|The Tetrachord (Ancient). is one of the earliest known scales; it is ancient. It was known in Babylon and Sumeria. This is a scale for the Kithara (Lyre), rather than a Harp. The Harp was only a minor instrument in Sumeria and shunned by the mainland Greeks. Tetrachord means four notes in Greek (cord can mean either a string or note in Greek, in this case it is note because the various tetrachord scales can be played quite well on a three string Greek Lyre). The first note of the tetrachord is called the keynote or tonic, and the fourth string is tuned to a perfect forth above the tonic. Any two equal strings whose length are in the ratio of 4:3 will produce a perfect fourth (also 498.05 cents). The intervals of the two notes between the first note and the fourth note are movable and are moved to allow the stringed instrument to play the various Greek Modes. There are three tetrachord modes: 1) the Enharmonic, 2) the Chromatic and 3) the Diatonic (ancient Greek definitions of these words are different from modern usage). The Enharmonic Mode is formed by two tones and two diesis. The Chromatic Mode consists of two semitones and a tone and a half. The Diatonic Mode is formed by two consecutive tones and a semitone. The movable notes will have to fit in this interval of 498 cents by the following rules:
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
however Ptolemy disagreed, he chose:
Enharmonic 386 74 38
Chroma malakon 316 120 62
Chroma syntonon 267 151 80
Diatonon malikon 233 182 83
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.
The of Aristonuxes of Tarentum (330 BC) is an 18-note scale that spans two octaves. This two-octave span is supposed to be the range of male voice. Rose, in his version of Vitruvius, stated that the lowest note is the A an octave plus two notes below middle C and the highest note was the A above middle C, but he gave no explanation of why he choose it to be this way. The is comprised of five Tetrachords and a few additional notes. Now refer to Figure 2, and I will try to walk through how to form this scale (This scale almost appears to be two scales push together so that the beginning and end are on the same scale note, A. This results in a slight jumble of notes about in the middle of the scale). The scale is comprised of 8 fixed notes and 10 movable notes. A technique that would be consistence with the technology at this time would be to use a Monochord. I have built one for this purpose.
Now we will start with the fixed pitch strings and build the Harp scale. Take the pitch of the tonic from some authoritarian source. Tune the pitch of the Monochord to unison with this source. Now tune the pitch of the Proslambanomenus string to unison with the Monochord. Note the Monochord string’s vibrating length. Reduce this vibrating length by 8/9 and use this new pitch to tune the Hypaton Hypate Harp String to unison. Note this vibrating length on the Monochord and reduce it by 3:4 to get the pitch for the Meson Hypate. Now tune the Meson Hypate Harp String to unison with the Monochord. If at any time the Monochord runs out of range for the string it is currently using, a string change will be required. With a string change, when one is going up the scale, just remove the current string and insert the next lighter string, then go back to one of the previously tuned strings on the Harp and pick up the correct pitch and continue. The Synhemmenon Mese string will be 3:4 the vibrating length of the string tuned to the Meson Hypate, so reduce the vibrating length of the string in the Monochord and then tune the Synhemmenon Mese string to unison with the Monochord. Again note the vibrating length on the Monochord and reduce it by 3:4 to get the pitch step size to get to the Synhemmenon Nete. Now tune the Synhemmenon Nete Harp string to unison with the Monochord.
Now it appears that a change in technique is required. Retune the Monochord back to unison with the Synhemmenon Mese. Note the vibrating length on the Monochord for the Synhemmenon Mese and then reduce this vibrating length to 1:2 to get the octave pitch. Now tune the Hyperbolaeon Nete Harp (highest) String to unison with the Monochord. Now note the current Monochord vibrating string length and increase the length by 4:3 to go down a perfect forth to the Diezbugmenon Nete. Tune the Diezbugmenon Nete Harp String to unison with the Monochord. Again, note the vibrating length of the Monochord string length
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.
Figure 2 The of Aristonuxes
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).
Confusion. To Quote Sachs, from The Rise of Music in the Ancient World, East and West, pg222: “With all the Greek modes: Aeolian, Boeotian, Dorian, Iastian, Ionian, Locrain, Lydian, Phryaian, with or without epithets; aneimene, chalara, hyper, hypo, mixo, syntono, in an ever changing and often contradictory terminology, impelled the methodical Greek philosophers to organize this chaotic multiplicity of modes into one consistent system and eliminate those modes that were not acceptable. As early as 400 BC the kithara had progressed to eleven strings. Which in some pentatonic sequence covered exactly two octaves and within had the possibility of representing the new system in its totality.”
The Greater Perfect Scale. The first section I extracted from Peter Fraser’s web site and added the step sizes. I then followed this with a quotation from Sachs, who had a better copy of Euclid to work from. Both are interesting.
Figure 3. The Greater Perfect Scale.
The Greater Perfect Scale (300 BC). Quoting Sachs, from The Rise of Music in the Ancient World, East and West, pg222-223: “The Greek Philosopher and Mathematician, Euclid first described the Perfect System or Systema teleion.
The Perfect Scale is a 15 note scale that covers just two octaves. However the perfect system was more than just a double octave; it was perfect as a unique attempt to organize the musical space from one center, (the note) a. The center stands in its original octave of Dorian structure, e’-e, which, by adding half an octave above and half an octave below, is extended to two octaves a’-A. This new unit could shift both up and down by half an octave either way and thus cover three octaves.
The notes added above and below the inner octave were named after the three extreme notes at either end of it: nete, paranete, trite, above; lichanos, parhypate, hypate below. Only the lowest note was given a name of its own; the added note, proslambanomenos. The organization of the two octaves was rather strange. Subdivision were neither octaves nor pentachords, but tetrachords throughout. This implied two conjuncts at either end of the inner octave and disjunction or diazeuxis in the middle. Reading downward, the arrangement results in the tetrachord hyperbolaion, ‘of the exceeding notes’, conjunct with the tetrachord diezeugmenon, which as the name said, was disjunct from the tetrachord meson, ‘of the middle’ notes; this in was conjunct with the tetrachord hypaton, ‘of the low’ (literally ‘high’) notes. The proslambanomenos, ‘the left over’ note.
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
I am not satisfied with Sachs use of letter to indicate the notes in his representation of the Greater and Lesser Perfect Scale. Also these letters give the wrong impression as to the intervals between the notes. The Perfect Scale is a diatonic scale, the intervals between notes are: (Diatonic=unequal step sizes) Tone Tone Tone Semitone Tone Tone Semitone, then next octave or Tone Tone Semitone Tone Tone Tone Semitone. However, the Greeks did not use instrumental music and only need an octave and a few notes to achieve all the range that they wanted. The purpose of Greek music was to back a male singer (they did not use female singers).
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:
The middle scales, based on disjunct tetrachords, have the fifth on the top and are plagal.
The hyper scales based on conjunct tetrachords, with an additional note above are likewise plagal.
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:
The fourths and fifths were made in tune enough to be used in vertical sonorities.
Two of the wholetones (intervals between) became enlarged to become a new interval named a minor third.
One of the fourths (intervals) became narrow enough to become a new interval named a major third.
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.
Strong tonality was developed in melodic compositions because the scales were made permanently uneven.
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. “
Basic Pentatonic Scale (Ancient) (quoting from Jorgenson, now page 25)”In this tuning there are two triads: namely the F Major and D Minor. There are two major sixths and one minor third which are wide by one Syntonic comma. There are two minor thirds which are narrow by a Syntonic comma. These intervals which are altered by one Syntonic comma are known as Pythagorean intervals. … This gives Pythagorean triads a certain rhythmic harmoniousness. …
This descending scale has intervals between notes in the ratio: F 27:32 D 8:9 C 27:32 A 8 :9 G 8:9 F. The cent values in round numbers are 294-204-294-204-204. The Basic Pentatonic scale contains pure major tones of the 8:9 ratio, pure fourths of the 3:4 ratio, pure fifths of the 2:3 ratio, minor sevenths of the 9:16 ratio and pure octave of the 1:2 ratio. There are no semitones, tritones or major sevenths. All the triads and sixths are altered by the ratio 80:81 Syntonic comma. The minor third ratio is 27:32; the major third ratio is 64:81; the minor sixth ratio is 81:128 and the major sixth ratio is 16:27.
The line of fifths in the Pentatonic Scale is F C G D A. The bearing octave is F below middle C to F above middle C. All fifths and fourths are pure. Major sixths and minor triads are wide, minor thirds are narrow. Tuning steps:
Tune middle C to tuning folk.
Tune F below middle C and F above middle C both pure to middle C.
Tune G below middle C pure to middle C.
Tune D above middle C to G below middle C pure.
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.
Tune A below middle C pure to D above middle C.
Test all intervals.
F G A C D F
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.
It has been stated that the Celts organized their music in a pentatonic scale. Since the Celts preferred to develop their memory and refrained from writing anything down, one has to rely on their enemies for any historical record of their customs.
It is highly likely that the gap scale Irish Harp music examples that are reported by Bunting and other collectors are prehistoric Pentatonic Scales transposed to historic Diatonic Scale or else works where the composer tried to make his music sound this way.
Enharmonic Scale of Archytas (380 BC) Still quoting Jorgenson, page 31: “This is a descending scale with the interval ratio of E 4:5 D 35:36 C 27:28 B 8:9 A 4:5 G 35:36 F 27:28 E. The cent value in round numbers for the interval between pitches is 386 - 49 - 63 - 204 - 386 - 49 - 63.”
Eratosthenes Diatonic (230 BC) is also the Diatonic Ditoniaion published by Claudius Ptolemy in 140 AD. Quoting Jorgenson, page 31:
“This scale is the basis of Western Music. In this tuning, the three basic primary triads are developed. These are the G Major, C Major and F Major triads. Also, the three lower relative minor triads E Minor, A Minor and D Minor are developed. The formation of these triads creates the basic C Major tonality.
This descending scale has intervals between notes as: E 8:9, D 8:9, C 243:256, B 8:9, A 8:9, G 8:9, F 243:256, E. The cent value in round numbers for the step size between pitches are 204-204-90-204-204-204-90. The above scale contains 243:256 ratio semitones, 512:729 ratio tritones and 128:243 major sevenths in addition to all the intervals contained in the Basic Pentatonic Scale.
NOTE in Jorgenson’s method, pure means that an upper partial of one pitch is in unison with an upper partial of the other note. Even though Jorgenson did not like electronic tuners, the electronic tuner method of tuning is faster than and as far as I can see better. In the electronic tuner method, just tune to the closest 12 Tone Equal Temperament note, then tune to the correct offset in cent. In the music below, the number above a pair of notes is the beat frequency of a pair of their upper partials.
Tuning the Eratosthenes Diatonic Scale: In the Eratosthenes diatonic scale there are six major and minor triads to the octave and all of them contain proportional beat speeds. All the triads are Pythagorean triads. The line of fifths are F C G D A E B. The bearing octave is E below middle C. All fifths and fourths are pure. Major sixths and minor thirds are narrow.
Tune middle C to the Tuning Fork (unison zero beat).
Tune F below middle C pure to middle C (a fifth, a pair of upper partials have a zero beat).
Tune G below middle C pure to middle C (a fourth, a pair of upper partials have a zero beat).
Tune D above middle C pure to G below middle C.
Tune A below middle C pure to D above middle C.
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.
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.
The Eratosthenes Diatonic Scale contains the following “Pythagorean” Triads:
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”
Chromatic Scale of Didymium (30 BC) Still quoting Jorgensen, page 31: “This is a descending scale with the interval ratio E 5:6 D 24:25 C 15:16 B 8:9 A 5:6 G 24:25 F 15:16 E. The cent value in round numbers for the interval between pitches is 316 - 70 - 112 - 204 - 316 - 70 - 112. Pure minor tones, minor thirds, major thirds, minor sixths and major sixths. There are pure ratio 24:25 semitones so small that they could be classified as large third tones.
Diatonic Syntonon of Ptolemy (140 AD) Also known as the Diatonic Scale of Didymium (30BC). Still quoting Jorgenson, page 31: “This is a descending scale with the interval ratio of E 9:10 D 8:9 C 15:16 B 8:9 A 9:10 G 8:9 F 15:16 E. The cent value in round numbers for the interval between pitches is 182 - 204 - 112 - 204 - 182 - 204 - 112.
In this scale all the ratio 64:81 Pythagorean major thirds and Pythagorean minor sixths were changed into pure 4:5 major thirds and pure 5:8 minor sixths. Three of the minor thirds and three of the major sixths were also made pure. The interval of a fourth between A D was altered by one Syntonic comma to have a new ratio of 20:27 which is an out of tune wolf interval. The fourth A D was ruined out of necessary in order to gain all pure thirds and sixths. There are both major and minor whole tones in this scale. The three primary triads: G, C, and F are in Just intonation. Ptolemy’s Diatonic Syntonon is the ancestor of the Common Model Just Tuning as outlines in Marpug’s Monochord No.1. The Ptolemy diatonic scale is more harmonious than the Eratosthenes diatonic scale, but lacks the utility of the Eratosthenes. This conflict between harmoniousness and utility lead to the difference between the Meantone temperament and the Well temperament.”
Diatonic Hemiolon of Claudius Ptolemy (140 AD) Still quoting Jorgenson, page 31: “This is a descending scale with the interval ratios of E 9:10 D 10:11 C 11:12 B 8:9 A 9:10 G 10:11 F 11:12 E. The cent value in round numbers for the interval between pitches is 182 - 165 - 151 - 204 - 182 - 165 - 151.”
Music Theory of Anicius Manlius Boethius (520 AD) Boethius read a reference that stated that Pythagoras had perfected a diatonic scale based on fifths (not possibly true) and Boethius tried to find this scale. Boethius found Eratosthenes Diatonic scale instead and incorrectly named it the Pythagorean Diatonic; so from this time on, to avoid additional confusion Eratosthenes scale is now called the Medieval Pythagorean Diatonic scale. Boethius also got caught in the confusion of whether Greek scales ascend or descend and he got it wrong. Next he got the names of the Greek Modes incorrect. But Borthius was not the only Medieval Music Theorist that got the Greek Modes names incorrect. The modes known as the Church Modes are the results of this mistake.
Medieval Pythagorean Scale (of Boethius) This scale is called in error the Medieval Pythagorean Diatonic Scale. This is wrong for two reasons: one, Boethius misnamed the Diatonic Scale of Eratosthenes and two this scale is more than a diatonic scale it has four chromatic notes added: Bb, Eb, F# and C#. But it is a very useful scale and is a good scale in which to compose.
Quoting Jorgenson, page 45: “This is the Eratosthenes diatonic scale with four chromatic tones added. There are still no wolf interval. In this tuning there are eighteen major and minor triads to the octave, and all of them contain proportional beat speeds. The C# minor, C# major, E major, F minor G# minor and Ab major triads still do not exist in this scale. The line of Just fifths is: Eb, Bb, F, C, G, D, A, E, B, F#, C#.
In this scale, there is a new kind of interval created which is known as a schisma interval. These intervals are each altered from Just intonation by the extremely small amount of one schisma, which in round numbers is only 2 cent in size. With Schisma intervals it is almost impossible to tell that they are different from Just interval. They are altered intervals, but they are never directly tempered by ear. Schisma intervals exist because they are the end result of lines or groups of other pure intervals. The diminished fourths: C# to F, F# to Bb and B to Eb all sound quasi pure because they are actually narrow major thirds. The augmented seconds: Bb to C#, and Eb to F# both sound quasi pure because they are actually wide schisma minor thirds.”
Tuning Steps for Medieval Chromatic Pythagorean Scale: (quote from Jorgenson, page 48)
Use the eleven steps for tuning the Eratosthenes scale (from this paper, page 25) then
Tune Bb below middle C pure to F below middle C.
Tune Eb above middle C pure to Bb below middle C.
Tune F# below middle C pure to B below middle C.
Tune C# above middle C pure to F# below middle C.
Medieval Modes and Music Theory of Guido of Arezzo (1050 AD) Developed four line staff and the Ut Re Mi Scale. Established the Church Modes as Dorian, Phrygian, Lydian, Mixolydian and maybe the Locrian (some authorities deny that the Locrian is a mode). No matter, the scales of Guido were different from the correct Greek modes. The correct knowledge was still readily available in Constantine (Eastern Roman Empire, official language was Greek), so it is a wonder that no one bothered to go obtain it? Christian Monks and people in trade were making the trip.