Termperament Morphing

[Kent Swafford]

I found Ed Foote¹s class at the KC Institute to be extremely fascinating. 
Ed¹s graphs which showed the changes of the widths of thirds through the 
circle of fifths in various temperaments was a revelation.

The various graphs of Jim Coleman¹s temperaments made clear that various 
temperaments have been developed that provide choices in the amount of 
key color from temperament to temperament.

The Valotti temperament from 1799 was interesting because the widths in 
cents of the 3rds progress evenly and symmetrically through the circle of 
5ths, successively wider from C to F# then narrower back to C. (The 
Coleman 16 is similar but doesn¹t try for perfect symmetry.)

I was particularly struck by Ed¹s observation that computerized tuning 
devices can make the historical temperaments easily available to us. My 
own experience with temperaments other than 12 tone equal temperament has 
been almost entirely on the computer. I have MIDI files that play 19, 24, 
and 48 tone to the octave equal temperament in addition to various 
historical 12 tone temperaments.

I started to think about the math functions built into Dean Reyburn¹s 
CyberTuner. Bob Bussell has pointed out (and Ed has picked up on) the 
fact that the averaging function can be used to immediately create a 
temperament that is half-way in between any two temperaments, say between 
equal temperament and one of the mean-tone temperaments. Ed pointed out 
that the average of equal temperament and a mean tone temperament creates 
a temperament that approaches a well temperament.

I realized that not only could you create a temperament that is half-way 
in between any two temperaments, but you could morph from one to the 
other in a virtually infinite number of intermediate steps. A single 
temperament could then become a whole class of related temperaments.

So I fired up RCT and spent a few minutes morphing between Equal 
Temperament and Valotti, in celebration I guess of the bicentennial of 
this temperament. I even created an exaggerated temperament with _more_ 
key color than Valotti. Each of these temperaments is symmetrical like 
the Valotti but vary in the amount of key color, measurable in the change 
in width of the thirds. The 120% 3rds range from 6.3 cents to 21.1 cents, 
while the 20% 3rds range only from 12.2 to 15.2, a _very_ mild unequal 
temperament.

Such temperament morphing might be useful in a couple of ways. First 
there is the ability to control key contrasts to any degree one wishes. 
Second, temperament morphs might be a way of correcting for the 
inharmonicity levels of the various instruments we tune. A criticism of 
electronic tuning of historical temperaments using cents deviations has 
been the fact that inharmonicity of the modern piano affects the outcome 
of the temperament. Well, fine, we can correct for that inharmonicity, if 
we choose.

Thanks, Ed, for this kick in the brain.

Kent Swafford

PS--I am attaching an RCT file with this post to be sent to the RCT list, 
and for the benefit of interested pianotech readers, below are the 
deviation numbers of some morphs.

The per centage numbers are as compared to the key contrasts of the 
³100%² Valotti temperament.

Standard disclaimers apply!

(Valotti-Young 120%)
C 	7.4
G 	4.9
D 	2.5
A 	0
E	 -2.5
B	-4.9
F# 	-7.4
C# 	-4.9
G# 	-2.5
D# 	0
A# 	2.5
F	 4.9
C	 7.4

Valotti-Young 1799 Well-Temperament		
C		5.9
G		3.9
D		2
A		0
E		-2
B		-3.9
F#		-5.9
C#		-3.9
G#		-2
D#		0
A#		2
F		3.9
C		5.9

(Valotti-Young 80%)
C		4.4
G		2.9
D		1.5
A		0
E		-1.5
B		-2.9
F#		-4.4
C#		-2.9
G#		-1.5
D#		0
A#		1.5
F		2.9
C		4.4

(Valotti-Young 67%)
C		3.9
G		2.6
D		1.3
A		0
E		-1.3
B		-2.6
F#		-3.9
C#		-2.6
G#		-1.3
D#		0
A#		1.3
F		2.6
C		3.9

(Valotti-Young 50%)
C		3
G		2
D		1
A		0
E		-1
B		-2
F#		-3
C#		-2
G#		-1
D#		0
A#		1
F		2
C		3

(Valotti-Young 33%)
C		2
G		1.3
D		0.7
A		0
E		-0.7
B		-1.3
F#		-2
C#		-1.3
G#		-0.7
D#		0
A#		0.7
F		1.3
C		2

(Valotti-Young 20%)
C		1.5
G		1
D		0.5
A		0
E		-0.5
B		-1
F#		-1.5
C#		-1
G#		-0.5
D#		0
A#		0.5
F		1
C		1.5