TUNING digest 1338

Re: Topic No. 1:

I bought a copy of Mathieu's book at a 10% discount at Border's in
Philadelphia.

Mathieu sent me a draft copy of part of the book about two years ago in
response to my contacting him on a referral from Ben Johnston. His
theoretical and notational concepts are very similar to my own, which I
call JustMusic (TM).

One of the fundamental ideas in Mathieu's conception is that we don't
have to search too hard to find where the simpler ratios are in the
pitch-continuum -- we can learn to sing them fairly easily because they
resonate in our body cavities and in the objects around us, giving a
quite direct experience of the unique power and sound of each of them as
we sing them.

I've bought a copy of the book and have perused it, but haven't had the
time for a proper reading, as I am working to finish writing my own.
Much of his notational philosophy is indeed very similar to mine.

=======================================
Re: Topic No. 2, "prime limits"

As you may read below, my tuning theories are founded upon the idea that
each prime number given or implied in a musical ratio (compared to some
reference) lends a unique and distinctive aspect to the sound of that
ratio.

Just as quantum physics has discovered that there is an uncertainty
principle involved in trying to describe what happens to sub-atomic
particles, after boiling ratios down to their irreducible components,
what is left is the series of prime numbers -- a series the ordering and
pattern of which no one has yet been able to discover. I believe it is
quite possible that the prime series exhibits the same kind of randomness
as sub-atomic physics, thus all of our human ordering and patterning of
musical sounds is based on a given property which is random. If there is
any weight to this idea, it may be worth exploring further -- it ties in
with Ancient Greek and Medieval European ideas about the "Music of the
Cosmos". I think it presents some interesting philosophical questions.

==========================================
Re: Topic No. 2, "a standard notational which can capture any tuning..."

This has been the whole focus behind the development of my theory.
Unfortunately, I haven't found a way to reconcile the differences between
just-intonation and temperament. However, I have narrowed it to those
two types of "accidental". Within each of these two broad families,
each respective notation can represent any given tuning in a clear,
simple manner, from one given unique reference point. This makes it
possible to level any number of different intonational philosophies to
the same analytical playing field.

I base both of my notational systems on the standard line-space/ letter
name/ sharp-flat European notation -- thus, trained musicians have no
difficulty finding the 12-equal (or perhaps more correctly, the
Pythagorean) note which falls closest to the desired frequency.

---------------------------------------------

For any equal temperament, this is supplemented by a root sign with the
appropriate roots and exponents of 2. For meantone, a combination of
roots and ratios is used. I'm limited to representing it in email
because it's difficult to show roots and powers, but as an example (just
replace the verbal description of the expression with mathematical
symbols):

Middle-C in 12-equal is:
the 12th root of 2 to the 0 power = 1, the unity.

E above Middle-C in 12-equal is:
the 12th root of 2 to the 4 power (= 400 cents = the 4th semitone
above C)

E above Middle-C in 1/4 comma Meantone is:
th 4/4 root (= 1 or identity root) of 5/4, which equals 5/4 itself

A above Middle-C in 1/4 comma Meantone is:
the 4/3 root of 5/2

D above Middle-C (i.e., the "mean tone") in 1/4 comma Meantone is:
the 4/2 root (= square root) of 5/2

G above Middle-C in 1/4 comma Meantone is:
the 4/1 root (= 4th root) of 5

Middle-C in 1/4 comma Meantone is:
the 4/0 root of 5 -- an impossiblity, which equals no root, which
equals 1, the unity.

------------------------------------

For just-intonation, the accidental consists of the series of prime
nubmers above 2 used as bases, with positive, zero, or negative exponents
for each prime. The idea is similar to Ben Johnston's notational
philosophy, but my reference scale is not the mixed diatonic using ratios
of 3 AND 5, but rather the 3-limit scale only.

In actual musical notation, it is least clumsly to eliminate all primes
which have a zero exponent as they are equal to 1 and do not affect the
product of the multiplication, and also to eliminate prime-base 3, as the
letter-name and sharp-flat aspect of the notation already represents the
3-Limit or Pythagorean scale.

Thus, as examples:

- the G a 3/2 above Middle-C is written: G 3^1 (3 to the 1st power)

- the E a 5/4 above Middle-C is written: E 5^1 (5 to the 1st power)

- the Eb a 7/6 above Middle-C is written: Eb 3^ -1 7^1 (3 to the minus
1, 7 to the first)

- the B# which has the monstrous ratio of 2025/1024 above Middle-C, and
which is the tritone of the tritone, is in JustMusic simply written: B#
3^4 5^2


In mathematical calculations, it is easiest to assume the series of prime
bases 3, 5, 7, 11... as a given, and notate only the exponents -- this
time including the zero exponents as place-holders. Interval
calculations can be done by simple Matrix Addition, rather than
complicated multiplications of fractions.

Also, by factoring out the powers of 2 as being unnecessary for the
purpose of determing pitch-CLASS, the ratios are made much simpler, and
the important prime components are more clearly exposed. Including the
powers of 2 gives an absolute frequency precisely. (Notating Partch's
music this way makes it a lot easier to analyze)

In my JustMusic system, powers of 2 may be represented in either of two
ways:

- 2 to the 0 power may equal 1 Hz, with each ascending "octave" of C
being the next higher power of 2. The notes audible to humans fall in
the approximate range of 2 to the 4th power (= 16 Hz) to a couple of
semitones above 2 to the 14th power (approximately 20,000 Hz).

- alternatively, 2 to the 0 power may represent "Middle-C", with the four
audible octaves below it being 2 to the -1, -2, -3, and -4 powers
respectively, and the octaves above middle C being the corresponding
positive exponents of 2. I find this method to be conceptually more
viable.

In addition to working on completing my book, I have just submitted an
intial article for publication. If anyone would like me to send a copy
via regular mail, let me know, and send a check for $3 to cover my
reproduction and mailing costs. Also, I'd like to enlist the help of
those capable to design a web site. Anyone interested in corresponding
may feel free to email or call anytime.

[By the way, Ezra Sim's notation is a 72-equal system, but his harmonic
basis is implied 37-limit (sometimes higher) just-intonation. As far as
notational systems go, his is clear and concise (simpler than mine), the
step-size is only 16 and 3/4 cents, and the 72-equal degrees fall mighty
close to all of the ratios he is representing. Johnny Reinhard uses a
multiplicity of notational concepts, because he is a "polymicrotonalist",
using a wide variety of both just and equal tunings, frequently in the
same piece.]

Joseph L. Monzo
monz@juno.com
4940 Rubicam St., Philadelphia, PA 19144-1809, USA
phone 215 849 6723
------------------------------------------------------------------------------------------------------------
On Sat, 28 Feb 1998 11:19:26 -0500 (EST) tuning@eartha.mills.edu writes:
> TUNING Digest 1339
>------------------------

>Topic No. 1
>
>Date: Fri, 27 Feb 1998 15:13:28 -0800 (PST)
>From: John Chalmers
>To: Alternative Tuning List
>Subject: Books noticed
>Message-ID:
>
>
>I was in a Barnes and Noble's bookstore today and saw a book which
>I think deserves more than the quick skim I had time to do.
>"Harmonic Experience" by W.A. Mathieu, 1998, Inner Traditions
>International, Rochester, VT USA $45.00.
>
>It is large harmony text based on both western and Indian scales and
>Just Intonation as well as 12-tet. Mathieu limits himself pretty much
>to the 5-limit, but does have one short chapter on 7, 11 and above.
>It has lots of ear and voice training examples in JI and discusses the
>construction of Indian and well as western modal scales. Good biblio
>as well (Harrison, Partch, Doty, Hopkin, etc.)
>
>...
>--John
>
>------------------------------
>
>Topic No. 2
>
>Date: Sat, 28 Feb 1998 09:38:10 -0600 (CST)
>From: mr88cet@texas.net (Gary Morrison)
>To: tuning@eartha.mills.edu
>Subject: Re: Fretless versus Fretted
>Message-ID:
>
>...
>
>>In Just Intonation, what does "limit" refer to?
>
> The limit is the largest odd or prime number (depending on who you
>ask)
>in the frequency ratios in the tuning. Most JI investigators find
>that
>each odd or prime outlines a new, broad class of harmony. I
>personally
>think that there's a lot of truth in that idea, but that there are
>many
>other factors that have to be considered, some with about equal
>weight.
>
>...
>
>
>What about a standard notation that can accurately capture any tuning?
>(freqeuncy,volume,time)
>
> Two people to talk to along those lines are Ezra Simms and Johnny
>Reinhard. Johnny is on the list, but I don't think Ezra is.
>
>------------------------------
>
>End of TUNING Digest 1339
>*************************
>


Joseph L. Monzo
monz@juno.com
4940 Rubicam St., Philadelphia, PA 19144-1809, USA
phone 215 849 6723

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