Temperament Morphing

[Kent Swafford]

Thanks for the interest in temperament morphing.

Some needed corrections are included below.

Kent Swafford wrote:

>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.

I was calculating 3rd widths with a spread sheet and it did exactly what 
I told it to do which was to exclude a whole column of numbers from the 
calculations. This incorrect calculation led me to think that the Valotti 
key colors are less than they actually are, which led me to to make an 
"exaggerated" Valotti 120% temperament. Such a temperament would be "over 
the line" and probably of little or no musical value.

>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.

Wrong calculations. 

The Valotti 120% thirds would range from 3.8 to 23.6 cents; like I said, 
"over the line and of little or no musical use".

The Valotti 20% thirds actually range from 11.7 to 15.7 cents.

>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!

This "Valotti-Young 1799 Well-Temperament" is the one included in RCT in 
the historical temperament window, and is a different temperament than 
the supremely symmetrical temperament that Ed Foote presented in his 
class.

>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