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3.1 The Greek Heritage

Knowledge of the ancient Greek musical system was available to medieval theorists mainly through the writings of Anicius Manlius Severinus Boethius (?480 - ?524 A.D.).  Boethius was possibly the last Roman writer who understood Greek.  Most of the musical theory he passed on was concerned with Pythogoreanism. [JOHNSTON]

In his book De Institutione Musica Boethius presents music as a numerical science in which consonance and the intervals to be permitted in melody and tuning are strictly determined by numerical ratios. [GROUT]  His work was a primary influence on the Medieval era.

Although Boethius was familiar with the Greeks musical letter notation he used Roman letters in De Institutione Musica.  To further confuse the issue he used A to represent the lowest note whereas in Greek notation alpha represented the highest note. [ABRAHAM]  This may possibly have led to a mistaken belief that Greek scales were of ascending rather than descending order.

3.2 Medieval Pythagorean Tuning

From the knowledge that Pythagoras devised a system of tuning based solely on perfect fifths, medieval theorists set about constructing a tuning on that basis.  Such a scheme is documented in 9th and 10th century organ building guides. [SCHULTER]

The scheme usually presented as the mathematical basis for Pythagorean tuning is as follows.  Starting from 1 / 1, the series of ascending fifths may be constructed arithmetically by successive multiplication of 3 / 2 as illustrated in the first column of the table below left.  To keep all the notes in the same octave, some of them need to be transposed down by a number of octaves as illustrated in the second and third columns.  The table below right shows the resultant series of fifths sorted by ascending order of pitch together with modern note names.

T 3.2.1 Series of Fifths
Original fifth Octaves to drop Transposed fifth
1 / 1 0 1 / 1
3 / 2 0 3 / 2
9 / 4 1 9 / 8
27 / 8 1 27 / 16
81 / 16 2 81 / 64
243 / 32 2 243 / 128
729 / 64 3 729 / 512
T 3.2.2 Sorted Series of Fifths
Sorted fifth Note name Interval to next
1 / 1 F 9 / 8
9 / 8 G 9 / 8
81 / 64 A 9 / 8
729 / 512 B 256 / 243
3 / 2 C 9 / 8
27 / 16 D 9 / 8
243 / 128 E 256 / 243
2 / 1 F 9 / 8

To arrive at such a tuning in practice one would alternately tune upward fifths and downward fourths so as to remain within the same octave.

This is probably not the method actually used by Pythagoras for at least two reasons.  First, Pythagoras believed that it should not be necessary to go beyond the third power of any number (see section 2.7 above) yet this method involves the fourth, fifth and sixth powers of 2 and 3.  Second, the method does not yield a strict diatonic scale in the sense that the interval in the middle is a semitone not a tone (dia tonic) and the first four note group is not a tetrachord as it does not span a perfect fourth.  Using a descending series of fifths leads to similar problems.  Furthermore, the system of both ascending and descending fifths mentioned in section 2.7 yields the ancient Greek Phrygian mode which is equivalent to the Medieval Dorian mode.  This is listed as Mode I in the Gregorian system, possibly because it arises first from the ancient Greek tuning system.

Another significant difference between a sequence of both descending and ascending fifths and a sequence of only ascending fifths is that the former method generates a third scale degree of a minor third whilst the latter generates a major third.  Both of these scale degrees differ from equivalent intervals in the series of harmonics, 6 / 5 for the minor third and 5 / 4 for the major third, by 81 / 80, roughly an eighth of a tone, an interval known as the syntonic comma or comma of Didymus.

The intervals of the scale are of two distinct sizes, 9 / 8 and 256 / 243.  The logarithm of the interval 9 / 8 (0.117783) is very roughly twice the logarithm of the interval 256 / 243 (0.052116) so the interval 9 / 8 sounds like a step twice the size of the step 256 / 243.  These are tone and semitone steps.

3.3 Modes

Because the diatonic scale has two semitone steps and five tone steps, different patterns of intervals arise by traversing it from different starting points.  These different starting points are referred to as modes of the scale.

Gregorian modes are identified by number and are named after Pope Gregory I (?540 - 604 A.D.).  The ancient Greek mode names were re-applied by some authors in the 10th century.  Possibly because they misinterpreted Boethius, the names of the modes are not consistent with the ancient Greek usage.  [GROUT]

Modal music of the Middle Ages and later has a final note for each mode with which the melody (usually) ends.  In authentic modes the final is the last note of the note range.  In plagal modes the final note is the fourth of the scale.  The Hypomixolydian mode, having no counterpart in the Greek system, differs from the Dorian mode only in its final.

In the following table the final is indicated in bold.  Unspaced letters show semitone intervals.

T 3.3.1 Medieval Modes
Authentic Plagal
I Dorian D EF G A BC D II Hypodorian A BC D EF G A
III Phrygian EF G A BC D E IV Hypophrygian BC D EF G A B
V Lydian F G A BC D EF VI Hypolydian C D EF G A BC
VII Mixolydian G A BC D EF G VIII Hypomixolydian D EF G A BC D

The only four and five note steps in the diatonic scale which are not true harmonic fourths and fifths are those between F - B and B - F.  This interval, equal to three whole tones and called a tritone, is regarded as most inharmonious.  In order to avoid this interval the note B was sometimes lowered by a semitone to a note called B flat (indicated in this article as Bb).  An alternative solution was to raise the F by a semitone to F sharp (F#).

The use of Bb, particularly in the Lydian and Dorian modes actually created two new modes.  The Lydian mode with Bb is equivalent to the modern major mode, as is the Mixolydian mode with F#.  Dorian mode with Bb is equivalent to the modern natural minor mode.  However, for a long time they were not regarded as new modes.

3.4 Guido of Arezzo

Guido of Arezzo (?995 - ?1050), a Benedictine monk for parts of his life, was a musician and very influential musical theorist.  He is credited with having greatly enhanced the emerging system of written musical notation and of having devised a musical scale structure known as the hexachord.

The hexachord was primarily conceived to facilitate the teaching of music and according to Guido it enabled a pupil to learn in five months what might previously have taken ten years.  The syllables ut, re, mi, fa, sol, la comprised a hexachord having the intervals tone, tone, semitone, tone, tone.  For example, it might correspond with the note sequence C D EF G A.  An important characteristic was that only a single semitone interval was included and this was in the middle between two three note groups.

Because of its simplified interval structure the hexachord was easy to sing.  The names of the scale degrees were derived from the first six lines of a Latin hymn to St. John the Baptist.  In a musical setting which may have been created by Guido for the purpose, each line started on the appropriate pitch for that note.  At a later time the seventh syllable si was added based on the initial letters of the last line.

Ut queant laxis    resonare fibris
Mira gestorum    famuli tuorum,
Solve polluti        labii reatum,
    Sancte Joannes.

That thy servants may freely sing forth the wonders of thy deeds, remove all stain of guilt from their unclean lips, O Saint John.

The notation is still in use in France and Italy for note names and persists in England in the song 'Do, a deer, a female deer...' used to teach children in the musical Sound of Music.  The tune may have changed but the utility of the hexachord structure as a teaching aid has stood the test of time.

The hexachord could be found at three places in the existing scale,  G A BC D E,  C D EF G A  or  F G ABb C D.  The natural form of B was called 'hard' and the flattened B was called 'soft' so the hexachord starting on G was called 'hard' (durum) and the hexachord starting on F was called 'soft' (molle).  These later became the names of the major (dur) and minor (moll) modes of today.  The hexachord starting on C was called 'natural'.

The lowest note on which a hexachord might start was bass G which was identified by the Greek letter gamma.  This hexachord was called gamma ut which is the origin of the word gamut, now taken to mean an entire range or compass.  A more narrow interpretation of the word gamut is the set of notes comprising the white notes plus Bb, the notes of musica recta.  In order to move beyond the six notes of the hexachord it was necessary to transfer to a new hexachord starting on the fourth or fifth degree of the old one.  This process, called mutation, involved treating a single note as belonging to both hexachords, a process analogous to modern key modulation.

3.5 Henricus Glareanus

Modal theory was revised by Heinrich Loris, also known as Henricus Glareanus, in his publication Dodecachordon published in 1547.  Glareanus added four new modes which were equivalent to the modified Dorian and Lydian modes with a Bb.  The new Aeolian and Ionian modes provided the same interval structure but did not require the use of Bb.  These were the precursors of the modern minor and major scale modes.

Glareanus also mentions the purely theoretical Lochrian and Hypolocrian modes.  Because these would require B as their final they were unusable on account of the tritone from B to F.

T 3.5.1 Extended Modes
Authentic Plagal
IX Aeolian A BC D EF G A X Hypoaeolian EF G A BC D E
( Locrian ) ( BC D EF G A B ) ( Hypolochrian ) ( F G A BC D EF )
XI Ionian C D EF G A BC XII Hypoionian G A BC D EF G

The older modal scales which were mainly applicable to melodic music are also called church modes or ecclesiastical modes.

3.6 Pythagorean Tuning of the Diatonic Major Scale

Suppose the system of tuning based on ascending fifths is transposed to start a fourth lower, starting with C rather than F.  This can be achieved by multiplying all the intervals by 2 / 3.  The notes F, G, A and B are also transposed up an octave (multiply by 2 / 1).

The fourth of the scale is a true fourth and there are two identical tetrachords separated by a tone, a strict diatonic scale.  The sequence of intervals in each tetrachord, tone, tone, semitone, is the same as the ancient Greek Lydian modes except that the scale is reversed, ascending not descending.  The pattern also matches the intervals of Guido of Arezzo's hexachord and the Ionian mode described by Henricus Glareanus.  It is the scale now known as the diatonic major scale.

T 3.6.1 Diatonic Major Scale - Pythagorean Tuning
Original fifth
(was called)
Transposed fifth
(now called)
Note name Interval to next step
3 / 2 1 / 1 C 9 / 8 tetrachord
27 / 16 9 / 8 D 9 / 8
243 / 128 81 / 64 E 256 / 243
1 / 1 4 / 3 F 9 / 8
9 / 8 3 / 2 G 9 / 8 tetrachord
81 / 64 27 / 16 A 9 / 8
729 / 512 243 / 128 B 256 / 243
2 / 1 C 9 / 8

In the following table all possible intervals of the diatonic major scale with Pythagorean tuning are analysed.  Colour is used to indicate similar intervals. 

If you have audio support in your browser you can left click to hear the notes in sequence and right click (hold on mobile, two fingers on MacBook) to hear the notes as a chord.

The first main column of the table repeats the tunings from the table above right.  Each main column of the table gives the interval from the note named at the head of the column (using modern note names) to the next lower note named at the left of each row.  If you would like to swap the rows and columns click the top left cell.

T 3.6.2 Analysis of Pythagorean Tuning - Modern Ionian (major) mode (ascending)
pivot C D E F G A B C
C 1 / 1 16 / 9 128 / 81 3 / 2 4 / 3 32 / 27 256 / 243 2 / 1
D 9 / 8 1 / 1 16 / 9 27 / 16 3 / 2 4 / 3 32 / 27 9 / 8
E 81 / 64 9 / 8 1 / 1 243 / 128 27 / 16 3 / 2 4 / 3 81 / 64
F 4 / 3 32 / 27 256 / 243 1 / 1 16 / 9 128 / 81 1024 / 729 4 / 3
G 3 / 2 4 / 3 32 / 27 9 / 8 1 / 1 16 / 9 128 / 81 3 / 2
A 27 / 16 3 / 2 4 / 3 81 / 64 9 / 8 1 / 1 16 / 9 27 / 16
B 243 / 128 27 / 16 3 / 2 729 / 512 81 / 64 9 / 8 1 / 1 243 / 128
C 2 / 1 16 / 9 128 / 81 3 / 2 4 / 3 32 / 27 256 / 243 1 / 1

Like Pythagorean tuning of the ancient Greek Phrygian mode all possible five note steps in the scale are true harmonic fifths (3 / 2), except that between B and the F, which is a semitone smaller.  All possible four note steps are true harmonic fourths (4 / 3), except that between F and the B which is a semitone larger.

The two note steps are all either whole tones (9 / 8) or semitones (256 / 243).  The three note steps are all either two tones (81 / 64) or tone plus semitone (32 / 27) .  All the six note steps are either four tones (128 / 81) or four tones plus a semitone (27 / 16) . All seven note steps are either five tones (16 / 9) or five tones plus a semitone (243 / 128).  The tuning of all these intervals is completely consistent throughout.