Re: twintone, paultone, 34

Due to my math skills this would probably be more appropriate
on the main list, but due to the topic here and what I think
my questions may elicit, I'll put it here.

First, some "Bob Valentine oriented definitions for EDOs".

Meantone := best(5/4) = octave_reduced( 4 * best(3/2) )
Schismic := best(5/4) = octave_reduced( 8 * best(4/3) )

diaschismic := not ( Meantone OR Schismic )

This last definition is weak, and I think I could go further
and say

diaschismic := ( best(5/4) < octave_reduced( 4 * best(3/2) ) ) AND
( best(5/4) > octave_reduced( 8 * best(4/3) ) )

which I think holds for all 10, 22, 34, 46, 58 etc (till the
periodicity 'blips').

Or, (and this is probably the RIGHT answer)

2 * best( 5/4 ) = octave_reduced( 4 * best( 4/3 ) )

Regarding WHY to use 34, the reasons I've been debating include...
1) large number of MOS with P5 (MOS can also be altered to be
nice scales with other properties, but good P5 is very nice)
2) good 5-limit
3) nice et identities
P5 divisible by 2, 4 and 5 (but not 6 or 7, making it different
from other ETs I intend using)
P4 divisible by 2 (and 7)
4) deconflates 9/8 and 10/9
5) consistent to 17-limit IFF you REMOVE 7 altogether
I believe that consistency is a nice feature, if you use an EDO
to approximate a JI system, the interval arithmatic should
match up

So, I have tried in the past to express it as a system in 3, 5,
and 13 (since 13 is notably absent from other EDOs I intend using)

3 5 13
[ 1 0 2 ] 2 * best( 13/8 ) = normalized( best( 4/3 ) )
[ -4 2 0 ] from above...
[ ] got another ? does it come out to 34 ?

thanks a lot,

Bob Valentine

In-Reply-To: <200201280951.LAA65774@ius578.iil.intel.com>
Robert C Valentine wrote:

> Or, (and this is probably the RIGHT answer)
>
> 2 * best( 5/4 ) = octave_reduced( 4 * best( 4/3 ) )

That looks right, but it's usually done as

best(5:4) = half_octave_reduced( 2 * best(4:3) )

because all diaschismics are divisible by two (in terms of notes to the
octave).

> So, I have tried in the past to express it as a system in 3, 5,
> and 13 (since 13 is notably absent from other EDOs I intend using)
>
> 3 5 13
> [ 1 0 2 ] 2 * best( 13/8 ) = normalized( best( 4/3 ) )
> [ -4 2 0 ] from above...
> [ ] got another ? does it come out to 34 ?

By combining 58 (which is fully 13-limit consistent) and 34, I get

best (16:13) = tritone_reduced(2*best(3:2))

and

best (16:13) = tritone_reduced(4*best(8:5))

but that gets quite hairy.

Graham

--- In tuning-math@y..., Robert C Valentine <BVAL@I...> wrote:
>
> Due to my math skills this would probably be more appropriate
> on the main list, but due to the topic here and what I think
> my questions may elicit, I'll put it here.
>
> First, some "Bob Valentine oriented definitions for EDOs".
>
> Meantone := best(5/4) = octave_reduced( 4 * best(3/2) )
> Schismic := best(5/4) = octave_reduced( 8 * best(4/3) )
>
> diaschismic := [SNIP]
> Or, (and this is probably the RIGHT answer)
>
> 2 * best( 5/4 ) = octave_reduced( 4 * best( 4/3 ) )

That's right! If you ever forget these things, just look up the
ratios:

Schisma = 32805:32768
Diaschisma = 2048:2025

Then you can always work out the equivalencies.

These apply not only to EDOs, but also to equal temperaments.

Now, can you figure out how "kleismic" is defined? Hint: the kleisma
= 15625:15552

You can always look up commas here:

http://www.xs4all.nl/~huygensf/doc/intervals.html

Hey Bob,

I think I got the answer you were looking for (before the tuningmath
digest arrives no less).

>
> So, I have tried in the past to express it as a system in 3, 5,
> and 13 (since 13 is notably absent from other EDOs I intend using)
>

try

3 5 13
[ 1 0 2 ] 2 * best( 13/8 ) = normalized( best( 4/3 ) )
[ -4 -2 0 ] the corrected diaschismic identity?
[ 0 4 1 ] Kewl!

This looks about right and gets a determinant of 34.

Now, could one of the lattice-sticians show a lattice with two
3-5 lattices joined by a 13/8. Try to go more in the 5 direction
as it stays "prime limit consistent" in 5 space more than 3 space.

> thanks a lot,
>
> Bob Valentine

is there an echo in here?