Notice: Undefined index: HTTP_ACCEPT_LANGUAGE in D:\htdocs\imanuelhab\languages.php on line 27
Imanuel's JI 1 ⋄ Imanuel's website ⋄

Imanuel's website

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.

Prime factors and frequencies

Frequencies (Hz)

D♭E♭G♭A♭B♭
CC♯DD♯EFF♯GG♯AA♯BC
261.6278294.3310.7329.9348.8370.6392.4414.2440465.1494.2523.3

Logarithmic frequency values (cents)

D♭E♭G♭A♭B♭
CC♯DD♯EFF♯GG♯AA♯BC
0105203.9297.5401.3498603702795.6900996.111011200
0100200300400500600700800900100011001200
053.9-2.51.3-232-4.40-3.910

Numerators

D♭E♭G♭A♭B♭
CC♯DD♯EFF♯GG♯AA♯BC
117919294173193716172
1731929217319372172
322
2
2

Denominators

D♭E♭G♭A♭B♭
CC♯DD♯EFF♯GG♯AA♯BC
1168162331221222991
2222333231133
22222233
22222
22

Intervals

117919294173193716172
1168162331221222991
37614825639133331276333
251991712612222851222
296434244371116434241878117129
23512719348885127191486413623
17181946492171819222352171817
16171843787161718209333161716

Shift intervals in table


Numerators (circle of fifths)

G♭D♭A♭E♭B♭
CGDAEBF♯C♯G♯D♯A♯FC
139371166834177638512256128
3337291717171919222
322222222
222222
222
222
222
222
22
2

Denominators (circle of fifths)

G♭D♭A♭E♭B♭
CGDAEBF♯C♯G♯D♯A♯FC
124112393131931
22112333333
233

Intervals

139371166834177638512256128
124112393131931
33148127639133763256333
229985126122512171222

Shift intervals in table by fifths


Numerators (circle of fourths)

B♭E♭A♭D♭G♭
CFA♯D♯G♯C♯F♯BEADGC
14161919171768232148182432
2219191717172937332
22222322
2222222
2222
2

Denominators (circle of fourths)

B♭E♭A♭D♭G♭
CFA♯D♯G♯C♯F♯BEADGC
139864392311111
33232332311
32223
2

Intervals

14161919171768232148182432
139864392311111
441714514452285199444
331283383339163874333

Shift intervals in table by fourths


Other tunings

Imanuel's JI 2Imanuel's JI 3Imanuel's 53-tone JI 1
Imanuel's 31-tone JI 1Imanuel's 31-tone JI 2Imanuel's Bohlen-Pierce JI 1
Gert Kramer's Divine 9Bohlen-Pierce JIImanuel's 72-tone JI 1
Imanuel's Pelog JI 1

Copyright © 2019 Imanuel Habekotté

Page rendered on 23-03-2019 at 14:35:22, it took 9.324251 seconds