Tuning-Math Digests messages 6931 - 6955

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Message: 6931

Date: Sun, 22 Jun 2003 12:59:35

Subject: Re: 12 equal to meantone conversion algorithm

From: Carl Lumma

Pretty simple, compared to this:

http://lumma.org/stuff/adaptive.txt *

-Carl


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Message: 6937

Date: Mon, 23 Jun 2003 10:27:09

Subject: Re: 12 equal to meantone conversion algorithm

From: Graham Breed

Gene Ward Smith wrote:

> I should point out that this is only true if the chromatic scale is 
> not harmonized. My algorithm, while very simple, does not work on the 
> crude level of note-for-note, but is based on note-sets. If the 
> chromatic passage was harmonized in a way which leads back to C, for 
> instance by C-A-D-cm-C-F-D-G-fm-F-C7-G7-C, back to C we would come.

I also have some code for 12-equal to meantone conversions, and I 
managed to write it without using complex numbers!  It's at 
http://www.microtonal.co.uk/gesualdo.zip - Ok *

I don't know how out of date that is, but I do have a more recent 
version on my Revo.  One difference is that I have found passages in 
Gesualdo that it doesn't convert correctly.  There's a total of 3 wrong 
chords.  From what I remember, these can be resolved by using a 
different gamut depending on whether he used a Bb in the key signature.

The gamut restrictions could possible be removed by using a flexible key 
center.  Take everything relative to the average number of steps on the 
spiral of fifths, calculated recursively as

total(n) = k*total(n-1) + note
number(n) = k*number(n-1) + 1
center(n) = total(n)/number(n)

Then, k=0 means it calculates each new chord looking at only the 
previous chord, and k=1 makes it remember all previous chords equally, 
which will give very conservative results.  I didn't implement this for 
the Gesualdo program, because it isn't a rule he was likely to follow, 
but it might work for an automated tuning program.


                    Graham


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Message: 6939

Date: Mon, 23 Jun 2003 13:49:59

Subject: Re: 12 equal to meantone conversion algorithm

From: Manuel Op de Coul

Have a look at this too:
ÖFAI technical reports *

Manuel


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Message: 6941

Date: Tue, 24 Jun 2003 06:07:59

Subject: Re: Interval Database Experiences

From: Dave Keenan

--- In tuning-math@xxxxxxxxxxx.xxxx "Porres" <decuritiba@y...> wrote:
> I just checked an explanation about continued fractions and it's 
> amazing, I see you're using convergents and semi-convergents on your 
> Excell table, it seemed at first kinda complicated, but I guess I'll 
> dig it...
> 
> but hey, what does it have to do with that pdf file ( Self Similar 
> Pitch Structures - Clampitt ) ??? I downloaded that once and couldn't 
> figure it out to, let me dig the traditional harmony first, I'm still 
> studying Bach's counterpoint technique.

Sorry to take so long to reply.

It is the same mathematics put to a different purpose. In Clampitt's
paper it is applied in the logarithmic pitch domain, whereas you're
using it in the linear frequency domain. Clampitt is finding ratios
representing fractions of an octave (or other interval of periodicity)
(i.e. degrees of equal temperaments) where you are finding ratios
representing frequency ratios (i.e. justly intoned intervals when the
numbers are small).

It's nice that the same mathematical tool has these two different
applications to tuning. You might say Clampitt is applying it to
melodic properties while you are applying it to harmonic ones.

> ha ha ha, good job on your notation research, specially by keeping up 
> the good humor, since you're involved in Scala, would you know of a 
> complete table of name intervals? I guess that would be useful... 

I'm not really "involved in Scala", but the file 'intnam.par' (and its
equivalents in other languages) that comes with Scala, is very useful.
With a little work it can be imported into a spreadsheet and sorted by
interval size or whatever.

-- Dave Keenan


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