## 2.1 Proportion and Harmony

The ancient Greek philosopher Pythagoras (?580 - ?500 B.C.) is generally credited with having discovered that musical intervals which are recognized as concordant are related by small integer ratios, an idea he may have acquired from Babylon. [ABRAHAM]  It is likely that he determined this result using a monochord, a single stringed instrument having a moveable bridge by means of which the string can be divided into two parts of variable proportion.

Strings with lengths in the ratio 2 : 1 produced the interval of an octave known to the ancient Greeks as diapason,   Those in the proportion 3 : 2 produced the interval of the fifth, known to the Greeks as diapente. Strings of equal tension with length in the proportion 4 : 3 produced the interval of a  fourth known to the Greeks as diatessaron.  The Greek word dia meant between, through or across.

All of these intervals are present between strings with relative lengths 2, 3 and 4.  Thus the most harmonious of intervals are contained in the number progression 1 : 2 : 3 : 4.  This reinforced the concept of spacial and musical harmony being related and the belief that the harmony of the entire universe was inherent in the mystical power of numbers.

Pythagoras himself left no written record of his work so it was via his pupil Philolaus that these observations have been passed on.  The first record of the use of a monochord to demonstrate this phenomena was by Euclid (c. 300 B.C.).

## 2.2 Tetrachord

The basic musical scale unit of ancient Greece was the tetrachord meaning literally four strings.  The first and fourth notes of the tetrachord were always tuned to the interval of a diatessaron (fourth) but the tuning of the other strings depended on the genus and mode of the music.  In the ancient Greek system notes of a scale were arranged in descending order.

In the diatonic genus the tuning of the other intervals comprised two tones and a semitone.  The chromatic genus comprised a minor third (three semitones) and two semitones.  The enharmonic mode comprised a major third (two tones) and two quarter tones. [EB]

Prior to Pythagoras there appears to be little evidence of a theoretical basis for the tuning of musical scales.  Pythagoras was involved with the science of harmonics which was separate from the practical art of music.  In the absence of a theoretical basis for the tuning of scales the actual tuning can only have been empirical and probably varied widely.

## 2.3 Diatonic Division of an Octave

The intervals of diatessaron or fourth (4 / 3) and diapente or fifth (3 / 2) when combined (multiplied) give the interval of a diapason or octave (2 / 1).  The interval between them (divide the larger by the smaller) is 9 / 8 which is the interval recognised as a whole tone.  Indeed, this division of the octave provides a theoretical basis for the tuning of a tone.  Furthermore, the division of the octave into two tetrachords separated by a tone is the basis of the diatonic scale, dia tonic, having a tone between the two tetrachords which are then referred to as diatonic tetrachords.

This principle is illustrated in a detail from the painting School of Athens by Raphael (1483 - 1520) on the wall of the Stanza della Segnatura in the Vatican.  In a diagrammatic representation of a lyra the strings are tuned in the relative proportions VI, VIII, VIIII, XII giving the intervals of an octave (diapason) between VI and XII, fifths (diapente) between VI, VIIII and VIII, XII and fourths (diatessaron) between VI, VIII and VIIII, XII.  The interval of a tone is present between VIII and VIIII. [WITTKOWER]

F 2.3.1 Detail from Raphael's School of Athens

 TONOS VI VIII VIIII XII DIATESSARON DIATESSARON DIAPENTE DIAPENTE DIAPASON

## 2.4 Greater Perfect System

Possibly the most common division of the tetrachord was into two tone steps plus what was left, a half a tone or semitone step, in descending order.  Using modern note names this might be represented as E - D - C - B (descending order).  If a second disjunct tetrachord separated from the first by a tone was added then a diatonic scale of one octave resulted:   E - D - C - B  A - G - F - E.

This was further extended above and below by the addition of two conjunct tetrachords which each shared one note with the existing tetrachords.  Theorists added a further tone to the bottom of the series to complete a two octave span.  This two octave disdiapason was called the Greater Perfect System. [EB]

 TETRACHORD TETRACHORD A G F E D C B A G F E D C B A TETRACHORD TETRACHORD

## 2.5 Ancient Greek Modes

The scale presented above has semitone intervals between C and B and also F and E.  All the rest of the intervals are whole tones.  A different scale pattern in terms of the position of tone and semitone intervals may thus be produced by traversing the scale from different starting points.

The ancient Greeks distinguished harmoniai which were scale patterns in the greater perfect system from tonoi which were the modes used in tuning and performance of a stringed instrument.  Tonos meant to stretch or tension.

The seven possible harmoniai are given in the following table. [EB]  Unspaced letters show semitone intervals.

 E D CB A G FE Dorian D CB A G FE D Phrygian CB A G FE D C Lydian B A G FE D CB Mixolydian A G FE D CB A Hypodorian G FE D CB A G Hypophrygian FE D CB A G F Hypolydian

## 2.6 Intervals in the Greater Perfect System

The number series 4 : 6 : 8 : 9 :12 :16 is often associated with the greater perfect system and the theory of proportion. [WITTKOWER]  The intervals between members of this series are fifth, fourth, tone, fourth and fourth.  These intervals give the tuning for the fixed intervals of the tetrachords of the greater perfect system.

 TETRACHORD TETRACHORD A G F E D C B A G F E D C B A 16 12 9 8 6 4 TETRACHORD TETRACHORD

The proportions of the greater perfect system are illustrated on the frontispiece of Theorica Musice published in 1492 by the musical theorist Franchino Gafurio (1451 - 1522).  The blacksmith's hammers, bells, glasses, weighted strings and pipes all bear the numbers 16, 12, 9, 8, 6 and 4.

F 2.6.1 Frontispiece to Theorica Musice, Franchino Gafurio, 1492

The upper left illustration depicts Jubal, the biblical father of music, and six blacksmiths with differing size hammers striking an anvil.  This relates to the story that the young Pythagoras was first moved to investigate musical intervals on hearing the notes produced by different size hammers at a blacksmith's shop.  The upper right illustration depicts Pythagoras testing the interval of an octave between bells of size 16 and 8 and between glasses filled in the proportion 16 and 8.  The lower left illustration shows Pythagoras testing intervals on a stringed instrument and the lower right illustration shows Pythagoras and his pupil Philolaus testing intervals by means of flutes.

Actually, the hammers, bells, glasses and flutes would produce an ascending series of intervals when arranged in the order 16, 12, 9, 8, 6, 4 whereas the weighted strings would produce a descending series.  The pitch of a string is proportional to the square root of its tension. [JOHNSTON]

## 2.7 Pythagorean Tuning

Pythagoras is credited with having devised a system of tuning based solely  upon the interval of a fifth, the next most consonant interval after unison and the octave.  One way in which he may have done so was by constructing a series of both descending and ascending fifths from the same starting point. [JOHNSTON]

From a starting point 1 / 1 the ascending fifths are produced by successively multiplying by 3 / 2 to give 3 / 2, 9 / 4, 27 / 16.  The descending fifths are produced by successive multiplication by 2 / 3 to give 2 / 3, 4 / 9, 16 / 27.  The process amounts to raising the numbers 2 and 3 to successively higher powers and the sequence is built upon powers of just these two numbers.  Pythagoras believed that it should not be necessary to go beyond the third power (cube) of a number as one cannot go beyond length, width and depth. [WITTKOWER]  This construction of a tuning system is consistent with such a view; schemes in which the fifths are either all ascending or all descending are not.

Although Pythagoras was primarily concerned with the theoretical science of harmonics this scheme is also consistent with practical tuning of an eight stringed instrument in the ancient Greek Phrygian mode.  This mode has symmetrical tone, semitone, tone intervals within each tetrachord.  The manner of tuning may be illustrated with modern note names.  First tune an octave between the two outer strings, D D.  The other end of each tetrachord is a fourth from one of these or a fifth from the other, D A G D (descending).  Two further notes lie a fourth from these middle notes, D C A G E D.  The final two notes lie a fifth from these D CB A G FE D.  Unspaced letters indicate semitone steps.

In the table below left the first column illustrates the descending and ascending series of fifths.  In order to keep all the pitches within the same (descending) octave it is necessary to transpose them up or down an octave or two if they would otherwise be outside the octave.  This is illustrated in the second column.  The table below right shows the fifths sorted into descending order of pitch with a corresponding modern note name.  The third of this scale is what we now call a minor third comprising one tone and one semitone.

 Original fifth Transposed fifth 27 / 8 27 / 32 9 / 4 9 / 16 3 / 2 3 / 4 1 / 1 1 / 1 2 / 3 2 / 3 4 / 9 8 / 9 8 / 27 16 / 27
 Sorted fifth Note name Interval to next 1 / 1 D 8 / 9 8 / 9 C 243 / 256 27 / 32 B 8 / 9 3 / 4 A 8 / 9 2 / 3 G 8 / 9 16 / 27 F 243 / 256 9 / 16 E 8 / 9 1 / 2 D 8 / 9

Thus was born the mathematical basis for musical tuning systems.

Analysis of all possible intervals within this tuning scale is given in the table below.  Colour is used to indicate similar intervals.  The scale is shown in descending order and the intervals are indicated as descending intervals in keeping with ancient Greek custom.

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 higher note named at the left of each row.  If you would like to swap the rows and columns click the top left cell.

 pivot D C B A G F E D D 1 / 1 9 / 16 16 / 27 2 / 3 3 / 4 27 / 32 8 / 9 1 / 2 C 8 / 9 1 / 1 128 / 243 16 / 27 2 / 3 3 / 4 64 / 81 8 / 9 B 27 / 32 243 / 256 1 / 1 9 / 16 81 / 128 729 / 1024 3 / 4 27 / 32 A 3 / 4 27 / 32 8 / 9 1 / 1 9 / 16 81 / 128 2 / 3 3 / 4 G 2 / 3 3 / 4 64 / 81 8 / 9 1 / 1 9 / 16 16 / 27 2 / 3 F 16 / 27 2 / 3 512 / 729 64 / 81 8 / 9 1 / 1 128 / 243 16 / 27 E 9 / 16 81 / 128 2 / 3 3 / 4 27 / 32 243 / 256 1 / 1 9 / 16 D 1 / 2 9 / 16 16 / 27 2 / 3 3 / 4 27 / 32 8 / 9 1 / 1

For a key to the colours using modern interval names, click the button.

All possible five note steps in the scale are true harmonic fifths (2 / 3), except that between F and the B below, which is a semitone smaller.  All possible four note steps are true harmonic fourths (3 / 4), except that between B and the F below. which is a semitone larger.  The size of these two intervals differs by a small amount known as the Pythagorean comma, a matter of great importance to which we shall return.

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

To Pythagoras, the discovery that musical intervals were related to arithmetic was of mystical significance and symptomatic of an underlying harmony throughout the cosmos.  Observing that there were the same number of heavenly bodies as there were notes in the musical scale he ascribed to the heavenly bodies orbits with musical proportions which he described as the Music of the Spheres.

Later Greek philosophers and writers including Plato (?427 - ?347 B.C.), Aristotle (384 - 322 B.C.) and Aristoxenus (c. 320 B.C.) were all influenced by ideas originating from the Pythagorean school.  Aristoxenus, however, believed that musical intervals should be judged by ear and not by mathematical ratio.

## 2.8 Ptolemy

An account of ancient Greek contributions to musical tuning would not be complete without mentioning the later Greek scientist Ptolemy (2nd c. A.D.).  He proposed an alternative musical tuning system which included the interval of the major third based on that between the 4th and 5th harmonics, 5 / 4.  This system of tuning was ignored during the entire Medieval period and only re-surfaced with the development of polyphonic harmony.

## 2.9 Pentatonic Scale

If the last descending and ascending fifths are omitted from the Pythagorean tuning sequence the resulting scale is a mode of the pentatonic scale.  This scale type occurs and predominates in many types of music including ancient Chinese, far eastern, and European folk music, for example Celtic music.

The form which arises most naturally from both ascending and descending fifths involves only two steps in each direction.  It is shown here in ascending order (reading down the table).  The thirds of the scale are minor thirds and, in the mode given, alternate with tone intervals and are symmetrically placed.

There are five modes of the pentatonic scale, one starting on each degree.  Using unspaced letters to indicate tone steps and spaced letters to indicate minor thirds the modes are DE GA C, E GA CD, GA CDE , A CDE G, CDE GA.

 Sorted fifth Step in sequence of fifths Note name Interval to next 1 / 1 0 D 9 / 8 9 / 8 2 E 32 / 27 4 / 3 -1 G 9 / 8 3 / 2 1 A 32 / 27 16 / 9 -2 C 9 / 8

In a sense the two members of the Pythagorean tuning which are absent are the two least satisfactory members of that series.  In Pythagorean tuning the awkward and inharmonious tritone (three tones) arises between the notes B and F (using modern note names).  As these members are not present in the Pentatonic scale there is no tritone.  All the fourths and fifths are perfect.

 pivot D E G A C D 1 / 1 16 / 9 3 / 2 4 / 3 9 / 8 E 9 / 8 1 / 1 27 / 16 3 / 2 81 / 64 G 4 / 3 32 / 27 1 / 1 16 / 9 3 / 2 A 3 / 2 4 / 3 9 / 8 1 / 1 27 / 16 C 16 / 9 128 / 81 4 / 3 32 / 27 1 / 1