tutorial on ancient Greek Tetrachord-theory

OOPS!

When I announced my webpage tutorial on ancient Greek
Tetrachord-theory here yesterday, I had intended to reproduce
the text and diagrams of the webpage in my post for the
benefit of those without internet access. Here it is.

===============================

This was expanded out of a private email I sent to Dan Stearns.
Thanks for the inspiration, Dan!

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

From: Joe Monzo <monz@juno.com>
To: stearns@capecod.net
Subject: 'lichanos' and 'pyknon'

On Tue, 14 Dec 1999 22:36:50 -0800 "D.Stearns" writes:

> Joe

>> I've been working on my Aristoxenus stuff all day today.
>> (Have you seen that yet? I'm interested in your opinion.
>
> Yes, and as it's both fairly massive

I know - it's grown *WAY* beyond what I originally thought
it would be, and still going...

> and chock full of "lichanos" and "pyknons" (etc.) that I'm
> going to have to give it a couple reads before I could offer
> anything resembling a sensible comment!

I suppose I took for granted that people interested in Aristoxenus
would already know the Greek musical terms. I'm going to
have to add something at the beginning laying that all out.

Here's a brief tutorial (perhaps when I'll just paste this
into my paper):

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

First of all, the basis for Greek scale construction was
the _tetrachord_ (= '4 strings'). Their theory (at
least Aristoxenus and after) was based on the lyre, and
not on any wind instruments.

(Aristoxenus criticizes those who base their theory on the
_aulos_, which was a sort of oboe. Kathleen Schlesinger wrote
a book, _The Greek Aulos_, which Partch admired, but which has
since been discredited, where she reconstructs ancient scales
based on measurements of holes in surviving ancient auloi.)

(Note: Kraig Grady has already registered dissent against
the arguments discrediting Schlesinger; the fact is that
*much* more research needs to be done on the question of
what scales the ancient _auloi_ actually produced.)

So the tetrachord designates 4 notes, two of which are fixed
and two are moveable.

The fixed notes are those bounding the tetrachord, which
are always assumed to be a 3:4 interval. It's the position
of the two moveable notes that was argued about so much,
and which makes this stuff so interesting to tuning theorists.

(BTW, John Chalmers's book _Divisions of the Tetrachord_ is
entirely about specifically this.)

Those various divisions are what determine the different
_genera_ (plural of _genus_ - the actual Greek word
is _genos_, but commentators writing in English generally
use the Latin form). There were 3 basic genera:
Diatonic (= 'thru tones'), Chromatic (= 'colored' or
'thru the shades'), and Enharmonic (= 'properly attuned').

Apparently the Enharmonic derived from the ancient scales
which were called _harmonia_, thus its name. That was
the one with 'quarter-tones'. The chromatic had a pattern
that more-or-less involved a succession of 2 semitones,
and the Diatonic is the one we're most familiar with,
using mainly 'whole tones' with a few semitones.

Aristoxenus said that there were also many different 'shades'
or 'colors' of all three genera, using different interval
sizes, but that the genus was specified according to some
vague overall 'feeling' about how it sounded. He specified
the measurements for 2 shades of Diatonic and 3 shades of
Chromatic, but while he described other shades of Enharmonic,
he gave measurements for only one.

The main thing to remember is that the names of the Greek
notes are based on their position within the tetrachord,
and that since two of the notes are moveable, it's really
better to *THINK* in the Greek way rather than try to
represent this stuff in our modern scale/note terms.

But that said, the easiest way for you to begin understanding
it is to outline the Diatonic using our letter-name notes.

The reference pitch in Greek theory was called _mese_
(= 'middle'), which we can call 'A'. The names of the
strings (= notes) came from their position on the lyre.

A confusing point: the names designated the string's
distance from the player, NOT its pitch; similar to
a guitar, where the string lowest in pitch (low E) is
the one at the top of the set of six strings, and also
the nearest to the player.

The Diatonic genus 'octave' scale would be:

E nete Furthest/Lowest
D paranete Next to 'nete'
C trite Third
B paramese Next to 'mese'
A mese Middle
G lichanos Forefinger
F parhypate Next to 'hypate'
E hypate Nearest/Highest

(Note that I use the 'octave' pitch-space here only to
illustrate the whole 'octave' scale and to help modern readers
understand. Aristoxenus spoke almost entirely in terms of
divisions of a tetrachord spanning a 3:4 'perfect 4th'.)

The distance from _nete_ to _paramese_ is a 3:4, and the
distance from _mese_ to _hypate_ is a 3:4, with an 8:9 'tone of
disjunction' between _paramese_ and _mese_.

There were other tetrachords in the complete systems,
and some were conjunct (the lowest note of the upper
tetrachord is the same as the highest note of the lower
tetrachord) while others were disjunct (with a tone between),
and some of them used the _nete/paranete/trite_ names,
while others used the _lichanos/parhypate/hypate_ names.

I'm not going to go into all that, as its irrelevant
to the specific thing I discuss in my paper, where Aristoxenus
uses one tetrachord to describe the divisions, and says that
the same divisions would occur in all other tetrachords of
the complete systems.

So we'll stick with the tetrachord _mese-lichanos-parhypate-
hypate_. The Greeks thought of their scales downward, the
opposite from us.

The tricky part is that the same names are used for the
Chromatic and Enharmonic genera, where we would have different
letter-names because of the varying interval sizes.

Aristoxenus specifically argues against this type of conception,
saying that the notes in the various genera should be named
according to their *function* in the scale. This is really
a lot like using Roman numerals (sometimes with accidentals)
to designate scale-degrees and chords, instead of letters or
Arabic pitch-class numbers.

So the fixed boundary-notes, _mese_ and _hypate_,
would be analagous to our 'A' and the 'E' below it.
_lichanos_ and _parhypate_ are the two moveable notes:

A mese
/
/ lichanos
3:4
\ parhypate
\
\ E hypate

The Diatonic genus is illustrated by this tetrachord:

A mese
/ > tone = 'major 2nd'
/ G lichanos
3:4 > tone = 'major 2nd'
\ F parhypate
\ > semitone = 'minor 2nd'
\ E hypate

The distinctive thing about this genus is the interval of
a tone between _mese_ and _lichanos_. This top interval is
nowadays known as the 'Characteristic Interval' of a genus.
Then the other intervals of the Diatonic (going downward)
are a tone between _lichanos_ and _parhypate_, and a semitone
between _parhypate_ and _hypate_.

Still with me?...

Here's the basic Chromatic genus:

A mese
/ > trihemitone = 'minor 3rd'
/ F# lichanos
3:4 > semitone = 'minor 2nd'
\ F parhypate
\ > semitone = 'minor 2nd'
\ E hypate

Here, the Characteristic Interval between _mese_ and _lichanos_
is one of 3 semitones, a 'trihemitone' (what we would call
a 'minor 3rd'). The other two intervals are both semitones.

This is where the _pyknon_ (= 'compressed') comes in. There
is no pyknon in the Diatonic, because a pyknon indicates
a group of two intervals that is smaller than half of the
total tetrachord-space, that is, < sqrt(4/3).

Aristoxenus's 'Relaxed Diatonic' had a _lichanos_ that we
could call 'Gv', that is, a 'quarter-tone' between 'G' and 'F#'.
This is the exact mid-point of the 3:4, and thus marks the
lowest shade of Diatonic, as well as the lowest genus without
a pyknon. All genera with a lower _lichanos_ were Chromatic
or Enharmonic, and had a pyknon.

Aristoxenus calls this particular shade of Chromatic the
'Tonic', because the pyknon from _lichanos_ to _hypate_ (F# to E)
is a 'whole tone'.

Here's the Enharmonic:

A mese
/ > ditone = 'major 3rd'
/ F lichanos
3:4 > enharmonic diesis = quarter-tone
\ Fv parhypate
\ > enharmonic diesis = quarter-tone
\ E hypate

Here, the Characteristic Interval between _mese_ and _lichanos_
is a 'ditone' (what we would call a 'major 3rd'), and the two
remaining intervals are 'enharmonic dieses', or 'quarter-tones'.

I said earlier that Aristoxenus describes other shades of
Enharmonic which he does not measure. He argues (without
saying anything about ratios) that the one with the true
ditone was used in the ancient style, which he is known to
have preferred, and that modern musicians use a higher
_lichanos_ to 'sweeten' it. This can only mean that he
preferred the 64:81 Pythagorean ditone, and criticized the
4:5 used by the 'moderns', as measured by Didymus. To tuning
theorists, it's one of the most interesting things in his book.

But by far what I've found to be most interesting over the
years is his descriptions of the two other shades of Chromatic,
the 'relaxed' and the 'hemiolic'.

There has been much confusion simply because Aristoxenus
never says anything about ratios, but his method of tuning
is patently Pythagorean (see my diagrams of 'Tuning by
Concords').

He calls the enharmonic diesis a '1/4-tone', and the smallest
chromatic diesis a '1/3-tone', and mentions '1/6-tones' and
'1/12'-tones in his comparisions of the various genera,
but as you can see from my mathematical speculations, the
numbers don't jive unless you assume that he was using
very loose terminology, where '1/4', '1/3', '1/6', and '1/12' are
only *approximations*.

Anyway, that should be enough for you to understand my paper.
Hope it helps.

I'll have to give Aristoxenus a break for a while to give
you any further ideas about L&s.

-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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