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Imanuel's JI 1 – A new just tuning

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 (.scl) Try it online

Scala file contents

! Imanuel's JI 1.scl
!
A new just tuning, more info at http://imanuelhab.mooo.com/the_strangest_scales/new-just-tuning-imanuel
 12
!
 17/16
 9/8
 19/16
 29/23
 4/3
 17/12
 3/2
 19/12
 37/22
 16/9
 17/9
 2/1

Prime factors and frequencies

Frequencies (Hz)

D♭ E♭ G♭ A♭ B♭
C C♯ D D♯ E F F♯ G G♯ A A♯ B C
261.6 278 294.3 310.7 329.9 348.8 370.6 392.4 414.2 440 465.1 494.2 523.3

Logarithmic frequency values (cents)

D♭ E♭ G♭ A♭ B♭
C C♯ D D♯ E F F♯ G G♯ A A♯ B C
0 105 203.9 297.5 401.3 498 603 702 795.6 900 996.1 1101 1200
0 100 200 300 400 500 600 700 800 900 1000 1100 1200
0 5 3.9 -2.5 1.3 -2 3 2 -4.4 0 -3.9 1 0

Numerators

D♭ E♭ G♭ A♭ B♭
C C♯ D D♯ E F F♯ G G♯ A A♯ B C
1 17 9 19 29 4 17 3 19 37 16 17 2
17 3 19 29 2 17 3 19 37 2 17 2
3 2 2
2
2

Denominators

D♭ E♭ G♭ A♭ B♭
C C♯ D D♯ E F F♯ G G♯ A A♯ B C
1 16 8 16 23 3 12 2 12 22 9 9 1
2 2 2 23 3 3 2 3 11 3 3
2 2 2 2 2 2 3 3
2 2 2 2 2
2 2

Intervals

1 17 9 19 29 4 17 3 19 37 16 17 2
1 16 8 16 23 3 12 2 12 22 9 9 1
3 76 148 256 391 3 3 3 3 1276 3 3 3
2 51 99 171 261 2 2 2 2 851 2 2 2

Shift intervals in table


Numerators (circle of fifths)

G♭ D♭ A♭ E♭ B♭
C G D A E B F♯ C♯ G♯ D♯ A♯ F C
1 3 9 37 116 68 34 17 76 38 512 256 128
3 3 37 29 17 17 17 19 19 2 2 2
3 2 2 2 2 2 2 2 2
2 2 2 2 2 2
2 2 2
2 2 2
2 2 2
2 2 2
2 2
2

Denominators (circle of fifths)

G♭ D♭ A♭ E♭ B♭
C G D A E B F♯ C♯ G♯ D♯ A♯ F C
1 2 4 11 23 9 3 1 3 1 9 3 1
2 2 11 23 3 3 3 3 3
2 3 3

Intervals

1 3 9 37 116 68 34 17 76 38 512 256 128
1 2 4 11 23 9 3 1 3 1 9 3 1
3 3 148 1276 391 3 3 76 3 256 3 3 3
2 2 99 851 261 2 2 51 2 171 2 2 128

Shift intervals in table by fifths


Numerators (circle of fourths)

B♭ E♭ A♭ D♭ G♭
C F A♯ D♯ G♯ C♯ F♯ B E A D G C
1 4 16 19 19 17 17 68 232 148 18 24 32
2 2 19 19 17 17 17 29 37 3 3 2
2 2 2 2 2 3 2 2
2 2 2 2 2 2 2
2 2 2 2
2

Denominators (circle of fourths)

B♭ E♭ A♭ D♭ G♭
C F A♯ D♯ G♯ C♯ F♯ B E A D G C
1 3 9 8 6 4 3 9 23 11 1 1 1
3 3 2 3 2 3 3 23 11
3 2 2 2 3
2

Intervals

1 4 16 19 19 17 17 68 232 148 18 24 32
1 3 9 8 6 4 3 9 23 11 1 1 1
4 4 171 4 51 4 4 522 851 99 4 4 4
3 3 128 3 38 3 3 391 638 74 3 3 3

Shift intervals in table by fourths



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