50-EDO (cf. Woolhouse) is easy in Csound

For all you Csound users out there:

Csound offers a very easy way to implement 50-EDO/tET,
which is nearly identical to the 7/26-comma meantone
considered optimal (according to one sum-of-least-squares
measurement) by Wesley Woolhouse and Paul Erlich.

On these tunings, see my posting in Onelist TD 446.6,
or the relevant part of my webpage on Woolhouse:
http://www.ixpres.com/interval/monzo/woolhouse/essay.htm#temp

My webpage and TD posting show the tiny 'errors' between
50-EDO/tET and the 7/26-comma meantone. This meantone
gives a very small 'error' for the primary 5-limit consonances
of 2:3, 3:4, 4:5, 3:5, 5:6, and 5:8, and, being a 'meantone',
eliminates the syntonic comma and thus the problem of
commatic shift/drift.

It is easy in Csound because one of Csound's pitch
representations, 'oct', is a logarithmic way of describing
pitches which essentially amounts to 100-EDO/tET. The
format is 'oct.dec', with the number before the
decimal point specifying the 'octave', and the number
after the decimal point specifying the 100-EDO/tET degree.

For example,
Csound calculates '8.75 oct' as 2^(((8*100)+75)/100),
which is equal to 8 'octaves' plus 2^(75/100) [= 900 cents]
above the low reference 'C', and which indicates the note
A-440 Hz. So 'oct.dec' really means '2^(((oct*100)+dec)/100)'.

All a Csound user has to do is restrict himself to using
only even-numbered decimals, to easily represent the notes
in 50-EDO/tET.

BTW, regarding the more commonly-used 'pch' pitch
representation, where the number before the decimal point
represents the 'octave' (as in 'oct'), and the number after
the decimal point represents the 12-EDO/tET pitch-class:

With '8ve' and 'pc' as variables (the Csound format being
'8ve.pc'), the representation really means 2^(((8ve*12)+pc)/12).

The Csound Manual says:

> [Barry Vercoe, Csound Manual, p 10]
> Microtonal divisions of the pch semitone can be encoded
> by using more thqn two decimal places.

So the 3rd decimal-place gives 10-cent increments:
2^(((8ve*120)+pc)/120), the 4th decimal-place gives
1-cent increments: 2^(((8ve*1200)+pc)/1200), etc.

I don't know how far one may carry the decimals in Csound's
'pch' format. Anyone?

-monz

Joseph L. Monzo Philadelphia monz@juno.com
http://www.ixpres.com/interval/monzo/homepage.html
|"...I had broken thru the lattice barrier..."|
| - Erv Wilson |
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Joe Monzo wrote:
[snip]
> 50-EDO/tET,
> which is nearly identical to the 7/26-comma meantone
> considered optimal (according to one sum-of-least-squares
> measurement) by Wesley Woolhouse and Paul Erlich.
>
> On these tunings, see my posting in Onelist TD 446.6,
> or the relevant part of my webpage on Woolhouse:
> http://www.ixpres.com/interval/monzo/woolhouse/essay.htm#temp
[snip]

This got me curious about this "optimal" temperament. Joe's page explains
that the goal is one size of tone, split into two different semitones, in
order to minimize mean square error on three intervals 4:3, 5:4, 6:5 (pardon
my notational bias in putting the "numerator" first). (As it turns out, the
mean-tone resulting versions of 3:2, 5:4, 6:5 all end up a bit flat, with
the 3:2 suffering most.) Then 50-EDO/tET includes a very close approximation
to that ideal mean-tone scale. Of course, such a scale approximates both 9:8
and 10:9 as the same tempered interval.

But I hadn't thought that through when I first looked at 50-EDO/tET
graphically, and was surprised to see that 9:8 (203.9c) falls just about
exactly halfway between two tempered steps (192c & 216c), i.e., in the crack
(204c). Hence, chords based on stacks of 3:2's would be problematic, but
that's not what meantone is for... Of course, meantone is also not about
higher prime harmonies like 11/8 (551.3c) and 13/8 (840.5c), but 50-EDO/tET
gives _excellent_ approximations to these (552c & 840c)!

- just my 866/865 ;-)

Oh, by the way, I have slightly improved my "Scale Slide Rule" EPS file, so
that now in comparing a JI scale to an EDO scale, the latter's tones are
(optionally) numbered for easy reference; I've updated the EPS files
scale.eps and scale2.eps, and will soon update the corresponding web page:
http://www.mbay.net/~anne/david/eps4ji/index.htm

David Canright http://www.mbay.net/~anne/david/