2</sub>([[7/4]]). It also has a very accurate 23rd harmonic, inherited from [[170edo]].		680edo retains a reasonable 3rd and 5th harmonic, though nowhere near the accuracy of the prior multiple [[612edo]]; as a multiple of [[34edo]], 680edo borrows that edo's accurate representation of the interval [[25/24]], implying that the error on prime 3 is approximately twice that on prime 5. However, 680edo is most notable for its approximation of the 7th harmonic, 680 being the denominator of a semiconvergent to log<sub>2</sub>([[7/4]]). It also has a very accurate 23rd harmonic, inherited from [[170edo]].
	{{Harmonics in equal\|680\|columns=21}}		{{Harmonics in equal\|680\|columns=11}}
			{{Harmonics in equal\|680\|columns=11\|start=12\|collapsed=true\|title=Approximation of prime harmonics in 680edo (continued)}}

	=== Subsets and supersets ===		=== Subsets and supersets ===

Lériendil at 00:50, 14 December 2024

2024-12-14T00:50:56Z

← Older revision		Revision as of 00:50, 14 December 2024
Line 4:		Line 4:
	=== Odd harmonics ===		=== Odd harmonics ===
	680edo retains a reasonable 3rd and 5th harmonic, though nowhere near the accuracy of the prior multiple [[612edo]]; as a multiple of [[34edo]], 680edo borrows that edo's accurate representation of the interval [[25/24]], implying that the error on prime 3 is approximately twice that on prime 5. However, 680edo is most notable for its approximation of the 7th harmonic, 680 being the denominator of a semiconvergent to log<sub>2</sub>([[7/4]]). It also has a very accurate 23rd harmonic, inherited from [[170edo]].		680edo retains a reasonable 3rd and 5th harmonic, though nowhere near the accuracy of the prior multiple [[612edo]]; as a multiple of [[34edo]], 680edo borrows that edo's accurate representation of the interval [[25/24]], implying that the error on prime 3 is approximately twice that on prime 5. However, 680edo is most notable for its approximation of the 7th harmonic, 680 being the denominator of a semiconvergent to log<sub>2</sub>([[7/4]]). It also has a very accurate 23rd harmonic, inherited from [[170edo]].
	{{Harmonics in equal\|680\|columns=20}}		{{Harmonics in equal\|680\|columns=21}}

	=== Subsets and supersets ===		=== Subsets and supersets ===

Lériendil: 680edo seems to excel at a large number of high primes

2024-12-14T00:50:11Z

680edo seems to excel at a large number of high primes

← Older revision		Revision as of 00:50, 14 December 2024
Line 4:		Line 4:
	=== Odd harmonics ===		=== Odd harmonics ===
	680edo retains a reasonable 3rd and 5th harmonic, though nowhere near the accuracy of the prior multiple [[612edo]]; as a multiple of [[34edo]], 680edo borrows that edo's accurate representation of the interval [[25/24]], implying that the error on prime 3 is approximately twice that on prime 5. However, 680edo is most notable for its approximation of the 7th harmonic, 680 being the denominator of a semiconvergent to log<sub>2</sub>([[7/4]]). It also has a very accurate 23rd harmonic, inherited from [[170edo]].		680edo retains a reasonable 3rd and 5th harmonic, though nowhere near the accuracy of the prior multiple [[612edo]]; as a multiple of [[34edo]], 680edo borrows that edo's accurate representation of the interval [[25/24]], implying that the error on prime 3 is approximately twice that on prime 5. However, 680edo is most notable for its approximation of the 7th harmonic, 680 being the denominator of a semiconvergent to log<sub>2</sub>([[7/4]]). It also has a very accurate 23rd harmonic, inherited from [[170edo]].
	{{Harmonics in equal\|680\|columns=17}}		{{Harmonics in equal\|680\|columns=20}}

	=== Subsets and supersets ===		=== Subsets and supersets ===