a 24 tone non-edo scale

Thanks all for the feedback over the weekend regarding 15 EDO
and possible notation variations,and help in general. Some of the
comments make me feel like a dwarf among giants, but that's O.K.

A scale came to me a number of months ago. (I call it the Szpak
Scale since I don't know who came up with it first.) If anyone
here can get some use out of it that's fine with me. It is very
closely related to 24 EDO. It includes all notes of the 12 EDO
scale.

Some advantages (of the Szpak Scale):

Using the tonics of 12 EDO:

subminor 3rd is less than 6 cents off ideal
neutral 3rd is less than 10 cents off ideal
sub 4th (470.781 cents) is less than 10 cents off ideal
11th harmonic is less than 10 cents off ideal
7th harmonic is less than 8 cents off ideal

The scale goes like this: (in cents)

0-60.88-100-160.88-200-260.88 etc. until 1200 cents

--- In tuning@yahoogroups.com, "stephenszpak" <stephen_szpak@h...>
wrote:
...
> A scale came to me a number of months ago. (I call it the Szpak
> Scale since I don't know who came up with it first.) If anyone
> here can get some use out of it that's fine with me. It is very
> closely related to 24 EDO. It includes all notes of the 12 EDO
> scale.
>
> Some advantages (of the Szpak Scale):
>
> Using the tonics of 12 EDO:
>
> subminor 3rd is less than 6 cents off ideal
> neutral 3rd is less than 10 cents off ideal
> sub 4th (470.781 cents) is less than 10 cents off ideal
> 11th harmonic is less than 10 cents off ideal
> 7th harmonic is less than 8 cents off ideal
>
> The scale goes like this: (in cents)
>
> 0-60.88-100-160.88-200-260.88 etc. until 1200 cents

So you have a scale with two stepsizes, L=100 s=60.88

And it's built sLsLsLs...

Just out of curiosity, which are your ideal references, Stephen?

For instance, your approximation for neutral third is 360.88 cents,
which is more than 10 cents away from

347.41 cents 11/9
350 cents (24-eq neutral third)
348.39 cents (31-eq neutral third)

I understand your sub-diminished 4th's reference is septimal fourth,
that is, 21/16. And your ideal subminor 3rd must be 7/6. So may be
it's 49/40 what you're talking about? 7-limit?

Max.

>

Hello Stephen!

The neutral third which historicly was and is used primarily by the arabic and persian informed cultures never relied on limits in generating there tunings. The variation of this interval throught this area of the world is a matter of personal
taste and from what i am told cultural/regonal identity. The idea that there is one and only one neural third is contrary to practice. It is not subjected to the same scutiny that the a harmonic would have. There is plenty of room for all types
of neutral thirds.

>
> From: "Stephen Szpak" <stephen_szpak@hotmail.com>
>
>
> Stephen Szpak writes:
>
> (Please note this is, I believe your first reply to my original message a
> few days ago.)
>
> I think we're thinking on 2 separate plains here. You are analyzing
> the Szpak Scale
> with "odd-limit" "prime-limit" and other terms, and I am looking at the
> scale regarding
> the harmonics it contains.
>
> If one uses the tonics of the 12 EDO system (ignoring the richness of the
> other half of
> the Szpak Scale) you have the ability to reach, access, use all of the
> first 12 harmonics.
> The 12 EDO scale does NOT have the 7th and 11th harmonics.
>
> 12 EDO 7th harmonic not available
> 12 EDO 11th harmonic not available
>
> 24 EDO 7th harmonic not available
> 24 EDO 11 harmonic available in 24 tonics
>
> Szpak Scale 7th harmonic available in 12 tonics (the 12 western
> tonic notes)
> Szpak Scale 11th harmonic available in 12 tonics (the 12 western
> tonic notes)
>
> The reason that the neutral 3rd may be slightly off (I guess it is off
> by more than 10 cents)
> if because of this. The 24 EDO scale, which is what the Szpak Scale is
> based on, is really
> 2 12 EDO scales that have been interlaced. If the first scale is
> untouched the 12 familiar
> notes of western music are intact. The second scale has to be shifted
> however, if the
> harmonic 7th is to be included.
> But here's the important point. If the second scale is shifted too
> much the 11 th harmonic
> is lost. The Szpak Scale includes the 7th and 11 harmonics BOTH at less
> than 10 cents off
> their ideal locations at the expense of losing other ratios. In fact the
> shift of the second
> scale can only exist (you can check the math if you really want to )
> within a 2.49 cent
> window. Otherwise the 7th harmonic is off by 10 cents or more OR the
> 11 th harmonic
> is off by 10 cents or more. I realize that the "10 cent" business is
> arbitrary on my part, but
> the line has to be drawn somewhere.
>
> 0 58.827 100 158.827 200 258.827 etc. until 1200
>
> 0 61.317 100 161.317 200 261.317 etc until 1200
>
> 0 60.88 100 160.88 200 260.88 etc until 1200
> (Szpak Scale)
>
> I suppose you might have realized all this already. Feel free to get
> back to me either way.
>
> Stephen Szpak
>
>

-- -Kraig Grady
North American Embassy of Anaphoria Island
http://www.anaphoria.com
The Wandering Medicine Show
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