Yahoo Tuning Groups Ultimate Backup tuning 'higher-prime' translated out of academese

Someone on the Mahler List had a question about my reference
to 'higher-prime', and my explanation to him ran to such
depth that I thought tuning newbies could benefit, so I'm
sending a copy here. As always, criticism is welcome.

I'll probably make a 'beginners' webpage out of this, for
my website.

---------------- forwarded message -------------------

On Sun, 31 Oct 1999 10:20:17 EST Steve Vasta writes:

> I don't understand this business of higher-prime whatever.
> Please retranslate out of "academese" into more easily
> understandable terms. Much thanks.

OK - I'll try, but it's really difficult because it's quite
technical stuff. Sorry, but I've been working in tuning
theory for 16 years, and this is so old-hat for me that
don't always 'start at the beginning'. In addition, until
I got on the Mahler List in June, just about everyone I
corresponded with via email was another tuning theorist,
so they know all this stuff too. Here goes, very briefly
(but still quite long):

NO SUCH THING AS 'ATONAL':
--------------------------

I believe, as did Schoenberg, that 'atonal' music, in the
literal meaning of the term, is an oxymoron, an impossibility,
because...

FREQUENCY
---------

A sound which does not produce a specific pitch, for example,
a cymbal crash or beating on a snare-drum, is one that does
not vibrate air molecules at a specific frequency, but rather,
vibrates the molecules in motions that are more-or-less
random. This is referred to technically as 'white noise'.

On the other hand, a sound which produces a specific pitch is
one that vibrates air molecules at a *specific* frequency, as
in our famous tuning-fork standard, A-440. This simply means
that the note which we ordinarily call 'A' (above 'middle-C')
vibrates air molecules back and forth at the rate of 440 cycles
per second. A cycle is simply the waveform of that particular
sound, which never varies (or varies so slightly that we
don't perceive a change in pitch) during the course of the
sound.

Therefore, in any group of different pitches (i.e., any
chord or melody), each pitch will bear a specific relation
to all the others, and it is expressible mathematically.
There are often several different ways of describing these
mathematical relationships - primarily dependant on the
type of tuning system used on the instruments, altho modern
research has found that the mathematical relationships
*implied* or *perceived* by individual listeners is a far
more complex matter than has previously been thought -
but the important point is that the numerical relationships
*always* exist, as long as the sound source is periodic.

This is precisely why Schoenberg and I cannot condone the
use of 'atonal' to describe music that does not follow
the 'rules' of traditional tonality. Schoenberg preferred
'pantonal' to describe his post-1907 compositions; I like
'extended tonality'.

RATIOS:
-------

Any quantities that can be compared by some type of integer
proportion are said to be 'rational', expressible by a
*ratio*. NOTE THAT RATIOS MAY BE DESCRIBED AND MANIPULATED
MATHEMATICALLY AS FRACTIONS.

This is one of the fundamental aspects of tuning theory.
Any interval (i.e., any two specific pitches) where the
pitches are in an integer relationship to each other, can
be described as a ratio. The two most well-known rational
types of tuning system are 'Pythagorean' and 'Just-Intonation'.

The fundamental ratio is that of the 'unison' interval,
which is 1:1.

The next simplest ratio is that of the 'octave', 2:1.

These are the only unambiguous ratios in tuning theory
(altho even the 2:1 for the 'octave' is suspect: modern
research has found that most listeners find the 'most
consonant octave' to be slightly larger than 2:1).

The 'perfect 5th' in rational tuning theories is 3:2.
The 'perfect 4th', its inverse or 'complement', is 4:3.

Now I must leave rational tuning theory and digress to
describe the scale in common use today...

12-EQ:
------

This is my abbreviation for the standard 12-tone equal-tempered
scale, which is the one normally tuned on pianos and fretted on
guitars. (but these days, not always... microtonality is
becoming more and more popular... but on your standard
store-bought instruments, this is what you get.)

Another popular abbreviation is 12tET.

In the 12-eq scale, the 'octave' is always tuned precisely
to a 2:1 ratio, and each 'step' in the 12-tone scale is tuned
to the 12th root of each progressive power of 2.

In other words, using the 'chromatic' scale starting on 'C'
as an example, we get the following mathematical relationships.
The scale is presented here from the top down. '^' means 'raised
to the power of', and is the ASCII equivalent of superscript
exponents. Raising any number to a fractional power is the
same as drawing a 'root' sign with the denominator as the root,
for example, '2^(1/2)' is the same as 'the 2nd root of 2^1',
which is the same as 'the square root of 2'. Also note that
any number to the '0th' power equals 1. (Use a fixed-width
font such as 'Courier' to make everything line up properly.)

C 2^(12/12) = 2^1 = 2
B 2^(11/12)
Bb/A# 2^(10/12) = 2^(5/6)
A 2^(9/12) = 2^(3/4)
Ab/G# 2^(8/12) = 2^(2/3) = 'cube root of 2 squared'
G 2^(7/12)
Gb/F# 2^(6/12) = 2^(1/2) = 'square root of 2'
F 2^(5/12)
E 2^(4/12) = 2^(1/3) = 'cube root of 2'
Eb/D# 2^(3/12) = 2^(1/4) = '4th root of 2'
D 2^(2/12) = 2^(1/6)
Db/C# 2^(1/12)
C 2^(0/12) = 2^0 = 1

The simplifications to the right are not really important
in tuning theory - I just put them there to help you understand.
You should note, however, that the 'tritone' is the
square root of 2, which means that it *equally* divides
the pitch-space of the 'octave'. Also note that the
ratio of the 'octave', 2^(12/12):2^(0/12), is reducible
to 2:1, as described in the previous section.

It is very important to tuning theory that *none* of the
other intervals in this scale are expressible as ratios.
Except for the 'octave' and its multiples, all other
intervals can be described as ratios only by using infinite
non-repeating decimals. THE 12-EQ SCALE CANNOT BE DESCRIBED
AS FRACTIONS.

As far as the original subject of this correspondence, note
that THE ONLY PRIME NUMBER INVOLVED IN THE RATIOS OF
THE 12-EQ SCALE IS '2', THE FIRST PRIME.

Historical note: the mathematics to describe the 12-eq scale
were not developed until the 1600s (at almost the same time
in China and Europe - China was first), so even if the concept
was 'in the air' prior to that, this scale could not be
accurately tuned until then. In fact, real precision was
not possible until the advent of modern digital electronics.
Rational systems, on the other hand, have been quite
accurately tuned by ear since ancient times.

SONANCE
-------

I believe, as did both Schoenberg and Harry Partch, that
consonance and dissonance are not diametrically opposed
aspects of sound, but are rather the opposite endpoints of
a continuum.

Schoenberg continued to use the two terms in the way most
theorists use them, which has caused much confusion in the
literature about Schoenberg's theories by now.

Partch improved on this a bit by always saying 'relative
consonance or dissonance' instead of using either term
separately.

I simply use the term 'sonance' to describe the whole
continuum, with the understanding that absolute consonance
means the 1:1 (unison) ratio, absolute dissonance is an
undefinable ratio which is too complex to comprehend
(perhaps it is 'white noise'), and the sonance of all
other musical ratios falls somewhere in between.

CENTS
-----

In his 1875 English translation of Helmholtz's _On the
Sensations of Tone_, Alexander Ellis described a small
interval called a 'cent', which was made equivalent to
1/100th of a 12-eq 'semitone'. This measurement has become
almost indispensable in the modern tuning literature.

The mathematical description of it is: 2^(1/1200). Thus,
a 'semitone' equals 100 cents, and an 'octave' equals 1200
cents.

The cent is very useful for comparing the sizes of intervals
which may be derived from a variety of different mathematical
procedures. It's a kind of 'yardstick'.

PYTHAGOREAN TUNING
------------------

Pythagorean tuning (named after the ancient Greek philosopher)
is based on a series of 'perfect 5ths' or 'perfect 4ths'.
This is because the 12th pitch in the series is very close
to the original starting pitch, about 23 cents (or approximately
1/4 of a semitone, or 1/8-tone) higher or lower (depending
on whether the series was generated by '4ths' or '5ths').
This small discrepancy is now known as the 'Pythagorean
Comma'.

Because of this near-unison, many theorists both ancient
and modern felt that a 12-tone scale could provide all the
pitches necessary for musical use, with the appropriate
'fudging' of the 23-cent difference. This is, historically,
why we still use a 12-tone scale today.

Also, both our musical notation and the concept of a 'circle
of 5ths' are ultimately derived from the Pythagorean tuning
system.

The important point concerning prime-numbers is that
PYTHAGOREAN TUNING USES ONLY THE PRIMES 2 AND 3 IN ITS RATIOS.
Thus, it is known as a '3-limit' system.

JUST-INTONATION
---------------

There are conflicting differences in the various definitions
of 'just-intonation', but primarily, it refers to the fact
that there are ratios which involve '5' as a factor which
are rather close in pitch to their Pythagorean 'cousins'
but which make for much more consonant harmonies. These
are mainly the intervals of '3rds' and '6ths', both major
and minor.

As an example:

The 'major 3rd' in the Pythagorean scale is derived from
a succession of 4 'perfect 5ths' upward or 'perfect 4ths'
downward:

(3:2)^4 = 81:16 = E
(3:2)^3 = 27:8 = A
(3:2)^2 = 9:4 = D
(3:2)^1 = 3:2 = G
(3:2)^0 = 1:1 = C

This 81:16 ratio actually describes the very large interval
of a 'major 17th' (i.e., 2 'octaves' plus a 'major 3rd').
If we reduce the interval by eliminating the 'octaves'
(i.e., the powers of 2), so that it really is a 'major 3rd',
its ratio becomes 81:64. Its interval size can also be
described as approximately 408 cents.

What this means acoustically is that the waveforms of the
two pitches will coincide at every 64th cycle of the lower
pitch and every 81st cycle of the higher pitch. As these
numbers are too high for the brain to comprehend their
relationship during listening, this is not a very consonant
sound.

The ratio 5:4, on the other hand, is obviously much more
consonant, since the waveforms coincide at every 5th cycle
of the higher pitch and every 4th cycle of the lower pitch.
This interval can also be described as approximately
386 cents. It is known as the 'just major 3rd', 'just'
being here a synonym for 'correct'.

The difference between the just and Pythagorean 'major 3rds'
is approximately 22 cents. This discrepancy is known as
the 'Syntonic Comma', or if just plain old 'comma' is used,
this is what is meant.

What I described here about the 'major 3rd' applies also to
the 'minor 3rd' as well as the 'major 6th' and 'minor 6th'.

Note that the size of the Syntonic Comma is almost the
same as that of the Pythagorean Comma, which means that
if both systems are extended far enough, to include a large
enough number of pitches, they are almost synonymous.
This idea has been the basis of many different kinds of
'tempered' scales other than the usual 12-eq one.

HIGHER-PRIME FACTORS
--------------------

This is where the disagreement come in about what is
definable as 'just-intonation'.

Many tuning theorists and composers use 'just-intonation'
to refer to *any* tuning system which uses only rational
pitch intervals, including those with prime-factors higher
than 5 (i.e, higher 'prime-limit' systems), while others
limit 'just-intonation' to meaning only that system using
factors of 2, 3, and 5, that is, a '5-limit' system.

At any rate, there was much disagreement from 1482 (when
Ramos first described a '5-limit' tuning system) up until
about Schoenberg's time, whether or not ratios involving
7 are implied in musical compositions, or indeed whether
ratios involving 7 can be considered musical at all.
(Schenker was a notable opponent of recognizing primes
higher than 5.)

Modern music, especially that derived from the blues, has
answered this question decisively in the affirmative.

Partly because of the influence of Schoenberg's theories
and compositions, and partly just because of the exhaustive
exploration that has taken place in the 20th century of the
intervallic and harmonic possibilities inherent in the 12-eq
scale, much music exists which implies (to a greater or lesser
degree) ratios which have prime-factors even higher than 7,
such as 11, 13, 17, and 19.

Most theorists are reluctant to admit that primes higher
than 19 are implied in actual music, and many won't even
go that far (since some prime-factors, particularly those
of 11 and 13, produce intervals quite far in pitch from
the familiar ones in the 12-eq, Pythagorean, and 5-limit
tunings), but some composers working in just-intonation
specifically use intervals which have higher prime-factors.
La Monte Young has probably gone further in this than any other composer,
using primes as high as 283 in his latest piece.

There are also *lots* of other types of tuning systems
besides equal-tempered and just-intonation, notably
'meantone' and 'well-temperament'; but there are even
others that are more complicated ('Lucytuning', Bohlen-
Pierce tuning, etc.). Much info about all aspects of
tuning can be found on my website.

-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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'higher-prime' translated out of academese

🔗Joe Monzo <monz@xxxx.xxxx>

🔗Janmichael Stubbs <wavingpalms@xxxxxxxxx.xxxx>

🔗Paul H. Erlich <PErlich@xxxxxxxxxxxxx.xxxx>

🔗Paul H. Erlich <PErlich@xxxxxxxxxxxxx.xxxx>