The most pure ratios are:
9/8 = Major Second.
6/5 = Minor Third.
4/3 = Perfect Fourth.
3/2 = Perfect Fifth.
Use the ratio 3/2 for C-G (perfect fifth).
Use the ratio 9/8 for C-D (major second) and F-G (major second).
This will cause the ratio of C-F (perfect fourth) and D-G (perfect fourth) to become 4/3.
The ratio of D-F (minor third) should ideally be 6/5 = 315.64 cents. However, this is not possible.
By using the arrangement I have described above, the ratio of D-F (minor third) would become 4/3 ÷ 9/8 = 4/3 × 8/9 = 32/27 = 294.13 cents.
D-F (minor third) will be 315.64 - 294.13 = 21.51 cents out of tune. This is called the syntonic comma.
Therefore, as you have rightly said, the solution is to widen D-F so that all the intervals except for the perfect fifth become equally out of just. Jason has made an interesting point. The aim is to create equal temperament within a just perfect fifth.
This can be achieved by widening D-F by 2/3 syntonic comma and by narrowing C-D and F-G by 1/3 syntonic comma.
D-F = 294.13 + 14.34 = 308.47 cents.
C-D and F-G = 203.91 - 7.17 = 196.74 cents.
The following interval will be pure:
Perfect fifth = C-G = 701.95 cents.
The following intervals will be equally out of just:
Major seconds = C-D and F-G = 196.74 cents.
Minor third = D-F = 308.47 cents.
Perfect fourths = C-F and D-G = 505.21 cents.
Offsets from just intonation in cents:
C = 0.00 - 0.00 = 0.00
D = 196.74 - 203.91 = −7.17
F = 505.21 - 498.04 = +7.17
G = 701.95 - 701.96 = −0.01
Offsets from 12-tone equal temperament in cents:
C = 0.00 - 0.00 = 0.00
D = 196.74 - 200 = −3.26
F = 505.21 - 500 = +5.21
G = 701.95 - 700 = +1.95
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Roshan Kakiya
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Original Message:
Sent: 08-21-2018 02:08
From: Cobrun Sells
Subject: Just Intonation Dilemma
I haven't done much study into Just Intonation, but I have had this dilemma for a while:
How do I tune the chord with notes 1, 2, 4, 5 so that 1 & 2, 1 & 4, 1 & 5, 2 & 4, 2 & 5, 4 & 5 are all Just ("not beating")? Is it possible? I've been able to generate the frequency of each tone on the software Audacity so that all intervals are pure except for the interval 2 & 4.
Is the solution to temper the frequency of 2 & 4 as to widen the interval enough so that 1 & 2, 1 & 4, 2 & 4, 2 & 5, 4 & 5 beat equally out of Just?
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Cobrun Sells
C.J. Piano Tuner
www.cjtuner.com
cobrun94@yahoo.com
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