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Tutorial on ancient Greek Tetrachord-theory

[Joe Monzo]

© 1999 by Joe Monzo

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

From: Joe Monzo
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 (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:

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. The notes bounding each tetrachord were fixed, and those inside it were moveable:

General schematic diagram of Diatonic genus "octave"

     E  nete          fixed
 /   D  paranete      moveable
 \   C  trite         moveable
   \ B  paramese      fixed
8:9 <
     A  mese          fixed
 /   G  lichanos      moveable
 \   F  parhypate     moveable
   \ E  hypate        fixed

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:

     A  mese          fixed
 /      lichanos      moveable
 \      parhypate     moveable
   \ E  hypate        fixed

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

     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.

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'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, less than half the square-root of 4/3, or
< (4/3)(1/2).

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 genus:

     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 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.