Topic: Harmonic waste

2 scales

Thread (4 messages)

From: genewardsmith (2010-12-22)
Subject: 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

From: Jacques Dudon (2010-12-23)
Subject: Re: Harmonic waste

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

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
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
>

From: genewardsmith (2010-12-23)
Subject: Re: Harmonic waste

--- In [email protected], 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.

From: Jacques Dudon (2010-12-24)
Subject: Re: Harmonic waste

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" ;)
- - - - - - -
Jacques