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Imanuel's Pelog 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 Pelog JI 1.scl
!
A new just tuning, more info at http://imanuelhab.mooo.com/pelog-new-just-tuning-imanuel
 9
!
 14/13
 7/6
 5/4
 15/11
 19/13
 19/12
 12/7
 13/7
 2/1

Prime factors and frequencies

Frequencies (Hz)

D♭ E♭ A♭
C C♯ D♯ E F G G♯ A B C
261.6 281.8 305.2 327 356.8 382.4 414.2 448.5 485.9 523.3

Logarithmic frequency values (cents)

D♭ E♭ A♭
C C♯ D♯ E F G G♯ A B C
0 96.2 200.2 289.7 402.7 492.7 596.7 699.8 803.8 900
0 100 200 300 400 500 600 700 800 900
0 -3.8 0.2 -10.3 2.7 -7.3 -3.3 -0.2 3.8 0

Logarithmic frequency values (cents, for 12-ET with octave equivalence)

D♭ E♭ A♭
C C♯ D♯ E F G G♯ A B C
0 128.3 266.9 386.3 537 657 795.6 933.1 1071.7 1200
0 100 300 400 500 700 800 900 1100 1200
0 28.3 -33.1 -13.7 37 -43 -4.4 33.1 -28.3 0
C C♯ D♯ E F G G♯ A B C
D♭ E♭ A♭

Numerators

D♭ E♭ A♭
C C♯ D♯ E F G G♯ A B C
1 14 7 5 15 19 19 12 13 2
7 7 5 5 19 19 3 13 2
2 3 2
2

Denominators

D♭ E♭ A♭
C C♯ D♯ E F G G♯ A B C
1 13 6 4 11 13 12 7 7 1
13 3 2 11 13 3 7 7
2 2 2
2

Intervals

1 14 7 5 15 19 19 12 13 2
1 13 6 4 11 13 12 7 7 1
19 247 72 52 22 364 168 70 210 38
13 168 49 35 15 247 114 48 143 26
1 1 1 1 1 13 6 2 1 2
19 247 72 52 22 28 28 35 210 19
13 168 49 35 15 19 19 24 143 13
19 247 72 52 22 28 28 35 210 19
13 168 49 35 15 19 19 24 143 13
5 195 114 76 132 169 24 196 98 10
4 154 91 60 105 133 19 156 78 8
1 1 1 4 3 1 1 4 2 2
5 195 114 19 44 169 24 49 49 5
4 154 91 15 35 133 19 39 39 4
5 195 114 19 44 169 24 49 49 5
4 154 91 15 35 133 19 39 39 4
14 91 30 60 209 247 144 91 14 28
13 84 28 55 195 228 133 84 13 26
1 7 2 5 1 19 1 7 1 2
14 13 15 12 209 13 144 13 14 14
13 12 14 11 195 12 133 12 13 13
14 13 15 12 209 13 144 13 14 14
13 12 14 11 195 12 133 12 13 13

Shift intervals in table

Numerators (circle of fifths)

D♭ A♭ E♭
C G C♯ G♯ D♯ A E B F C
1 19 28 19 14 48 10 104 240 32
19 7 19 7 3 5 13 5 2
2 2 2 2 2 3 2
2 2 2 2 2
2 2 2 2
2 2 2
2

Denominators (circle of fifths)

D♭ A♭ E♭
C G C♯ G♯ D♯ A E B F C
1 13 13 6 3 7 1 7 11 1
13 13 3 3 7 7 11
2

Intervals

1 19 28 19 14 48 10 104 240 32
1 13 13 6 3 7 1 7 11 1
19 364 247 84 144 70 104 1680 352 608
13 247 168 57 98 48 70 1144 240 416
1 13 1 3 2 2 2 8 16 32
19 28 247 28 72 35 52 210 22 19
13 19 168 19 49 24 35 143 15 13
19 28 247 28 72 35 52 210 22 19
13 19 168 19 49 24 35 143 15 13

Shift intervals in table by fifths

Numerators (circle of fourths)

E♭ A♭ D♭
C F B E A D♯ G♯ C♯ G C
1 15 13 5 24 14 19 112 152 16
5 13 5 3 7 19 7 19 2
3 2 2 2 2 2
2 2 2 2
2 2 2 2
2

Denominators (circle of fourths)

E♭ A♭ D♭
C F B E A D♯ G♯ C♯ G C
1 11 7 2 7 3 3 13 13 1
11 7 2 7 3 3 13 13

Intervals

1 15 13 5 24 14 19 112 152 16
1 11 7 2 7 3 3 13 13 1
15 143 35 48 98 57 336 1976 208 240
11 105 26 35 72 42 247 1456 152 176
1 1 1 1 2 3 1 104 8 16
15 143 35 48 49 19 336 19 26 15
11 105 26 35 36 14 247 14 19 11
15 143 35 48 49 19 336 19 26 15
11 105 26 35 36 14 247 14 19 11

Shift intervals in table by fourths

Copyright © 2020 Imanuel Habekotté

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