Ancient Greek Tetrachords

When it comes to ancient Greek music, we've learned about Pythagorean tuning, and a little bit about Archytas' tetrachords and method of geometric means. Next we're going to learn about ancient Greek tetrachords generally.

We have tetrachord descriptions from ancient Greek writers like Archytas, Didymus, Eratosthenes, and Ptolemy. Their tetrachords have four notes and span a perfect fourth, as you'd expect. The last relative interval, between the third and fourth notes, defines the tetrachord's type. The last interval will be roughly sized like a major second ("Diatonic" tetrachords), or it will be sized like a minor third ("Chromatic" tetrachords"), or it will be sized like a major third, ("Enharmonic" tetrachords).

A diatonic tetrachord (ending in a major second) is more specifically sized something like this:

[m2, M2, M2]

with minor second sized ratios including [256/243, 16/15, 28/27, 21/20, 12/11]. The major seconds in a diatonic tetrachord might be [8/7, 9/8, 10/9, 11/10]. I think 12/11 and 11/10 are quite neutral sounding, not minor and major, but I'm a little late to voice my objections to the authors.

For a chromatic tetrachord (ending in a minor third), the intervals are sized like this:

[semitone, semitone, m3]

Now, one of the semitones is going to be a minor second and one is going to be an augmented unison, but they weren't consistent about which one, or even about whether a larger or smaller ratio goes first. Options for the first spot include [28/27, 16/15, 20/19, 28/27, 22/21, and 256/243]. Options for the second spot include [15/14, 25/24, 19/18, 243/224, 12/11, and 2187/2048].  The minor third has options of [6/5, 32/27, and 7/6].

For an enharmonic tetrachord (ending in a major third), the intervals are sized like this:

[super particular quarter tone, super particular quarter tone, major third].

The quarter tone options get pretty wild: [46/45, 40/39, 39/38, 36/35, 32/31, 31/30, 28/27, 24/23].  The major third has options of [5/4, 19/15, and 81/64].

We have just one example of each tetrachord type (Diatonic, Chromatic, Enharmonic) from the authors Archytas, Eratosthenes, and Didymus, but Ptolemy gave us a bunch.

I first learned about these from Robert Erickson's "The Musical System of Archytas". That source had a few typo though. When I searched for corrections, I found basically the same information in "The Harmonics Of Ptolemy" by James Frederick Mountford and in "Divisions Of The Tetrachord" by John Chalmers.

Let's go through them quickly. Archytas used septimal harmony in his tetrachords:

Diatonic: [Sbm2, SpM2, AcM2] # [28/27, 8/7, 9/8]

Chromatic: [Sbm2, SpAcA1, Grm3] # [28/27, 243/224, 32/27]

Enharmonic: [Sbm2, Sp1, M3] # [28/27, 36/35, 5/4]

Didymuys used 5-limit harmonic for two of his tetrachords and then busts out some 31-limit super particulars for his quarter tones:

Diatonic: [m2, M2, AcM2] # [16/15, 10/9, 9/8]

Chromatic: [m2, A1, m3] # [16/15, 25/24, 6/5]

Enharmonic: [None, None, M3] # [32/31, 31/30, 5/4]

It's cool, but I don't have interval names for those.

Eratosthenes gave a Pythagorean diatonic tetrachord and then used 19-limit ratios for his other two:

Diatonic: [Grm2, AcM2, AcM2] # [256/243, 9/8, 9/8]

Chromatic: [FaA1, Rsm2, m3] # [20/19, 19/18, 6/5]

Enharmonic: [ReA1, FaPr1, Rsd4] # [40/39, 39/38, 19/15]

Okay.

We can consider Pythagorean versions for all three types of tetrachord:

Diatonic: [Grm2, AcM2, AcM2] # [256/243, 9/8, 9/8]

Chromatic: [Grm2, AcAcA1, Grm3] # [256/243, 2187/2048, 32/27]

Enharmonic: [Grm2, AcM3] # [256/243, 81/64]

The Pythagorean Enharmonic's Grm2 should be split in two parts, but there isn't a standard tuning for the division. They might not have even had a division there. Whatever you pick, it's going to be pretty ugly. For example, you might use

(531441/524288) and (134217728/129140163) @ 23c and 66c

or

(282429536481/274877906944) and (70368744177664/68630377364883) @ 47c and 43c

It's not great, is it? 

Ptolemy wrote about four diatonic tetrachords. They all have extra greek words to indicate the tonal color.

He repeated the Pythagorean/Eratosthenes diatonic and the Archytas diatonic:

Diatonic Ditoniaion (Eratosthenes, Pythagorean) : [Grm2, AcM2, AcM2] # [256/243, 9/8, 9/8]

Diatonic Toniaion (Archytas) : [Sbm2, SpM2, AcM2] # [28/27, 8/7, 9/8]

And also three new ones:

Diatonic Malakon: [SbAcm2, M2, SpM2] # [21/20, 10/9, 8/7] # The soft diatonic.

Diatonic Homolon: [DeAcM2, Asm2, M2] # [12/11, 11/10, 10/9] # The equable diatonic.

Diatonic Syntonon: [m2, AcM2, M2] # [16/15, 9/8, 10/9] # The intense diatonic.

The equable one has the same relative intervals as my Georgian folk tetrachord, but in reverse order. Nice. These are respectively 7-limit, 11-limit, and 5-limit.

Ptolemy also gave us two Chromatic tetrachords:

Chromatic Malakon: [Sbm2, SpA1, m3] # [28/27, 15/14, 6/5]

Chromatic Syntonon: [AsSpGr1, DeAcM2, Sbm3] # [22/21, 12/11, 7/6]

which are 7-limit and 11-limit. Finally he gave us just one enharmonic tetrachord:

Enharmonic: [Nb1, Dsm2, M3] # [46/45, 24/23, 5/4]

which is 23-limit.