Topic: More on 14/11 exotic temperaments

1 scales

Thread (1 messages)

From: Gene Ward Smith (2003-03-17)
Subject: More on 14/11 exotic temperaments

If we have a circle of fifths of the form

[a, a, a, b, a, a, a, b, a, a, a, 128/(a^9*b^2)]

we get corresponding thirds of the form

[1/4*a^3*b, 1/4*a^3*b, 1/4*a^3*b, 1/4*a^3*b, 1/4*a^3*b, 1/4*a^3*b, 
1/4*a^3*b, 1/4*a^3*b, 32/(a^6*b^2), 32/(a^6*b^2), 32/(a^6*b^2),
32/(a^6*b^2)]

If we set b = 3/2, the thirds become

[3/8*a^3, 3/8*a^3, 3/8*a^3, 3/8*a^3, 3/8*a^3, 3/8*a^3, 3/8*a^3, 3/8*a^3,
128/(9*a^6), 128/(9*a^6), 128/(9*a^6), 128/(9*a^6)]

If now 128/(9*a^6) = 14/11, then a = (704/63)^(1/6). Substituting that
value for a in the above gives us a circle of fifths

[a, a, a, 3/2, a, a, a, 3/2, a, a, a, 21*sqrt(77)/121]

and corresponding major thirds

[sqrt(11/7), sqrt(11/7), sqrt(11/7), sqrt(11/7), sqrt(11/7),
sqrt(11/7), sqrt(11/7), sqrt(11/7), 14/11, 14/11, 14/11, 14/11]

(704/63)^(1/6) is a fifth of 696.43 cents, about a seventh of a cent
flatter
than 1/4-comma meantone, and sqrt(11/7) is a major third of 391.25 cents,
which is sharp by sqrt(176/175), or 4.93 cents. The system is
evidently quite
practical, and interesting if you have any use for pure 11/7 intervals.

Here is the temperament in cents:

! smith-exotic1.scl
Exotic temperament featuring four pure 14/11 thirds and two pure fifths
12
!
86.061694
198.385340
310.708979
391.246017
503.569655
589.631356
701.955001
782.492032
894.815678
1007.139316
1093.201017
2/1