The Tetrachord (Ancient)

Quoting Sachs The Rise of Music in the Ancient World, East and West some Greek Modes and Shades:

“according to Aristonuxes: interval from tonic from second note to third third to fourth(fixed)

Enharmonic 398 49 49

Chroma malakon 364 66 66

Chroma hemiolion 348 74 74

Chroma toninion 298 99 99
Diatonon malakon 249 149 99

Diatonon syntonon 199 199 99

Enharmonic 386 74 38

Chroma syntonon 267 151 80

Diatonon toninon 204 233 62

Diatonon syntonon 182 204 198

Diatonon ditoniaion 204 204 90

Diatonon homalon 182 165 151”

Tetrachords can be extended to make larger scales with more range. Consecutive tetrachords can be arranged two ways 1) the conjunct is where two Teteachords are connected and share one string (for a total of seven strings) or 2) the disjunct where the two Tetrachords are separate and there is a tone ( 9:8 string length ratio) separation between the two Tetrachords (for a total of eight strings). Again remember that the Tonic and the Forth are the fixed strings or notes. The two strings between the tonic and forth are movable pitches; however, the sum of their step size stays equal to a tone (9:8 step size) plus a semi-tone (16:15 step size).

The scale The following on the scale is from the Ten Books of Architecture by Vitruvius. The version available to us today is a summery of a summary of a translation of a translation of a copy of a copy (after all the average life of a copy is about a hundred years, so there had to be many copies and some copyist had to make “improvements”). Vitruvius translated Aristonuxes Greek into Latin and summarized. Vitruvius was copied and copied, with each copier changing it a little. Then Dr. Morris Morgan translated the copy from Valentine Rose’s Latin into English and cut the section on Aristonuxes short saying what Vitruvius Summery just contained an obvious interpolation. Valentine Rose’s copy added a grand staff to the section on Aristonuxes to help explain the  Refer to Figure 2, the scale of Aristonuxe as quoted from Vitruvius, Book V, Chapter IV, page XX. These scales are also called the hexatonic and the octatonic scales.

and increase its length further by 4:3 to go down to the Diezbugmenon Paramese. Tune the Diezbugmenon Paramese Harp string to unison with the Monochord. This concludes tuning the fix pitch strings, but as noted before this gives a break in the middle of the scale between the Synmemmenon Tetrachord and the Diezbugmenon Hexichord.

The Synhemmenon Nete (highest note of the lower Tetrachord) is a pitch step and a half above the Diezbugmenon Paramese (the lowest note of the higher Tetrachord).

Figure 3. The Greater Perfect Scale.

Naming of the notes of the Greater Perfect Scale

The somewhat cryptic remark “literally high” refers to a strange inversion in Greek terminology: nete, the highest note (in our current terminology) means low in Greek (or neat, short or small). Likewise, hypate, the lowest note means high (or long or tall) in Greek.
Similarly the Greeks constructed a lesser perfect scale or systema teleion elatton on the basis of the old heptad of two conjunct tetrachord. It is comprised of only an eleventh, from d’ to A. The top tetrachord did not exist, and the disjunct tetrachord diezugmenon was replaced by a conjunct tetrachord synemmenon:

Naming of notes of the Lesser Perfect Scale

Quoting Sachs The Rise of Music in the Ancient World, East and West, pg225: “Ptolemy, who lived in the second century AD, admitted seven keys, the centers of which ascended diatonically from e to d’. With an attempt to keep “a” as the common center.

There is a higher group of hyper scales, a lower group of hypo scales, and a middle group without epithets. At first sight, all of them are similar Dorian keys; but the modal structures are fundamentally different in the three groups:

1. 
   The middle scales, based on disjunct tetrachords, have the fifth on the top and are plagal.
   
2. 
   The hyper scales based on conjunct tetrachords, with an additional note above are likewise plagal.
   
3. 
   The hypo scales based on conjunct tetrachords, with an additional note below, have the fourth on top, or rather, should have the fourth on top and be authentic. “
   

Quote from Sachs The Rise of Music in the Ancient World, East and West, pg234: “Aristides Quintilianus expressly states that voices did not span more than two octaves. For this reason he adds that Dorian was the only key sung in its total range. … Lyre players had similar limitations. … The consequences are that music space, vague and shapeless in our music, became a palpable reality in Greece. Each key had its own center, to be sure; but also music space as a whole had its immovable center which, being the pitch tone was never neglected. As a result, every melody had two foci; every note or group of notes gravitated toward two different centers at once, toward the center of the individual key and toward the center of the immovable perfect system. The first bearing was called dynamis or mobile force, and the second, thesis or stationary force. A note changed its dynamis according to its key; its thesis was immovable. The note e’ for example, was in all melodies nete kata thesin or highest stationary note, whatever the key; but in Mixolydian, it was also mese katra dynamin or mobile center, and in Lydian trite kata dynamin or mobile third from above. The mobile and stationary functions coincided only in Dorian.

Look at a Greek melody from the dynamic center - b in Phrygian or d’ in Mixolydian - finds the modal structure that we call Dorian; descending, the sequence is to step through two whole tones and a concluding semitone to the (dynamic) hypate. Things look different from the stationary center. The player, adjusting his (Lyre to an) a and tuning the outer strings to e’ and e (the usual range of melodies), realized that each transposition of the (Dorian) scale altered the structure of his octave, since it shifted the semitone to places where previously whole tones had been. The central rectangle in the above diagram encases the resulting structure of all eight keys.
The Phrygian key, by one tone higher than the Dorian, sharped by two notes of the central octave, c’ and f; thus the two central tetrachords become e’ d’ c#’ and a g f# e: the semitone moved to the middle of the tetrachord and the original Dorian octave changed to the Phrygian species. In the same way, the Lydian key sharped d’ c’ g f and shifted the semitones to the upper ends of the tetrachord that formed the central octave. Dorian as a key (Dorios tonos) created a Dorian mode (Doristi harmonia) in the perfect system; a Dorian mode in the perfect system could only originate if the melody followed the Dorian key. And the same is true for Phrygian, Lydian, and the rest. Key and mode conditioned each other and rightly were given the same tribal names.”
Now quoting from Jargenson, page 24: “Tuning to Fifths and Fourths are the easiest intervals outside of the octave to tune because they exist next in order after the octave in the Harmonic series. The ease of tuning Fifths and Fourths along with their greater utility determined that Fifths and Fourths were to be used for creating most of the practical scales from the beginning of history to the present.
When the practice of tuning the pentatonic scale by means of a circle of fourths and fifths became established, the following changes took place:

1. 
   The fourths and fifths were made in tune enough to be used in vertical sonorities.
   
2. 
   Two of the wholetones (intervals between) became enlarged to become a new interval named a minor third.
   
3. 
   One of the fourths (intervals) became narrow enough to become a new interval named a major third.
   
4. 
   The new intervals listed above and their inversions resulted in minor thirds, major thirds, minor sixths, and major sixths all being added to the basic harmonic vocabulary of fourths and fifths; however, there were still no semitones, tritones or major sevenths to make singing difficult.
   
5. 
   Strong tonality was developed in melodic compositions because the scales were made permanently uneven.
   
6. 
   One major triad along with its lower relative minor triad was discovered.
   

Evidentially the phenomenon of Just Intervals was discovered and used for tuning the fourths and fifths for two reasons: first, it is much easier to tune an interval without beats than it is to temper an interval and guess how many beats it should contain; second, the fifths and thirds can not be completely Just at the same time, but fourths and fifths were chosen to be pure because the vertical sonorities of the thirds and sixths were not yet important in harmony. The Just fifth tuning practice must have been centuries old when Pythagoras began his investigation into music. All the major and minor sixths and also the major and minor thirds are each altered by one Syntonic comma from Just because all the fifths are Just. The latter sixths and thirds are known as Pythagorean sixths and thirds. Since these intervals are altered by the extreme of one Syntonic comma each, their beat speeds are too fast to be objectionable because they seem to disappear into a single tone. Pythagorean sixths and thirds have a tone color or sound uniquely their own. “

1. 
   Tune middle C to tuning folk.
   
2. 
   Tune F below middle C and F above middle C both pure to middle C.
   
3. 
   Tune G below middle C pure to middle C.
   
4. 
   Tune D above middle C to G below middle C pure.
   
5. 
   Test the intervals for the purpose of finding possible mistake which should be corrected. In the figure the numbers above the intervals indicate the beat speeds in beats per second in round numbers.
   
6. 
   Tune A below middle C pure to D above middle C.
   
7. 
   Test all intervals.
   

Basic Pentatonic 1/1 27/32 4/3 101/64 16/9 2/1

(descending) 0 294 498 792 996 1200 cent value from tonic.

(step size) 27/32 8/9 27/32 8/9 8/9

Celtic Pentatonic 1/1 9/8 4/3 3/2 16/9 2/1

0 203.9 498.0 701.9 996.0 1200 cent value of tones from tonic.

1. 
   Tune middle C to the Tuning Fork (unison zero beat).
   
2. 
   Tune F below middle C pure to middle C (a fifth, a pair of upper partials have a zero beat).
   
3. 
   Tune G below middle C pure to middle C (a fourth, a pair of upper partials have a zero beat).
   
4. 
   Tune D above middle C pure to G below middle C.
   
5. 
   Test
   

6. 
   Tune A below middle C pure to D above middle C.
   
7. 
   Test
   

8. 
   Tune E above middle C and E below middle C both pure to A below middle C. This creates the interval EF which is the first semitone. The inversion FE is the first major seventh.
   

9. 
   Test
   

1. 
   Tune B below middle C pure to E above middle C and also check to see that it is pure to the E below middle C. This creates the interval FB which is the first tritone.
   

11. Test

The Pythagorean Triads are proportional as follows:

minor root = 0:1:1

minor first inversion = 1:1:0

minor second inversion = 1:0:1

major root position = 0:3:2

major first inversion = 3:4:0

major second inversion = 4:0:3”

Use the eleven steps for tuning the Eratosthenes scale (from this paper, page 25) then

12. 
    Tune Bb below middle C pure to F below middle C.
    
13. 
    Test
    

12. 
    Tune Eb above middle C pure to Bb below middle C.
    
13. 
    Test
    

12. 
    Tune F# below middle C pure to B below middle C.
    
13. 
    Test
    

12. 
    Tune C# above middle C pure to F# below middle C.
    
13. 
    Test