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Conference Morsel: Fine Tuning

(An excerpt from the interest session “Fine Tuning Your Choir,” presented by James Franklin during the 2014 ACDA Southwestern Division Conference)
 
If we want to have a more finely tuned choir, we need to have an understanding of how intervals might sound.  We are so entrenched in a modern system of tuning, i.e., equal temperament, that many of us are not even aware that there are other tuning possibilities. 
 
Let’s compare Equal Temperament with its forefathers of tuning, Just and Pythagorean.  (Note: In equal temperament a semitone is divided into 100 cents such that C  à  C#  =  100 cents, C#  à  D  =  100 cents, and so on.)
 
System
C
D
E
F
G
A
B
C
Just
0
204
386
498
702
884
1088
1200
Equal T
0
200
400
500
700
900
1100
1200
Pyth.
0
204
408
498
702
906
1110
1200
 
Since Pythagorean tuning is built on a system of perfectly tuned fifths, we should use that value (702) as our guide.  Therefore, a true P5 should ring slightly higher than a P5 played on a piano.
 
For the fourth scale degree, the subdominant or fifth below the tonic, we arrive at a cent value of 498 by subtracting a P5 (702) from an octave (1200).
 
The cent value for the second scale degree (204) comes from stacking two fifths on top of one another.
 
          C  à  G  à  D  =  1404 cents
          1404  –  1200  =  204
 
Since just intonation is built on the concept of really pure thirds, we use that system to find a truer value for intervals of a third.  In equal temperament the major third (C à E) is 400 cents, but in just intonation the same major third is only 386 cents.  A true M3, a just M3, is actually lower than we might imagine.
 
Let’s review.
 
Scale Degree
ET Cent Value
Target Cent Value
Adjustment
5
700
702
Slightly Higher
4
500
498
Slightly Lower
2
200
204
Slightly Higher
3
400
386
Fairly Lower
 
 
on May 31, 2014 10:23am
Could someone with more vocal and theoretical knowledge than I have, please explain the relationship between vibrato and tuning?  How is it even possible to fine tune when a group of singers are all varying their pitches slightly?  Is proper vibrato a pulse in dynamic level rather than a variation in sound frequency?  Could vibrato have arisen historically as harmony became more complex and musicians groped toward the pitch adjustments that would eventually result in equal temperament?  Thank you.
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