680edo

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← 679edo680edo681edo →
Prime factorization 23 × 5 × 17
Step size 1.76471¢ 
Fifth 398\680 (702.353¢) (→199\340)
Semitones (A1:m2) 66:50 (116.5¢ : 88.24¢)
Consistency limit 9
Distinct consistency limit 9

680 equal divisions of the octave (abbreviated 680edo or 680ed2), also called 680-tone equal temperament (680tet) or 680 equal temperament (680et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 680 equal parts of about 1.76 ¢ each. Each step represents a frequency ratio of 21/680, or the 680th root of 2.

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 log2(7/4). It also has a very accurate 23rd harmonic.

Approximation of prime harmonics in 680edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53
Error Absolute (¢) +0.000 +0.398 +0.157 -0.002 -0.730 -0.528 -0.838 +0.722 -0.039 -0.754 +0.259 -0.756 -0.239 +0.247 -0.213 +0.025
Relative (%) +0.0 +22.5 +8.9 -0.1 -41.4 -29.9 -47.5 +40.9 -2.2 -42.7 +14.7 -42.8 -13.5 +14.0 -12.0 +1.4
Steps
(reduced)
680
(0)
1078
(398)
1579
(219)
1909
(549)
2352
(312)
2516
(476)
2779
(59)
2889
(169)
3076
(356)
3303
(583)
3369
(649)
3542
(142)
3643
(243)
3690
(290)
3777
(377)
3895
(495)

Subsets and supersets

Since 680 factors into 23 × 5 × 17, 680edo has subset edos 2, 4, 5, 8, 10, 17, 20, 34, 40, 68, 85, 170, and 340.

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