Byzantine scale and Ets

I wrote!:

> I don't like ro ignore the 225/224,
>
> I can assure you that I can feel the difference between 15/14 and 16/15
> which is 225/224,
> and the difference between 28/27 and 25/24 which is 225/224.
>
> Rami Vitale

Dears,

As you may noticed I'm not so good in English, so my messages may have many mistakes.

Paul Erlich wrote:

> Hi Rami,
>
> This may be true, but there is a difference between _tempering out_ the 225:224, and
> _ignoring_ the 225:224. For a similar example, look at the Western European diatonic scale in
> meantone temperament (used 1500-1800), where the 81:80 is tempered out. Although 81:80
> is 22 cents, in meantone temperament, no consonant interval is off JI by more than 6 cents.Dears

Dave Keenan wrote:

> Rami,
>
> Yes the difference between 16/15 and 15/14 etc. is perceptible, but if
> we have an interval that is halfway between the two, then the
> difference between that and either 16/15 or 15/14 is only 3.9 cents
> (you'd probably find more variation than that between singers). Here's
> the optimal tempering of the steps in the 19 note version of your
> scale, which still includes those five 7-note Byzantine scales, and is
> itself a more even scale.

OK. I understand that.

But look into this situation.

Look to these two tetrachords:

15/14 7/6 16/15

16/15 7/6 15/14

Although they are somehow similar, but every one of them has a special independent taste.

When you temper-out the 225/224, these deferent scales ( tetrachords ) will be the same, so, something is missing here.

Another thing, if the studier of the scale or the musician ( and who is not a musicologist ), will see different intervals presented equal, then we are leading him to incorrect conclusions. This happened a lot in the Western, Arabic and Byzantine music.

I think when dividing an octave into small equal intervals there are three important considerations:
1.. An interval ( such as 28/27 ) must be presented always in one number wherever it appears in the scales.
2.. Two different intervals ( such as 15/14 & 16/15 ) must be presented differently, and of course the number expresses 15/14 must be larger than the one expresses 16/15.
3.. If we have two adjoining intervals ( a & b ) such as ( 12/20 & 15/14 ) , and if ( a * b =c ) such as ( 12/20 * 15/14 = 9/8 ), then the number expresses "c" must be equal to [ then number expresses "b" plus the number expresses "c" ].
These three considerations are important beside the mathematical accuracy in order to give the studier right scene about the whole scale, and to let him conclude right conclusions without involving with the hard-to-understand intervals presented in the form of a/b.

It is really hard and complicated problem, but is it solvable? I think in Byzantine music it is, I will send the solve I see soon.

Rami Raoul Vitale
Mathematics student
Lattakia-Syria

--- In tuning@y..., "Rami Vitale" <alfred1@s...> wrote:
> I wrote!:
>
> > I don't like ro ignore the 225/224,
> >
> > I can assure you that I can feel the difference between 15/14 and 16/15
> > which is 225/224,
> > and the difference between 28/27 and 25/24 which is 225/224.
> >
> > Rami Vitale
>
>
> Dears,
>
> As you may noticed I'm not so good in English, so my messages may have many mistakes.
>
> Paul Erlich wrote:
>
> > Hi Rami,
> >
> > This may be true, but there is a difference between _tempering out_ the 225:224, and
> > _ignoring_ the 225:224. For a similar example, look at the Western European diatonic scale in
> > meantone temperament (used 1500-1800), where the 81:80 is tempered out. Although 81:80
> > is 22 cents, in meantone temperament, no consonant interval is off JI by more than 6 cents.Dears
>
> Dave Keenan wrote:
>
> > Rami,
> >
> > Yes the difference between 16/15 and 15/14 etc. is perceptible, but if
> > we have an interval that is halfway between the two, then the
> > difference between that and either 16/15 or 15/14 is only 3.9 cents
> > (you'd probably find more variation than that between singers). Here's
> > the optimal tempering of the steps in the 19 note version of your
> > scale, which still includes those five 7-note Byzantine scales, and is
> > itself a more even scale.
>
> OK. I understand that.
>
> But look into this situation.
>
> Look to these two tetrachords:
>
> 15/14 7/6 16/15
>
> 16/15 7/6 15/14
>
> Although they are somehow similar, but every one of them has a special independent taste.

Well 15:14 * 7:6 = 5/4, which is a note you can harmonically lock into against the drone.

With 16:15 as the bottom interval, you don't get that 5/4.

But that doesn't prove anything. How do you know the second step is exactly 7:6?

> When you temper-out the 225/224, these deferent scales ( tetrachords ) will be the same, so, something is missing here.

It's not a fair comparison . . . see above.
>
> Another thing, if the studier of the scale or the musician ( and who is not a musicologist ), will see different intervals presented equal, then we are leading him to incorrect conclusions. This happened a lot in the Western, Arabic and Byzantine music.

OK, I'm willing to grant you this. But how do you know the precise ratio 15:14 is being sung against the 1/1 drone?

>
>
> I think when dividing an octave into small equal intervals there are three important considerations:
> 1.. An interval ( such as 28/27 ) must be presented always in one number wherever it appears in the scales.
> 2.. Two different intervals ( such as 15/14 & 16/15 ) must be presented differently, and of course the number expresses 15/14 must be larger than the one expresses 16/15.
> 3.. If we have two adjoining intervals ( a & b ) such as ( 12/20 & 15/14 ) , and if ( a * b =c ) such as ( 12/20 * 15/14 = 9/8 ), then the number expresses "c" must be equal to [ then number expresses "b" plus the number expresses "c" ].

I call this "consistency".

> These three considerations are important beside the mathematical accuracy in order to give the studier right scene about the whole scale, and to let him conclude right conclusions without involving with the hard-to-understand intervals presented in the form of a/b.
>
> It is really hard and complicated problem, but is it solvable? I think in Byzantine music it is, I will send the solve I see soon.

How about 171-tET?