Imanuel's website

Description

These "just" scales are based on equal temperament. All intervals are aimed to be within a schisma (32805/32768, ~1.95 cents) of the approximated interval from equal temperament.

I wrote a program that evaluates fractional approximations to the intervals by repeatedly incrementing the numerator of the fraction. The denominator is increased accordingly. These approximations are assigned a "distance" (or, inverse score) by dividing the larger of the two intervals (either the approximated equal temperament interval, or the approximation itself) by the smaller one. The program for a specific interval will stop when a sufficiently good approximation is found (I chose a distance below a schisma to be "sufficiently good").

One other factor however is also introduced, which often times makes these distances larger than that schisma in the end: the interval encompassing both the harmonics and the subharmonics (the product of the numerator and denominator) should be smaller than 8 octaves (2^8 = 256).

For getting the prime-limit scales, I could simply skip the approximations where either the numerator or the denominator had prime factors above the designated value. For example a 19-limit by definition has no prime factors above 19 in the numerator or denominator.

Try it now / Get Scala file

Scala file contents

! Imanuel's 31-tone JI 1.scl
!
A new just tuning, more info at http://imanuelhab.mooo.com/new-31-tone-just-tuning-imanuel
 31
!
 38/37
 21/20
 15/14
 12/11
 19/17
 8/7
 7/6
 6/5
 11/9
 5/4
 14/11
 17/13
 4/3
 15/11
 7/5
 10/7
 19/13
 3/2
 23/15
 11/7
 8/5
 18/11
 5/3
 12/7
 7/4
 25/14
 11/6
 15/8
 21/11
 37/19
 2/1

Prime factors and frequencies

Other tunings