Septimagic comma

Interval information

Ratio	537824/531441
Factorization	25 × 3-12 × 75
Monzo	[5 -12 0 5⟩
Size in cents	20.66952¢
Names	septimagic comma, magi comma
Color name	sz54, saquinzo 4th
FJS name	[math]\displaystyle{ \text{ddd4}^{7,7,7,7,7} }[/math]
Special properties	reduced
Tenney norm (log2 nd)	38.0563
Weil norm (log2 max(n, d))	38.0735
Wilson norm (sopfr(nd))	81
Comma size	small
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537824/531441, the septimagic comma or magi comma is a no-fives 7-limit comma of measuring 20.7 cents. It is the difference between five 14/9s and two 3/1s.

Temperaments

Tempering out this comma gives the rank-3 septimagic temperament in the full 7-limit, and the magi temperament in its minimal prime subgroup of 2.3.7. Magi is the no-five restriction of the magic temperament.

Septimagic

Subgroup: 2.3.5.7

Comma list: 537824/531441

Mapping: [⟨1 0 0 -1], ⟨0 5 0 12], ⟨0 0 1 0]]

Mapping generators: ~2, ~243/196, ~5

Optimal tunings:

• WE: ~2 = 1199.822 ¢, ~243/196 = 380.604 ¢, ~5 = 2786.312 ¢
• CWE: ~2 = 1200.000 ¢, ~243/196 = 380.638 ¢, ~5 = 2786.523 ¢

Optimal ET sequence: 19, 41, 104c, 126, 145, 167, 186

Badness (Sintel): 11.758

Etymology

The word septimagic is a contraction of septimal magic into a single word.

See also

• Septimagic clan