Byzantine scales and prime intervals

Rami Vitale wrote:

> 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.

Paul Erlich wrote:

>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?

As I said by the taste, some singers are that accurate.

Rami Vitale wrote:

> 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" ].

Paul Erlich wrote:
> I call this "consistency".

> How about 171-tET

Dear Paul,

Yes 171 is somehow good, but for instance:
21/20 * 49/48 = 16/15
when:
12 + 5 =? 16
Believe me it is a serious problem since you cannot present the scales ( the Byzantine scales as I gave ) as a series of numbers.

I have another solution which I think is more practically convenient, but it is more complicated than picking a number from a list.

First of all I use prime intervals.

What is prime Intervals? they are some intervals such as 2/1 3/2 5/4 7/6 11/10, when presented in the form high_frequency/low_frequency, the numerator is a prime number and the denominator is 1 less than the numerator.

Have you heard about them? I don't know if someone else have used them.

Every interval is composed by some of these intervals by multiplication and dividing, like:
28/27 = 7/6 * 2/3 * 2/3 * 2/1 ( composed by 7/6 & 3/2 & 2/1 )
We can also compose this interval as:
28/27 = 7/6 * 2/3 * 2/3 * 2/1 * 11/10 * 10/11

But since 11/10 & 10/11 are opposed and 11/10 * 10/11 =1 then we omit them.
When you omit such opposed intervals, every interval is composed by prime intervals in just one way.
e.g. 9/8 = 3/2 * 3/2 * 1/2, you cannot use another prime intervals to compose 9/8!

All intervals in the Byzantine scales are composed by some of theses prime intervals: 2/1, 3/2, 5/4, 7/6
I will shorten and say if the measurements of these intervals are accurate then everything is OK.

But they must be accurate also relative to the comma we are using, I mean:
3.2 + 4.2 = 7.4 in the 36 per octave is better than
6.4 + 8.4 = 14.8 in the 72 per octave

because approximately:
3 + 4 = 7

when:
6 + 8 ?= 15

So, what is the number between 1! and 300, which gives accurate measurements to these prime numbers in cents and relative to the comma we use?
In fact no one does! 171 and 212 are somehow good but unfortunately not good enough

But 53 is a noticeable number because it is small and gives accurate measurements for 5/4 & 3/2 but not for 7/6:

Interval in the 53 per octave
2/1 53
3/2 31.003
5/4 17.062
7/6 11.787 ( 12 - 0.213 )

So what, big deal, we put a symbol on 12 as (,) to notice that the 7/6 measurement is smaller a bit than 12 and let's see what will happen.

For instance 28/27 will be:
7/6 * 2/3 * 2/3 * 2/1 = 28/27
12, - 31 - 31 + 53 = 3,

We consider (,) as a little bit bigger than a negative quarter of the comma we are using, and (') as a little bit bigger than a positive quarter of the comma we are using

When we add a number with (,) we add (,) to the result - when we subtract a number with (,) we add (')
When adding two numbers with (,) we add (,,) to the result and when we subtract two numbers with (,) we add ('') to the result
and so on.

The Byzantine scale will be:

21/20 50/49 21/20 64/63 49/48 36/35 25/24 36/35 49/48 64/63 21/20 50/49 21/20 21/20 50/49 21/20 64/63 49/48 36/35 25/24 36/35 49/48 64/63

4, 1'' 4, 1' 2,, 2' 3 2' 2,, 1' 4, 1'' 4, 4, 1'' 4, 1' 2,, 2' 3 2' 2,, 1'

The diatonic scale will be:

9 8 5 9 9 8 5

It is a complicated method for musicologists, but it is very easy to musicians and I think it is very useful because:
1.. It is OK 100% with the three considerations above. ( e.g. 50/49 & 49/48 are presented differently : 1'' & 2,, )
2.. They are small and easy to remember numbers
3.. The diatonic scale has no (,) and no ('), why? because it is composed by only 2/1 & 3/2 & 5/4
Every scale composed with only these prime intervals has no (') and (,) and is joyful!
Every one composed with 7/6 has (') or (,) and also it is somehow sad! So (,)s are meaningful!
4.. If you want mathematical accuracy ( if you want to play this scale ) consider (') as positive quarter of the comma and (,) as negative quarter of the comma, you will get the 212 per octave which is very accurate, and you will get this without any need to remember other numbers.
This method is good for Byzantine scales and for all scales composed by 2/1 3/2 5/4 7/6 but not for others

Rami Vitale

--- In tuning@y..., "Rami Vitale" <alfred1@s...> wrote:
> Rami Vitale wrote:
>
> > 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.
>
> Paul Erlich wrote:
>
> >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?
>
> As I said by the taste, some singers are that accurate.
>
> Rami Vitale wrote:
>
> > 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" ].
>
> Paul Erlich wrote:
> > I call this "consistency".
>
> > How about 171-tET
>
> Dear Paul,
>
> Yes 171 is somehow good, but for instance:
> 21/20 * 49/48 = 16/15

This is not correct. 21/20 * 49/48 = 343/320.

21/20 * 64/63 = 16/15.

256/245 * 49/48 = 16/15.

> when:
> 12 + 5 =? 16
> Believe me it is a serious problem since you cannot present the
scales ( the Byzantine scales as I gave ) as a series of numbers.

I think 171-tET will turn out to be just wonderful for your purposes
once you get your arithmetic right. :) It's consistent in the 7-
limit, at LEVEL 8.

--- In tuning@y..., "Rami Vitale" <alfred1@s...> wrote:

> > 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.

This simply means you don't want to use approximations, so you
needn't worry about any non-just tuning system on principle. In
practice, two intervals may be so close you cannot tell them apart,
in which case you might want to rethink your principle.

> > 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 think what you want is a logarithm, such as cents; however I wonder
what your condition "adjoining" means?

> > How about 171-tET

This is only an approximate logarithm, but it is so close in the
7-limit one could well ask why it isn't good enough.

> Yes 171 is somehow good, but for instance:
> 21/20 * 49/48 = 16/15
> when:
> 12 + 5 =? 16

No, but 12 + 5 = 17. We have h_171(21/20) = 12, h_171(49/48) = 5,
and h_171(21/20 * 49/48) = h_171(343/320) = 17.

The 171-et keeps very good track of 7-limit intervals, and it is
hardly likely you need anything better.

> Believe me it is a serious problem since you cannot present the
scales ( the Byzantine scales as I gave ) as a series of numbers.

Isn't this what you do below?
> When you omit such opposed intervals, every interval is composed by
prime intervals in just one way.
> e.g. 9/8 = 3/2 * 3/2 * 1/2, you cannot use another prime intervals
to compose 9/8!

I've never heard of this, but I really like it. In case anyone is
wondering, the claim of unique factorization into Rami numbers is
true, since for any prime p the corresponding Rami number factors as
p^1 q_i^e^i, where the q_i are the primes less than p. The
transformation matrix is therefore upper triangular with ones along
the diagonal, and hence is unimodular; and so the transformation is
invertible.

> All intervals in the Byzantine scales are composed by some of
theses prime intervals: 2/1, 3/2, 5/4, 7/6

This simply says they are JI scales in the 7-limit. That entails that
the 171 division will work very, very well for your purposes!
> So, what is the number between 1! and 300, which gives accurate
measurements to these prime numbers in cents and relative to the
comma we use?

171.

> In fact no one does! 171 and 212 are somehow good but unfortunately
not good enough

I suspect 171 is much, much better than you are giving it credit for
being.

> But 53 is a noticeable number because it is small and gives
accurate measurements for 5/4 & 3/2 but not for 7/6:

> Interval in the 53 per octave
> 2/1 53
> 3/2 31.003
> 5/4 17.062
> 7/6 11.787 ( 12 - 0.213 )

Let's try these in the 171 octave:

2/1 171
3/2 100.0286
5/4 55.0497
7/6 38.0291

Given how close these are to integer values, and because 171 is quite
a lot bigger than 53, this should work. Simply use 171, 100, 55 and
38 after Rami factorization and convert to the 171-et.

> So what, big deal, we put a symbol on 12 as (,) to notice that the
7/6 measurement is smaller a bit than 12 and let's see what will
happen.
>
> For instance 28/27 will be:
> 7/6 * 2/3 * 2/3 * 2/1 = 28/27
> 12, - 31 - 31 + 53 = 3,

In 171 we have (2/1)^1 * (3/2)^(-2) * (7/6)^1, and we get
171 - 2*100 + 38 = 9. If take instead the log base 2^(1/171) of
28/27, we get 8.9793 instead of 9, which means we were 0.197 cents
sharp. That's not bad!

----- Original Message -----
From: Paul Erlich <paul@stretch-music.com>
To: <tuning@yahoogroups.com>
Sent: Friday, September 07, 2001 2:59 PM
Subject: [tuning] Re: Byzantine scales and prime intervals

> --- In tuning@y..., "Rami Vitale" <alfred1@s...> wrote:
> > Rami Vitale wrote:

> > Rami Vitale wrote:
> >
> > > 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" ].
> >
> > Paul Erlich wrote:
> > > I call this "consistency".
> >
> > > How about 171-tET
> >
> > Dear Paul,
> >
> > Yes 171 is somehow good, but for instance:
> > 21/20 * 49/48 = 16/15
>
> This is not correct. 21/20 * 49/48 = 343/320.

Yes of course it is a strange mistake made by me

> I think 171-tET will turn out to be just wonderful for your purposes
> once you get your arithmetic right. :) It's consistent in the 7-
> limit, at LEVEL 8.

Once again what does 7-limit at LEVEL 8 means?

171-tET is good, accurate, it presents 50/49 & 49/48 equaly but it is not a
big deal.

But is it better than 53 with (')s?
The most important thing which makes me think that 53 with (')s is better,
is that it tells you when an interval is copmosed with 7/6 by giving an (')
or (,) on the number which expresses the interval.
In Byzantine music, and I think in any musical form it is an important
thing. As I mentioned before I notice that every scale has some intervals
composed by 7/6 ( i.e. 15/14 6/5 28/27 9/8 15/14 6/5 28/27 & 28/27 9/8 8/7
9/8 28/27 9/8 8/7 ) is a sad scale,
when every scale has no such intervals is joyfull ( i.e. 9/8 10/9 16/15 9/8
9/8 10/9 16/15 ). ( I like to hear comments on this )
So in practical use 53 with (')s makes it possible to understand the scales
( Byzatine scales for instance ) and even to make some theories on them
without the need to use intervals presented in relative fraquency forms.

Another thing,
I think these conversations are very usefull, and I insist on my opinions to
see if they are really bad or good. I want be here for a week.

Rami

----- Original Message -----
From: <genewardsmith@juno.com>
To: <tuning@yahoogroups.com>
Sent: Friday, September 07, 2001 4:58 PM
Subject: [tuning] Re: Byzantine scales and prime intervals

> --- In tuning@y..., "Rami Vitale" <alfred1@s...> wrote:
>
> > > 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.
>
> This simply means you don't want to use approximations, so you
> needn't worry about any non-just tuning system on principle. In
> practice, two intervals may be so close you cannot tell them apart,
> in which case you might want to rethink your principle.

Yes I want to use approximations, but to the limit which doesnot affect the
studying of the scales I'm using, I mean this principle can be applied to a
specific scale but to all any unspecified scales, else this principle cannot
be possible.

> > > 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 think what you want is a logarithm, such as cents; however I wonder
> what your condition "adjoining" means?

No it is not only about using logarithms such as cents, because of the
approximations, for instance:
21/20 * 15/14 = 9/8
in cents:
84 + 119 != 204

As I said above you must think that you must apply this principles in some
way to the scale you are using, not to any scale,
so "adjoining intervals" means adjoining intervals in the scale you want to
apply this principles to.

> > > How about 171-tET
>
> This is only an approximate logarithm, but it is so close in the
> 7-limit one could well ask why it isn't good enough.
>
> > Yes 171 is somehow good, but for instance:
> > 21/20 * 49/48 = 16/15
> > when:
> > 12 + 5 =? 16
>
> No, but 12 + 5 = 17. We have h_171(21/20) = 12, h_171(49/48) = 5,
> and h_171(21/20 * 49/48) = h_171(343/320) = 17.
>
> The 171-et keeps very good track of 7-limit intervals, and it is
> hardly likely you need anything better.

Yes of course! It was a strange mistake made by me.

171 is good... enough but I do think that 53 with (') as I mentioned has
some chances compared with 171 ( see the next message )

> > Believe me it is a serious problem since you cannot present the
> scales ( the Byzantine scales as I gave ) as a series of numbers.
>
> Isn't this what you do below?

mmm...Not if you understand me, I meant what I mentioned here above about
logarithms and cents.

> > When you omit such opposed intervals, every interval is composed by
> prime intervals in just one way.
> > e.g. 9/8 = 3/2 * 3/2 * 1/2, you cannot use another prime intervals
> to compose 9/8!
>
> I've never heard of this, but I really like it. In case anyone is
> wondering, the claim of unique factorization into Rami numbers is
> true, since for any prime p the corresponding Rami number factors as
> p^1 q_i^e^i, where the q_i are the primes less than p. The
> transformation matrix is therefore upper triangular with ones along
> the diagonal, and hence is unimodular; and so the transformation is
> invertible.

Thank you a lot for doing this, I must rerereread to understand it clearly.

> > All intervals in the Byzantine scales are composed by some of
> theses prime intervals: 2/1, 3/2, 5/4, 7/6
>
> This simply says they are JI scales in the 7-limit. That entails that
> the 171 division will work very, very well for your purposes!
> > So, what is the number between 1! and 300, which gives accurate
> measurements to these prime numbers in cents and relative to the
> comma we use?
>
> 171.

What does JI scales in the 7-limit means? I think sombody here must lead me
to a dictionary for tuning terms.

> > In fact no one does! 171 and 212 are somehow good but unfortunately
> not good enough
>
> I suspect 171 is much, much better than you are giving it credit for
> being.

OK OK! see the next message...

Rami

--- In tuning@y..., "Rami Vitale" <alfred1@s...> wrote:
>
> > I think 171-tET will turn out to be just wonderful for your
purposes
> > once you get your arithmetic right. :) It's consistent in the 7-
> > limit, at LEVEL 8.
>
> Once again what does 7-limit at LEVEL 8 means?

The basic 7-limit consonances are 1:1, 8:7, 7:6, 6:5, 5:4, 4:3, 7:5,
and their octave inversions and extensions. Consistency at level 8
means that you can combine eight of these intervals and the result
will always agree between JI and 171-tET. For your purposes, level 3
or 4 would probably be sufficient.
>
> 171-tET is good, accurate, it presents 50/49 & 49/48 equaly but it
is not a
> big deal.

I hope not! The difference, 2401:2400, is only 0.7 cent.

--- In tuning@y..., "Rami Vitale" <alfred1@s...> wrote:

> I think sombody here must lead me
> to a dictionary for tuning terms.

http://www.ixpres.com/interval/dict/