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on
ancient Greek Tetrachord-theory
From: Joe Monzo
On Tue, 14 Dec 1999 22:36:50 -0800 "D.Stearns"
> Joe
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
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 (a sort of small harp),
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,
where she reconstructs ancient scales
based on measurements of holes in surviving ancient auloi.
The work's major hypothesis, that the Greek modes
each had a particular numerical determinant and
a characteristic set of rational intervals produced
by the spacing of aulos finger-holes, has since
been discredited, but her theory remains an interesting
avenue for future exploration.)
So the tetrachord designates 4 notes, of which two are fixed
and two are moveable.
The fixed notes are those bounding the tetrachord, which
are always assumed to be the interval
of the Pythagorean
'perfect 4th', with the
ratio 3:4.
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 topic.)
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; this is 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:
(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.
The notes bounding each tetrachord were fixed, and those
inside it were moveable:
General schematic diagram of Diatonic genus "octave"
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 (or in other words, the systems
have "tetrachordal similarity", which is a common feature
of scales all around the world).
So we'll stick with a generic example of the tetrachords
having the note names
mese - lichanos - parhypate - hypate.
The Greeks thought of their scales downward, the
opposite of the way we do.
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
latter 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' a
'perfect 4th' below it.
lichanos and parhypate are the two moveable notes:
As in the lower tetrachord in the scale illustrated a few paragraphs
above, the Diatonic genus is illustrated by this tetrachord:
Intervallic structure of diatonic genus
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.
Thus, the genus was given the name "diatonic", which in
Greek means "thru tones",
because it is the only genus which
has 2 more-or-less equal
"whole-tone"
intervals in each
tetrachord, in addition to the "tones of disjunction"
separating various tetrachords; the overwhelming majority
of between-degree intervals in this genus are "whole-tones".
Still with me? ... now lets move on to the other genera.
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, less than half the square-root
of 4/3, or
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, the Characteristic Interval between mese and lichanos
is a
'ditone'
(what we would today 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,
possibly tending toward
12edo
(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
To: stearns@capecod.net
Subject: 'lichanos' and 'pyknon'
>
>> 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
> going to have to give it a couple reads before I could offer
> anything resembling a sensible comment!
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
E nete fixed
/
/ D paranete moveable
3:4
\ C trite moveable
\
\ B paramese fixed
8:9 <
A mese fixed
/
/ G lichanos moveable
3:4
\ F parhypate moveable
\
\ E hypate fixed
A mese fixed
/
/ lichanos moveable
3:4
\ parhypate moveable
\
\ E hypate fixed
A mese
/ > tone = 'major 2nd'
/ G lichanos
3:4 > tone = 'major 2nd'
\ F parhypate
\ > semitone = 'minor 2nd'
\ E hypate
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
< (4/3)(1/2).
Here's the Enharmonic genus:
A mese
/ > ditone = 'major 3rd'
/ F lichanos
3:4 > enharmonic diesis = quarter-tone
\ Fv parhypate
\ > enharmonic diesis = quarter-tone
\ E hypate
Joseph L. Monzo Philadelphia monz@juno.com
http://www.ixpres.com/interval/monzo/homepage.html
|"...I had broken thru the lattice barrier..."|
| - Erv Wilson |
--------------------------------------------------
Updated:
1999.12.16
2000.06.24
2001.10.31
2003.02.08
By Joe Monzo
|
or try some definitions. |
I welcome
feedback about this webpage:
|