4 TUNING INTO THE RENAISSANCE

4.1 The Emergence of Polyphony

Possibly the earliest form of music in which notes were sounded together other than in unison or at octaves was the practice of organum in which the melody was followed by an accompaniment, a practice dating from about the 9th and 10th centuries. In parallel organum the interval between the parts was a fixed fourth or fifth.  In free organum the interval varied but the consonant intervals fourth, fifth, octave and unison were used at key points in the melodic phrase.  In England the practice of using major thirds in organum developed in the late 12th century.

Development of the motet, a form of sacred song usually for three voices, helped carry the development of polyphony through the 13th to 15th centuries.  The birth of Ars Nova, the new art of the 14th century renaissance saw an increasing shift in musical emphasis from melody to counterpoint and harmony.  With this shift came an increasing use of the imperfect consonances of thirds and sixths.   But it was not until the 16th century that the efficacy of Pythagorean tuning was called into question.
 

4.2 Chromatic Scale

Extensions of the modal system to include Bb and F# have been mentioned above (section 3.3).  Further exploration of the modal system gradually introduced other semitone steps, particularly Eb, C# and G#.  Treating these new notes as modifications to existing notes enables a scale to still include all the note names but with one or more modified by an accidental, for example F G A Bb C D E F.  In musica ficta these accidentals (other than Bb) were understood by convention rather than notation as composers were reluctant to use notes outside of the system authorised by Guido of Arezzo. [GROUT]

These semitone modifications to the tonal scale led to a division of the octave into 12 semitone steps, a chromatic scale.  The theorist Jacobus of Liege remarked circa 1325 that 'keyboards now have all the whole-tones divided into their unequal semitones'. [SCHULTER]

Pythagorean tuning was still in use and the logical way to tune the new notes was as a fifth from existing notes, so Bb may be tuned as fifth down from the F above and F# as a fifth up from B below.

T 4.2.1 Pythagorean Diatonic Tuning with Added Bb and F#

Transposed fifth	Note name	Interval to next step
1 / 1	C	9 / 8
9 / 8	D	9 / 8
81 / 64	E	256 / 243
4 / 3	F	2187 / 2048
729 / 512	F#	256 / 243
3 / 2	G	9 / 8
27 / 16	A	256 / 243
16 / 9	Bb	2187 / 2048
243 / 128	B	256 / 243
2 / 1	C	9 / 8

This results in two slightly different sizes for the semitone step.  The steps from A to Bb or F# to G have the value 256 / 243, the same as the existing semitone steps between E and F or B and C.  This interval is called a Pythagorean diatonic semitone.  The semitone steps between Bb and B or F and F#, which do not represent steps of any diatonic scale and would rarely be sounded successively in music of the time, have the value 2187 / 2048.  This interval is called a Pythagorean chromatic semitone or apotome.  The two intervals differ by 531441 / 524288, a small interval known as the Pythagorean comma, about an eighth of a tone.  This is the amount by which a full cycle of 12 fifths differs from 7 octaves, a matter to which we shall return in the section on temperament.

The remaining semitones of a full chromatic scale may be added.  Eb may be derived by tuning down a fifth from Bb.  C# may be derived by tuning a fifth up from F# and transposing down an octave.  G# may be derived by transposing up a fifth from C#.

T 4.2.2 Early Pythagorean Chromatic Tuning (Eb x G#)

Transposed fifth	Note name	Interval to next step
1 / 1	C	2187 / 2048
2187 / 2048	C#	256 / 243
9 / 8	D	256/ 243
32 / 27	Eb (D#)	2187 / 2048
81 / 64	E	256 / 243
4 / 3	F	2187 / 2048
729 / 512	F#	256 / 243
3 / 2	G	2187 2048
6561 / 4096	G#	256 / 243
27 / 16	A	256 / 243
16 / 9	Bb	2187 / 2048
243 / 128	B	256 / 243
2 / 1	C	256 / 243

A problem with this tuning is that the fifth between G# and D# (Eb) is 262144 / 177147 which differs from a true fifth of 3 / 2 by 531441 / 524288, the Pythagorean comma.  This fifth is known as the wolf fifth because when played on Gothic organs it reminded listeners of howling wolves. The tuning is known as Eb x G# from the position of the rogue interval.  Although the interval does not howl much in the table above using pure tones the howling would have been much more pronounced in organs pipes with many overtones to their timbre.

From around 1400 Pythagorean tuning was probably modified to treat all the chromatic notes as flats. [SCHULTER]  There is still a wolf fifth, now between B and F# but the tuning is better suited to musical styles employing more thirds and sixths.  Arithmetically, these more harmonious intervals are reflected in the smaller numbers of the ratios of the revised tunings implying agreement of harmonics at an earlier point in the series. This tuning is known as F# x B. [SCHULTER]

T 4.2.4 Late Pythagorean Chromatic Tuning (F# x B)

Transposed fifth	Note name	Interval to next step
1 / 1	C	256 / 243
256 / 243	Db	2187 / 2048
9 / 8	D	256/ 243
32 / 27	Eb	2187 / 2048
81 / 64	E	256 / 243
4 / 3	F	256 / 243
1024 / 729	Gb (F#)	2187 / 2048
3 / 2	G	256 / 243
128 / 81	Ab	2187 / 2048
27 / 16	A	256 / 243
16 / 9	Bb	2187 / 2048
243 / 128	B	256 / 243
2 / 1	C	256 / 243

4.3 Just Intonation

Although almost all the fourth and fifths of Pythagorean tuning correspond exactly with harmonic ratios the thirds and sixths do not.  As mentioned above (section 3.2), the Pythagorean major third, 81 / 64, differs from the true harmonic major third, 5 / 4, by a small amount, 81 / 80, known as the syntonic comma or comma of Didymus.  The Pythagorean minor third, 32 / 27, differs from the true harmonic minor third, 6 / 5, by exactly the same amount.  The same holds true for major and minor sixths.

As early as around 1300 the English theorist Walter Odington pointed out that major and minor thirds have intervals close to 5 / 4 and 6 / 5 and that singers tend towards these intervals. [SCHULTER]  In 1482 the Spanish theorist Bartolomé Ramos de Pareja proposed that tuning should be modified to incorporate more harmonious thirds and sixths.  The scheme that he proposed was a form of mean tone temperament which is discussed in the next section (5.2).

The influential theorist Franchino Gaffurio (1451 - 1522) uncovered the tuning system devised by the ancient Greek Ptolemy which used natural harmonic thirds.  Gaffurio himself was a conservative and opposed to the introduction of a new tuning system.  The new scheme was championed primarily by Lodovico Fogliano in Musica Theorica of 1529 and Gioseffo Zarlino (1517 - 1590) in Institutioni Armoniche published in 1558. [GROUT]

The new tuning is usually referred to a just tuning or just intonation.  More generally the term just intonation is taken to mean any tuning system based on (small) natural numbers and thus includes Pythagorean tuning.  In the system championed by Zarlino, the major third becomes 5 / 4, the major sixth becomes 5 / 3 and the major seventh becomes 15 / 8 all based on the relation of the 5th harmonic to other harmonics.  Zarlino expressed the series as an harmonic progression of numbers from 180 to 90. [WITTKOWER]

T 4.3.1 Just Tuning - Zarlino

Interval from tonic	Note name	Interval to next step	Zarlino's harmonic series
1 / 1	C	9 / 8	180
9 / 8	D	10 / 9	160
5 / 4	E	16 / 15	144
4 / 3	F	9 / 8	135
3 / 2	G	10 / 9	120
5 / 3	A	9 / 8	108
15 / 8	B	16 / 15	96
2 / 1	C	9 / 8	90

This system involves two different sized tone steps, 9 / 8 as in previous tunings and the smaller 10 / 9 which is called a just minor tone.  The difference between them is the syntonic comma, 81 / 80.  The semitone has become 16 / 15 and is called a just diatonic semitone. It is very slightly smaller than a Pythagorean chromatic semitone.

This tuning system compromises one of the fourths and fifths, that between D and A.  All of the major thirds and minor sixths are consistent but one of the minor thirds, D to F, and the corresponding major sixth, F to D are also compromised.  As there are two different sized tone steps so also there are two different minor sevenths.

So long as one remains in the Ionian mode, equivalent to the modern major scale, or the Aeolian mode, equivalent to the modern minor scale, then this tuning works well.  Perhaps this is why Zarlino gave precedence to the Ionian mode and listed it first whereas Glareanus, using Pythagorean tuning, had seen it as the least satisfactory and listed it last.
 

4.4 Chromatic Just Tuning

Just tuning may readily be extended to the full chromatic scale by adding a minor third (6 / 5), minor sixth (8 / 5) and minor seventh (9 / 5) all derived from the 5th harmonic.  The minor second (16 / 15) is the interval which already exists between the just third and the fourth.  F# is calculated as a fifth above the just B.  All of these tunings use smaller numbered ratios than the corresponding Pythagorean tunings and therefore have coincident harmonics lower in the series.  The semitone steps, however, are of four different sizes.

T 4.4.1 Chromatic Just Tuning

Interval from tonic	Note name	Interval to next step
1 / 1	C	16 / 15
16 / 15	Db	135 / 128
9 / 8	D	16 / 15
6 / 5	Eb	25 / 24
5 / 4	E	16 / 15
4 / 3	F	135 / 128
45 / 32	F#	16 / 15
3 / 2	G	16 / 15
8 / 5	Ab	25 / 24
5 / 3	A	27 / 25
9 / 5	Bb	25 / 24
15 / 8	B	16 / 15
2 / 1	C	16 / 15

There are quite a number of wolves to be seen.  There are wolf fifths between Bb - F and D - A having the value 40 / 27 with a decimal value of about 1.4815, rather less than the ideal 3 / 2 or 1.5.  There is also a wolf fifth between F# - C# with the value 1024 / 675 or about 1.5170, larger than the ideal fifth.  There are corresponding wolf fourths at A - D, F - Bb and C# - F#.

More disturbingly for a system which seeks to perfect thirds and sixths there are wolf thirds between A - C#, B - D#, E - G# and F# to A#.  These wolves have the value 32 / 25 or 1.28 and differ significantly from the ideal 5 / 4 or 1.25.  There are corresponding wolf minor sixths.  Perception of the tuning of thirds and sixths is less critical than that of fifths and fourths but these are getting on for quarter of a tone out.

There are also wolf minor thirds (and major sixths) of two different sizes.  The minor thirds of 32 / 27 have a decimal value of about 1.1852 and those of 75 / 64 have a decimal value of about 1.1719.  These both differ significantly from the ideal 6 / 5 or 1.2.

Zarlino was undoubtedly aware of the limitations of just tuning and was amongst the earliest theorists to advocate a system of equal temperament.  In 1588 he reported that Abbot Girolamo Roselli praised such a symmetrical temperament as "spherical music". [SCHULTER]  The mathematical foundation for such a tuning system is based in the theory of logarithms which was not published until a quarter century after Zarlino's death.

Other historical alternatives to just intonation are discussed in the next section of this essay. 
 

4.5 Arithmetic and Harmonic Means

Zarlino observed that the arithmetic mean 3 between 2 and 4 divides an octave into a fifth and a fourth, 2 : 3 : 4.  (Or 6 : 9 : 12.)  Alternatively, the harmonic mean 8 between 6 and 12 divides the octave into a fourth and a fifth, 6 : 8 : 12.  Similarly, the arithmetic mean 5 between 4 and 6 divides a fifth into major and minor thirds, 4 : 5 : 6, whereas the harmonic mean 12 between 10 and 15 divides the fifth into minor and major thirds, 10 : 12 : 15.  Furthermore, the arithmetic mean of a major third, 4 : 5 or 8 : 10, divides it into major and minor tones, 8 : 9 : 10.  Zarlino saw this result as 'truly miraculous'. [WITTKOWER]

(To calculate an arithmetic mean add the figures and divide by the number of them, for example the arithmetic mean of  6 and 12 is given by ( 6 + 12) / 2 = 9.  To calculate an harmonic mean divide the sum of the reciprocals of the figures by the number of then and take the reciprocal of the result, for example, the harmonic mean of 6 and 12 is given by 2/( 1/6 + 1/12) = 2/(3/12) = 24/3 = 8.)
 

4.6 The Common Chord and Sestina

The first, third and fifth scale degrees when sounded together came to be recognised as the common chord.  This combination of notes was perhaps first found in the 14th century English faburden style of organum in which the middle part was a fourth below the melody and the bass a third below that.  Transposing the top note down an octave yields the common chord.

As observed by Zarlino, the intervals of a major third, 5 / 4, and a minor third 6 / 5, when combined (multiply) amount to a fifth, 3 / 2.  (The same holds true in Pythagorean tuning.)  The common chord may be either a minor third on top of a major third or a major third on top of a minor third. 

16th century theorists recognised a sestina as a chord comprising the intervals octave, fifth, fourth, major third, minor third, for example C C G C E G, having pitches related in the proportions 1 : 2 : 3 : 4 : 5 : 6. [EB62]

T 4.6.1 Sestina

1 : 2 : 3 : 4 : 5 : 6

4.7 Subharmonics

The most prominent component of a musical tone is the fundamental.  In the same way that a tone may contain higher harmonic frequencies which are multiples of the fundamental it may also contain lower frequencies which are sub-multiples of the fundamental.  These are known as subharmonics.  An alternative view is to regard subharmonics as the series of pitches of which a tone might itself be an harmonic even if they are not present in the tone.  In practice they may be present in small proportions.

This implies a series of descending pitches which are reciprocally related to the ascending harmonics, F, F/2, F/3, F/4, F/5, F/6, etc.  This series contains the same intervals as the series of ascending harmonics.  The interval between the first two members of the series is an octave.  The interval between the second and third subharmonics is a perfect fifth (3 / 2) and between the third and fourth the inversion of this, a perfect fourth, (4 / 3).  The interval between the fourth and fifth subharmonics, (5 / 4) is a major third and the interval between the fifth and sixth, (6 / 5) is a minor third.

The 4th, 5th and 6th harmonics have the relative frequency ratios 4 / 4 : 5 / 4 : 6 / 4 and provide the tuning for the just major scale.  The 4th, 5th and 6th subharmonics have the relative frequency ratios 1 / 4 : 1 / 5 : 1 / 6, or arranged in ascending order over a common denominator, 10 / 60 : 12 / 60 : 15 / 60.  The arithmetic and harmonic means calculated by Zarlino thus have a natural relation to the harmonic and subharmonic series. [MOORE]  One could argue that the major scale mode arises from the harmonic series and that the minor scale mode arises from the subharmonic series.