Harmonic waste

"Harmonic waste" seems to be a useful concept in constructing circulating scales, but I think the name is very tendentious and misleading. If you optimize a scale by a method such as least squares, you often end up with so-called "waste", which by the nature of how it is constructed must be doing something and hence not be in any reasonable sense of the word "waste".

For an example, take either Vallotti-Young or Werckmeister III, the six intervals of the 5-limit diamond (aside from 1 or 2) and a tolerance of ten cents. You will end up with the exact same scale (in the case of Werckmeister III, after an extra iteration.) It's a temperament extraordinaire, and so is full of "waste". In fact, it has three major thirds in remote keys which are +- 2 cents of 14/11. I realize this would not be to everyone's taste, but please let's not call it "waste"! After all, tunings rather similar to this were in actual use: compare to rameau-french.scl in the Scala archives, "Standard French temperament, Rameau version (1726)". Then look at rousseau3.scl, "Standard French temperament Rousseau-3".

! val-werck.scl
Vallotti-Young and Werckmeister III, 10 cents least squares optimized
12
!
87.16412
192.01793
296.87174
384.03586
502.60070
582.31031
696.70413
792.01793
887.33173
1001.72555
1081.43517
1200.00000

This same thing was discussed a few years ago by Kalle Aho and others
on this list and all agreed on the same unadequation of the word
"waste". Does someone remembers what alternatives were proposed ?
Perhaps "deviation" ? (that's the one I am using in french myself).
However in a 12-tones-temperament if it occurs only in remote keys
this "waste" cannot be more perfect.
You perhaps know that in the Baroque period, Rameau and others were often using the meantone forbidden keys for "estrangetés" and other
entertainment pieces.
From the Ethno collection here is an example of a meantone full of
deviations that actually ends up very naturally on a Mills Gamelan
slendro tuning (slightly improved) on black keys - no waste at all ! :

! slendro_m-mean.scl
!
Wilson meantone from Bb to F# extended in a Slendro M on black keys
12
!
2689/2576
359/322
55/46
803/644
215/161
32/23
481/322
36/23
537/322
288/161
1199/644
2/1
! Erv Wilson Meta-meantone, x^4 = 2x + 2
! Slendro_M -c version = 385 7 63 9 2689 (> 63 7 9 385),
Dudon 1992

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Jacques

Gene wrote :

> "Harmonic waste" seems to be a useful concept in constructing
> circulating scales, but I think the name is very tendentious and
> misleading. If you optimize a scale by a method such as least
> squares, you often end up with so-called "waste", which by the
> nature of how it is constructed must be doing something and hence
> not be in any reasonable sense of the word "waste".
>
> For an example, take either Vallotti-Young or Werckmeister III, the > six intervals of the 5-limit diamond (aside from 1 or 2) and a
> tolerance of ten cents. You will end up with the exact same scale
> (in the case of Werckmeister III, after an extra iteration.) It's a
> temperament extraordinaire, and so is full of "waste". In fact, it
> has three major thirds in remote keys which are +- 2 cents of
> 14/11. I realize this would not be to everyone's taste, but please
> let's not call it "waste"! After all, tunings rather similar to
> this were in actual use: compare to rameau-french.scl in the Scala
> archives, "Standard French temperament, Rameau version (1726)".
> Then look at rousseau3.scl, "Standard French temperament Rousseau-3".
>
> ! val-werck.scl
> Vallotti-Young and Werckmeister III, 10 cents least squares optimized
> 12
> !
> 87.16412
> 192.01793
> 296.87174
> 384.03586
> 502.60070
> 582.31031
> 696.70413
> 792.01793
> 887.33173
> 1001.72555
> 1081.43517
> 1200.00000
>

--- In tuning@yahoogroups.com, Jacques Dudon <fotosonix@...> wrote:

> From the Ethno collection here is an example of a meantone full of
> deviations that actually ends up very naturally on a Mills Gamelan
> slendro tuning (slightly improved) on black keys - no waste at all

A very cool concept, Jacques--thanks for providing the scale. But of course, it's a modified meantone, not a circulating temperament, so "harmonic waste" doesn't really apply.

Gene wrote :

> But of course, it's a modified meantone, not a circulating > temperament, so "harmonic waste" doesn't really apply.

Right. I was confusing my simpler notion of "deviation" (= in respect of a necessary tempering, has a tuning any "deviations" for h3, h5, h7 etc. ?) with the notion of fifths, or major thirds "harmonic waste" which is a measurement (according to Claudio Di Veroli) of the sum of the "errors" of these intervals minus the normal error if they were pure fifths or major thirds, respectively one pythagorean comma, or four diesis (128/125).

> "Harmonic waste" seems to be a useful concept in constructing > circulating scales, but I think the name is very tendentious and > misleading. If you optimize a scale by a method such as least > squares, you often end up with so-called "waste", which by the > nature of how it is constructed must be doing something and hence > not be in any reasonable sense of the word "waste".
>
> For an example, take either Vallotti-Young or Werckmeister III, the > six intervals of the 5-limit diamond (aside from 1 or 2) and a > tolerance of ten cents. You will end up with the exact same scale > (in the case of Werckmeister III, after an extra iteration.) It's a > temperament extraordinaire, and so is full of "waste". In fact, it > has three major thirds in remote keys which are +- 2 cents of > 14/11. I realize this would not be to everyone's taste, but please > let's not call it "waste"! After all, tunings rather similar to > this were in actual use: compare to rameau-french.scl in the Scala > archives, "Standard French temperament, Rameau version (1726)". > Then look at rousseau3.scl, "Standard French temperament Rousseau-3".
>
> ! val-werck.scl
> Vallotti-Young and Werckmeister III, 10 cents least squares optimized
> 12
> !
> 87.16412
> 192.01793
> 296.87174
> 384.03586
> 502.60070
> 582.31031
> 696.70413
> 792.01793
> 887.33173
> 1001.72555
> 1081.43517
> 1200.00000

What were temperaments "extraordinaires" exactly ? I often heard this term but I can't find a definition of it. I suppose they came after the "temperaments ordinaires", and would have summed less harmonic wastes. Was it their main goal ? It's true that Werckmeister III and Valotti-Young's three larger thirds are 81/64 and according to the definition of harmonic wastes, sum up to zero waste wether in fifths or in thirds. In a a certain context it can be seen as a quality. In other contexts, accepting "5-limit harmonic wastes" can open to other qualities, among which more diversity. I would not say yours is "full of waste" anyway, but rather "full of taste" ;)
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Jacques