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 72-tone JI 1.scl
!
A new 72-tone just tuning, more info at http://imanuelhab.mooo.com/new-72-tone-just-tuning-imanuel
 72
!
 94/93
 50/49
 34/33
 26/25
 21/20
 18/17
 31/29
 27/25
 12/11
 11/10
 10/9
 37/33
 17/15
 63/55
 37/32
 7/6
 33/28
 25/21
 6/5
 40/33
 49/40
 21/17
 91/73
 29/23
 14/11
 9/7
 35/27
 38/29
 37/28
 151/113
 31/23
 34/25
 11/8
 43/31
 7/5
 41/29
 10/7
 49/34
 16/11
 25/17
 43/29
 223/149
 53/35
 29/19
 37/24
 14/9
 11/7
 27/17
 77/48
 34/21
 49/30
 33/20
 5/3
 37/22
 56/33
 12/7
 45/26
 110/63
 30/17
 41/23
 9/5
 20/11
 11/6
 37/20
 43/23
 17/9
 40/21
 25/13
 33/17
 49/25
 93/47
 2/1

Prime factors and frequencies