Re: Your Message to tuning

Weird music scale tuning of 192 notes

Used Scala to create a musical scale of 192 notes. Equal tempered. Got some weird results, looks like the scale is almost perfectly spaced cents wise by separations of a quarter ascending. Can someone take a look at this, try this
and tell me what is going on, what this means?

Thanks

---------------------------------
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--- In tuning@yahoogroups.com, dar kone <zarkorgon@...> wrote:
>
> Weird music scale tuning of 192 notes
>
> Used Scala to create a musical scale of 192 notes. Equal tempered.
Got some weird results, looks like the scale is almost perfectly
spaced cents wise by separations of a quarter ascending. Can someone
take a look at this, try this
> and tell me what is going on, what this means?

I'm not clear what you are saying. 192-et is 192 notes equally spaced at
intervals of exactly 6.25 cents.

--- In tuning@yahoogroups.com, "Gene Ward Smith" <genewardsmith@...>
wrote:

> I'm not clear what you are saying. 192-et is 192 notes equally
spaced at
> intervals of exactly 6.25 cents.

It's a good hemiwuer temperament tuning, and for all patent vals,
gives the least error tuning for tempering out 3136/3125. It's also
divisible by 12. So, it has some talents.

--- In tuning@yahoogroups.com, dar kone wrote:
>
> Weird music scale tuning of 192 notes
>
> Used Scala to create a musical scale of 192 notes. Equal tempered.
Got some weird results, looks like the scale is almost perfectly
spaced cents wise by separations of a quarter ascending. Can someone
take a look at this, try this and tell me what is going on, what
this means?

Hi "dark one"!

A scale of 192 notes per octave, equal tempered, has steps each
exactly one-sixteenth of a semitone:

1 octave = 12 semitone = 12 x 100 cent = 1200 cent, and therefore
1/192 octave = 12/192 semitone = 1/16 semitone = 100/16 cent
= 6.25 cent

So it consists of notes at:
0.00
6.25
12.50
18.75
25.00
31.25
...
1175.00
1181.25
1187.50
1193.75
1200.00
cents

So the *fractional* part of each note's cents value must be a
multiple of 0.25, ie a quarter of a cent.

Personally, I don't see anything weird in this pattern - it's what
you'd expect in n-EDO, whenever 1200/n has fractional part 0.25.

That is, when 1200/n - [1200/n] = 0.25
or 1200 - n[1200/n] = 0.25 n, or, writing k=[1200/n],
1200 = n(k+1/4)
or 4800/n = 4k+1 for some positive integer k,
when n = 4800/(4k+1).

The first 12 values of k, 4k+1 and corresponding n are:

1 5 960.00 <--- 960-EDO, using your scale step /5
2 9 533.333'
3 13 369.23076...
4 17 282.35294...
5 21 228.571428...
6 25 192.00 <--- your scale, 192-EDO
7 29 165.51724...
8 33 145.4545'
9 37 129.729729'
10 41 117.07317...
11 45 106.666'
12 49 97.959183...

Of these 12, only the first and the sixth have an integer value of
n, 960 and 192 respectively. Are there more like this? I don't
know, but you might spreadsheet the first 200 values of k to find
out what happens. Still, the other cases would correspond to an
equal division of m octaves, for some m which divides the value of
(4k+1).

So your scale 192-EDO is, numerically, rather special; along with
960-EDO, it's one of the only two high-EDO (greater than 100-EDO)
scales whose steps increase by a quarter-cent in their fractional
part. But that's not particularly special musically, since the
division of the octave into 1200 cents is a rather arbitary choice,
and many other octave division schemes have been proposed and used
in the past. For example, one *might* argue that a division of the
octave into 2!3!5!7!11!13!17!19! equal parts was desirable since it
supports the repeated divsion of octave divisions into an integer
number of parts in many different ways ... (but I wouldn't).

Does this answer your question?

Regards,
Yahya

--- In tuning@yahoogroups.com, "yahya_melb" <yahya@...> wrote:

> Of these 12, only the first and the sixth have an integer value of
> n, 960 and 192 respectively. Are there more like this?

The divisors of 1200 are 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24,
25, 30, 40, 48, 50, 60, 75, 80, 100, 120, 150, 200, 240, 300, 400,
600, and 1200. Each of these has an integer quanitity of step size in
terms of cents. The divisors of 4800 are 1, 2, 3, 4, 5, 6, 8, 10, 12,
15, 16, 20, 24, 25, 30, 32, 40, 48, 50, 60, 64, 75, 80, 96, 100, 120,
150, 160, 192, 200, 240, 300, 320, 400, 480, 600, 800, 960, 1200,
1600, 2400, 4800. These will come out as *.5, *.25, *.75 or *.00.

Hi Gene,

--- In tuning@yahoogroups.com, "Gene Ward Smith" wrote:
>
> --- In tuning@yahoogroups.com, "yahya_melb" wrote:
>
> > Of these 12, only the first and the sixth have an integer value of
> > n, 960 and 192 respectively. Are there more like this?
>
> The divisors of 1200 are 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24,
> 25, 30, 40, 48, 50, 60, 75, 80, 100, 120, 150, 200, 240, 300, 400,
> 600, and 1200. Each of these has an integer quanitity of step size in
> terms of cents. The divisors of 4800 are 1, 2, 3, 4, 5, 6, 8, 10, 12,
> 15, 16, 20, 24, 25, 30, 32, 40, 48, 50, 60, 64, 75, 80, 96, 100, 120,
> 150, 160, 192, 200, 240, 300, 320, 400, 480, 600, 800, 960, 1200,
> 1600, 2400, 4800. These will come out as *.5, *.25, *.75 or *.00.
>
That's half the answer. "dar kone" specifically noted a pattern
whereby [the fractional part of] the steps increased by 0.25 cent
ascending; this requires that the n steps per octave are each = (k +
0.25) cent, or (4k+1)/4 cent; thus, that n(4k+1) = 4800, ie that n =
4800/(4k+1) for some integer k. Letting k=1, 2, 3, ... we get the
series I showed, and only when k=1 or 6 do we get an integer value for
n with low k. These values correspond to the divisors 5 and 25 of
4800.

From your list of divisors, all larger ones except 75 are even, hence
not of form 4k+1, and 75 itself has form 4k+3. Thus the 192-EDO and
960-EDO scales are the only ones [the fractional part of] whose steps
increases by one-quarter cent ascending.

Regards,
Yahya