Thanks for your thoughts.

Mathematical music theory, and the computer, may provide some alternatives 
here. In the first place, the mathematical theory of voice-leading and chord 
relations has been developed in ways that do indeed apply to non 12-tone 
equally tempered scales. The starting point, of course, assuming that an 
octave is size 12, is to define a pitch-class as a real number under 12 
equivalence, not an integer under 12 equivalence.

That done, various scales and tuning systems can be brought within the 
computer, and the voice-leading operations that produce what Westerners love 
in harmony and counterpoint can be applied to those systems as well. Instead 
of using 12-tone equal temperament to accomodate modulation, the tuning 
system can pivot around common tones or sub-scales - adaptive tuning. This 
is quite simple on computers and rather difficult on physical instruments 
with fixed pipes, harps, pianos, or fretted strings.

Regards,
Mike

----- Original Message ----- 
From: "Richard Dobson" 
To: 
Sent: Saturday, February 14, 2009 6:59 AM
Subject: [Csnd] Re: Re: Re: Re: Re: Re: Csound for ancient Greek music


> Chuckk Hubbard wrote:
>> I've never felt comfortable when modern musicians state that harmony
>> never existed until Western music.  It often coincides with an
>> "explanation" of why 12-tone equal temperament is better than just
>> intonation.
>>
>
> Of course it ~existed~; it was just expressed and understood differently.
>
> Western harmony and polyphony is an extraordinary achievement, without 
> which we can scarcely live. But it has meant the loss of some things of 
> value, as (in the vast majority of instances) it depends on a more or less 
> rigid set of fixed pitch classes, whereas with a purely horizontal 
> formalization, pitches need be neither rigid nor fixed.
>
> As I say to my 12-tone friends when I am improvising on bansuri -"you have 
> your twelve semitones; I have all the spaces in between as well". It is a 
> lot more than simple portamento. Indeed, many western vernacular styles do 
> in many ways seek to combine both : 'bent' notes in jazz, the various 
> vocal kicks and wobbles  of contemporary vocalizing, borrowed variously 
> from Qawwali, Fado, Flamenco etc, but over chord changes. But little or 
> none of that can be written down, which in the West conventionally makes 
> it somehow less important  to some minds  than material that ~can~ be 
> written down.
>
> As soon as western composers invented modulation and keyboards (and staff 
> notation), they were basically stuffed from a "harmony" point of view, as 
> it led to the above-mentioned temperament problem in which (especially if 
> you include the piano's stretched octave), everything is more-or-less 
> systematically out of tune. Needless to say, singers  and players who can 
> bend pitch do so, taking them far away from anything  a practical keyboard 
> can manage. But, it is all to easy to make the result chaotic, as everyone 
> tries continually to adjust to each other vertically, while also trying to 
> make a coherent job of the horizontal. A lot of the time, it has to be 
> said, it doesn't quite work, and our resultant sense of "harmony" is more 
> a case of willful suspension of disbelief than of anything founded in 
> either mathematics or acoustics!
>
> On the bright side, it all means that even today, theory and practice are 
> sufficiently divergent to ensure the continued employment of both 
> theorists and (I hope!) musicians for many years to come.
>
> Richard Dobson
>
>
>
>
>
>
>
>
> Send bugs reports to this list.
> To unsubscribe, send email sympa@lists.bath.ac.uk with body "unsubscribe 
> csound"
> 

michael.gogins@gmail.com wrote:
> Thanks for your thoughts.
> 
> Mathematical music theory, and the computer, may provide some 
> alternatives here. In the first place, the mathematical theory of 
> voice-leading and chord relations has been developed in ways that do 
> indeed apply to non 12-tone equally tempered scales. The starting point, 
> of course, assuming that an octave is size 12, is to define a 
> pitch-class as a real number under 12 equivalence, not an integer under 
> 12 equivalence.
> 
> That done, various scales and tuning systems can be brought within the 
> computer, and the voice-leading operations that produce what Westerners 
> love in harmony and counterpoint can be applied to those systems as 
> well. Instead of using 12-tone equal temperament to accomodate 
> modulation, the tuning system can pivot around common tones or 
> sub-scales - adaptive tuning. This is quite simple on computers and 
> rather difficult on physical instruments with fixed pipes, harps, 
> pianos, or fretted strings.
> 

Only problem is that I suspect it's non-algorithmic. In particular, 
there are very likely multiple alternative best-fit solutions, and no 
"perfect" solutions (unless you use a lot of vibrato!). The choice of 
solution may come down to taste, some ingredient-X that cannot be 
modelled, or toss of a coin.

Richard Dobson