Hiro raised an interesting question:
how Just is a Just-tempered D whistle if you don’t play the D major scale?
I had tinkered with some calculations, converting hertzes to cents, etc. and might as well share them…
For the sake of simplicity:
- We’ll assume every instrument or scale is tuned to the common modern A (440 Hz)
- We’ll check only the basic 3 major scales playable on the D whistle: D, G, A. Transpose to other fundamentals, bell tone, if you wish.
- We shan’t deal for now with the accidentals in a given scale, to avoid confusing the matters with the chromatic/diatonic semitones issue.
- We’re using a standard low D whistle, where accidentals (C, G#) are cross-fingered for best (or worst) tuning.
Also–sorry–I don’t know how to enter cleanly formated tables with this online editor…
Frequencies for the different tunings
(rounded to one hertz)
a) Equal temperament b) Just D scale c) Just G scale d) Just A scale
Tone Equal justD justG justA
D_ 294 293 293 293
E_ 330 330 326 330
F# 370 367 367 367
G_ 392 391 391 ***
G# 415 *** *** 413
A_ 440 440 440 440
B_ 494 489 489 495
C_ 523 *** 521 ***
C# 554 550 *** 550
d_ 587 587 587 587
NB: *** semitones accidental in a given scale. Equal temperament as a reference.
Comparing the tuning of Equal temperament and Just tuning.
Compared to (1) Just D, (2) Just G, (3) Just A
the equal temperament is unjust by:
(in positive cents if sharp, negative if flat)
Tone justD justG justA
D 2 2 2
E -2 20 -2
F# 16 16 16
G 4 4 **
G# ** ** ••
A 00 00 00
B 18 18 -4
C •• 6 **
C# 14 ** 14
d 2 2 2
•• NB: let’s assume for the cross-fingerers the C-nat (oxx ooo) is meant for G scale, and some kinda (xxo xxx) G# for A scale. Half-fingerers, having an absolute ear, will get by… :roll:
This table shows the F# is sharp in every played scale. Also, the equal-temperament tuned whistle is systematically sharp when “un-just”…
How just are G and A scales when played on the D just-tuned whistle
The just-tempered D whistle is “unjust” by (cents) when playing in (1) G scale (2) A scale.
D 0 0
E 22 0
F# 0 0
G 0 **
G# ** ••
A 0 0
B 0 -22
C •• **
C# ** 0
d 0 0
One can see the D just-tune whistle is only off:
- in G scale, for the E, sharp by 22 cents (note the same happens to this scale with the equal-tempred whistle
- in A scale, for the B, flat by 22 cents (compared to the equal tempered whistle playing it sharp, the difference may be striking–a quartertone)
Also, an equal-tempered whistle is systematically sharp by comparable values for at least two tones in the scale.
Conclusion:
On theoretical grounds, there seems no disadvantage of getting a whistle tuned to D just temperament, as far as flexibility with different scales are concerned.