syntonic and skhismatic notation

I mentioned some of this before in a post about quartertones
that should have been in TD1153, but wasn't.

I class all just intervals and approximations thereof based
on syntonic meantone temperament using the letters t, s, r, q
and p for major seconds, minor seconds, augmented unisons,
diminished seconds and "commas" respectively. The syntonic
comma p5 is dimensionless in meantone temperament. I usually
think of the dimensions of intervals in terms of q and r. I
now use the term "demisemitone" for intervals with dimensions
of q and r-q, although I would rather call them quartertones.

A chromatic semitone is a kind of r, and an enharmonic diesis
q. For 7/4 to be an augmented sixth, a septimal comma has
dimensions of q.

In terms of fifths, octave reduced, t=2 steps, s=-5, r=t-s=7,
q=s-r=-12. The # symbol means "raise this note by r" and the
/ symbol can mean raising by q.

These symbols are consistent when 0<=q<=r, where 0 is a unison.
This is true for all meantone temperaments where the fifth
is between that for 19tet and 12tet. When the fifth is
sharper than 7/12 oct, a skhismatic approximation is better.
12tet is the scale that is both a syntonic and skhismatic
meantone, and can be defined as such using Fokker's method.

The symbols / and \ can then be added to a 12 note piano roll
view to denote boosts of q from the following scale:

r s s r s r s r s s r s
C C# D Eb E F F# G G# A Bb B C'

There is a one to one mapping between this and the standard
staff notation. The symbols # and b can still be used if
they are found to be convenient.


Over the past couple of days, I have extended this idea to
syntonic intervals using Greek letters. The major tone is a
kind of tau, and a diatonic semitone is a kind of sigma. Then,
rho=tau-sigma and pi=sigma-rho. In Pythagorean terms, sigma's
are apotomes and rho's are limmas.

A minor tone has dimensions of tau-pi or 2rho. A chromatic
semitone has dimensions rho-pi, and an enharmonic diesis is
2pi. The syntonic and Pythagorean commas are both pi's, and I
have also defined the septimal comma to be a pi.

The piano roll view can also be adapted to skhismatic scales
with / and \ denoting boosts of pi from a scale like this:

rho rho sigma rho sigma rho sigma rho rho sigma rho sigma
C D E F G A B C

The capital letters here name keys rather than notes, as
the standard note names do not work with skhismatic scales.
This is compatible with the syntonic version in that the
same scale will occupy the same positions in both notations,
but with different modifiers. They are also compatible with
12 note keyboards.


One day, I intend to implement these views in software, so
this isn't all abstract. It is also easier to get dynamic
tuning to work with 2-D than just scales.

Now, does anyone know of a 2-D approximation (linear
temperament) that covers 22tet? Approximating the
enharmonic diesis (7 0 -3)H and syntonic comma (-4 4 -1)H
to be equal doesn't work. If it did, it would cover 12, 22,
34 and 46tet. It is, incidentally, significant that these
are all even numbers.



SMTPOriginator: tuning@eartha.mills.edu
From: kollos@cavehill.dnet.co.uk
Subject: Re: 1/4 tones and 1/n "comma" meantones - naming
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About JC's request on the Microtonal Blues Band, here's info I posted once
before. There were two LP's (both still sitting in my closet) that are long
out of print, and doubt have been transferred to CD (though 8-tracks may
still be around):

- "Journey To Bliss - Emil Richards and the Microtonal Blues Band"
(Impulse Stereo A-9166)

- "Spirit of 1976 - Emil Richards and the Microtonal Blues Band / Live at
Donte's"
(Impulse Stereo A-9182)

To be upfront about it, these are not excursions into microtonal
composition per se; Emil and company used microtonal instruments,
virtually all of these percussion (and mostly made by Emil, I think) to add
colors to the music. They had been influenced by Hari Har Rao, who had
recently begun teaching Indian music in the LA area at the time. Most of
the expansion of the jazz idiom came in the form of odd meters. They were
also being influenced by the Maharishi himself (with LP #1 having a
particularly striking oil painting of him on the back), and more than a
certainty being influenced by burning vegetation of an illegal nature
(except for medicinal purposes, of course).

It's up to the individual whether to check them out as microtonal
resources, but if you want a retro insight into the late 60's (when I
believe these were recorded) they can't be beat. Side Two of LP #1 is
entitled "Journey To Bliss: A Meditation Suite in Six Movements".
Priceless stuff, I assure you.

Emil was turned on (hmmm...) to the Indian thing by Don Ellis, and for a
while they had a group called the Hindustani Jazz Sextet. About that
group, I know nothing. Oh, hell, about a lot of things I know nothing.

>I imagine others on this list would like the information as well.

Well, John, we'll 'just' see about that!

Cheers all,
Jon
*------------------------------------------------------------------*
Jonathan M. Szanto | Nearly one year old, Partch's place online
Corporeal Meadows | is really showing signs of urbane sprawl...
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SMTPOriginator: tuning@eartha.mills.edu
From: rapoport@mcmail.CIS.McMaster.CA
Subject: quartertones
PostedDate: 17-08-97 03:20:12
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I wonder if the "Microtonal Blues Band" your friend refers to is LaMonte Youngs
Forever Bad Blues Band?? If so the CD is available from them at 275 Church St.
NYC 10013 vox 212 925-8270 fax 212 226 7802

Regards,

Allen Strange

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