Chapter 2: sintono versus diatono

From theory to practice.

It is worth saying again: We shall fail if we search through Zarlino's work for the key to just intonation in practice. His concerns were always with theoretical aspects ²⁵. When calculating intervals following the principles laid down by Ptolemy, he produced something entirely abstract and speculative. And while he set out principles of intonation for singers, he gave no precise guidance as to how they should or could implement these principles. We can certainly wish that he had followed on with practical advice on the problems of intonation. Unfortunately, at the time of the Renaissance, there was a great gulf between the high-flown speculative theories of people like Zarlino and, on the other hand, the elementary practical instruction manuals which were beginning to develop, but generally only dealt with rudiments for beginners.

In this great gulf separating written theory and practice there must be buried a major part of what makes up the craft of the best interpreters of Renaissance music. We need to delve into it to find the answers to our questions. This expedition is dangerous, and the results uncertain. Nevertheless we believe it to be worth the trouble; we have not yet drawn from Zarlino all that he might be able to tell us. In particular the superposition on his theory of modern notions of scale and tonality, and the frequent recourse to the image of the piano keyboard, have had a very confusing effect on the understanding of his theory. We prefer to forget these current ideas and to follow his steps of reasoning, pressing his own ideas further. This may lead us even further than he himself was able to do, but it clearly could have been done by those of his readers in the sixteenth century who were looking for guidance in the practice of just intonation. In this way, we can try to arrive at a usable model; perhaps subject to conjecture, but we hope plausible.

Stage one: The tetrachord

With its mythological origins in the lyre of Mercury and of Orpheus ²⁶, the tetrachord is one of the foundations of ancient Greek musical theory. As its name indicates, it has four strings ²⁷ of which the two outer form an interval of a fourth ²⁸, or a ratio of 4:3. The tuning of the two central strings depends on the genus, diatonic, chromatic or enharmonic, as well as the species, or broadly speaking the chosen tuning principle. Schematically, and from lower pitch to higher, a diatonic tetrachord is formed by a semitone followed by two whole tones; a chromatic tetrachord by two semitones (which may be unequal) followed by a larger interval completing the fourth; and an enharmonic tetrachord by two small intervals which form, taken together, a semitone divided more or less equally. The fourth is then completed by a major third.

In the spirit of the Renaissance, Zarlino did not limit himself to Boethius and to Pythagorean tetrachords. He had recourse to other ancient sources, in particular to Ptolemy ²⁹, and this allowed him to compare, for each of these three genera, several species characterised by specific ratios or proportions. Above all, the diatonic genus is of interest to us, and the comparison of two particular species applied to the tetrachord. Zarlino derived these species from the work of Ptolemy, and called them respectively the diatonico diatono (which we will abbreviate to diatono) and the diatonico sintono (or sintono for short).

This diatono is none other than the Pythagorean diatonic ³⁰: it leads to a tetrachord in which a semitone of 256:243 is followed by two whole tones, each of 9:8.

Figure 2.1: The diatonic tetrachord; with diatono tuning

These ratios can be found in the numbers calculated by Zarlino, from below and moving upwards: 8192, 7776, 6912, 6144. As he worked in terms of the length of a string, they will of course be the inverse of those obtained when working in terms of frequency, which are shown in our parallel table. In this the scale degrees of the tetrachord are numbered upwards in the first column. In the next column are the ratios of the frequencies in relation to the lowest degree; and finally the value of this interval in cents ³¹. The main characteristics of this tetrachord are therefore a "small" semitone (s) of 90 cents (the equal temperament semitone is by definition 100 cents) and two "large" tones of 204 cents (T).

In the case of the diatonic tetrachord sintono, which is "the one the moderns use in their harmonies" ³² , Zarlino gives, working upwards, the numbers 48, 45, 40 and 36. Between these we can find a "large" semitone of 16:15 (S), or 112 cents; a major tone (T) of 9:8 and a minor tone (t) of 10:9, or 182 cents.

Figure 2.2: The diatonic tetrachord with sintono tuning

The sintono is distinctive in that the two intermediate degrees of the tetrachord appear a comma higher than those of the diatono, with the result that the two thirds, minor and major, appearing between the degrees 1-3 and 2-4 respectively are pure thirds. Since the four degrees of the diatono tetrachord are taken, by definition, from the same series of pure fifths, the two intermediate degrees of the sintono must form part of a series which is higher by a comma. Already at this stage we can see that the sintono allows us, moving by degrees, to choose between two distinct series of fifths so as to achieve pure thirds. One single tetrachord does not take us far, but at least it provides an embryonic basis for the diatonic framework we need.

Stage two: The monochord and the sistema massimo

The monochord is essentially a single long string divided by a movable bridge, used from ancient times for experiments with musical intervals. Using this concept, we can calculate the proportions of the device called by Zarlino the sistema massimo ³³ , a theoretical instrument of Classical Antiquity of sixteen strings organised into five tetrachords fixed in relation to each other (Figure 2.3 and Figure 2.4).

At its base is a single separate string (the proslambanomenos). Above this, starting a tone higher, are two superimposed tetrachords. These are joined, in that the uppermost string of the lower tetrachord, the hypaton, is identical to the lowermost string of the second, or middle, tetrachord, the meson. The uppermost string of the middle tetrachord is the central string, or mese. Following on, a tone higher, is the disjoint tetrachord (diezeugmenon), so called because one tone separates it from the middle tetrachord. Finally, the high-pitch tetrachord (hyperboleon) is itself contiguous with the disjoint tetrachord.

In addition, there is, starting with the mese, the conjoint tetrachord (synemennon), where the second degree provides a new semitone in between the mese and the disjoint tetrachord. This allows avoidance of the tritone which could appear between the second degree of the middle tetrachord (parhypate meson) and the first degree of the disjoint tetrachord (paramese). It is important to note here that, in the diatono, this tetrachord only provides one additional string, that of the semitone. Its first, third, and fourth degrees coincide with strings already present. It is not simply a question here of adding a single string, in the same way that one might add a black note between two white notes on a keyboard. On the contrary, this semitone fits precisely into a tetrachord which is itself an essential part of the complete system. We have here a first example of the way in which a modular structure, in this case a system of tetrachords, can serve as a diatonic framework.

In the sistema massimo, the extreme strings of each tetrachord are fixed. Changes are allowed only in the intermediate strings, depending on the definition of the tetrachord. Consequently, if we limit ourselves here to the diatonic genus, the species defines the precise tuning of the intermediate strings of each tetrachord. In this way Zarlino could provide an accurate calculation, for each species, of the tuning for the corresponding sistema.

He began by doing this for the diatono ³⁴ (Figure 2.3), i.e. for the traditional Pythagorean diatonic. In order to be able to give integral values to all the strings, he gave the value 9216 ³⁵ to the lowest note (proslambanomenos). Progression by degrees was solely by semitones of 256:243 (s) and by tones of 9:8 (T). Zarlino did not hesitate to criticise this system, in spite of it having been generally accepted for several centuries. He found the major and minor thirds of 81:64 and 32:27 profoundly unsatisfactory. For him a real consonance could only correspond to a ratio which was a simple multiple (double, triple, quadruple etc.) or a ratio of the form (N+1):N where N is a small integer (3:2, 4:3, 5:4, 6:5, etc.). Since the Pythagorean thirds do not satisfy these conditions, they are inevitably dissonant, which appeared to be in contradiction to the current musical practice, where thirds were in fact consonances ³⁶.

In this way, Zarlino established his preference for a system based on the sintono, characterised by the fact that it contained only relations of the genus superparticularis, i.e. relations of the form (N+1):N - 16:15 for the semitone (S), 10:9 and 9:8 for the minor tone (t) and the major tone (T) respectively, 6:5 for the minor third, 5:4 for the major third, and 4:3 for the fourth. Starting from a proslambanomenos of 864, he could give a whole number to all the strings ³⁷. Globally, this system barely differs from that based on the diatono. One major difference remains, though. In the classic diatono the third degree of the disjoint tetrachord (paranete diezeugmenon) becomes identical with the fourth degree of the conjoint tetrachord (nete synemennon). In the sintono case, it is lower by a comma, as shown by the numbers 320 and 324 calculated by Zarlino (Figure 2.5). In our diagram (Figure 2.6), we can see that the difference between a major tone and a minor tone is responsible for this effect.

We now find ourselves in the presence of a phenomenon analogous to that of the "split note" in the Helmholtz scale. However, although this split note is a destructive phenomenon in the case of of a major scale in the eyes of modern theory, here this does not apply. The modules of the system, i.e. the tetrachords, are not affected in the sense that their ratios one to another are maintained, both for the extreme strings and stepwise from degree to degree. Further, this system, which seems to contain redundancy in the Pythagorean version because certain strings have different names while having identical lengths (and therefore frequencies), has in fact a separate name for each of those strings separated by a comma (paranete diezeugmenon and nete synemennon). Each of them maintains perfectly well-defined relations with the other strings of the tetrachord concerned, and the system as a whole can function without problems.

Stage three: The hexachord and the Guidonian hand

In its syntonic version (i.e. based on the sintono of Ptolemy-Zarlino), the complex system of sixteen strings of the Greeks could have been adapted to the ideals of just intonation if it had been found to be of some practical utility for the music of the Renaissance. This was evidently not the case. From being a practical system in Antiquity, it had become, in the Middle Ages and in the Renaissance, primarily a theoretical framework and support for much speculation about intervals. It had lost contact with practical music. In fact it had been replaced by a more extended system of 22 strings which presented a similar modular structure. The module here is not the tetrachord but the hexachord, traditionally attributed to Guido d'Arezzo.

As Zarlino explained ³⁸, the hexachord has at its centre the semitone which was found at the bottom of the tetrachord. This indicates that it consists of an extension downwards of the tetrachord by two degrees, each of one tone.

	Scale degrees	Ratios	Cents
{	La	27:16	905.87
{	T
{	Sol	3:2	701.96
The tetrachord	T
{	Fa	4:3	498.04
{	s
{	Mi	81:64	407.82
	T
	Re	9:8	203.91
	T
	Ut	1	0

Figure 2.7: The diatono hexachord

From low pitch to high, the degrees of the hexachord are traditionally given the ritualistic syllables ut, re, mi, fa, sol and la, also credited to Guido d'Arezzo (and also of course to the hymn Ut queant laxis, Resonare fibris, etc.), which can conveniently be used in place of the Greek names.

Zarlino went on to calculate the ratios of the Pythagorean hexachord (Figure 2.8). His six degrees are given length numbers, for example, 10368, 9216, 8192, 7776, 6912 and 6144. Between these we can find four whole tones of 9:8 surrounding a semitone of 256:243. The complete system of 22 strings, often called the Guidonian Hand in tribute to Guido d'Arezzo ³⁹, carries seven hexachords. Since the hexachord is an extension of the tetrachord, the Guidonian Hand is itself an extension of the Greek system, and includes it. In our diagram (Figure 2.9), the strings and the tetrachords of the Greek system appear in grey.

We see, on the left, the keys (in fact the letters A to G), which provide a fixed framework for the whole set of hexachords ⁴⁰. These "keys" predate the present usage of the letters in anglophone countries to designate actual notes. They appear in capitals for the lower octave (A to G), in lower case for the central octave (a to g), and in doubled lower case for the highest octave (aa to ee, the "Hand" being limited upwards as well as downwards).

The five tetrachords of the Greek system can be found unaltered in the Guidonian Hand, where they keep the same ratios as in the original system. They have simply been converted into hexachords by adding two degrees downwards. This extension of the lowest tetrachord has required the addition of an additional string, one tone below the proslambanomenos (A). This is the gamma, which has given its name to the modern French gamme and English gamut.

The Greek system has been further extended upwards by adding two hexachords, starting on "f" and on "g". The two hexachords starting on "C" and "c" are named natural; the three based on a "G" are called hard and the two based on an "F" are called soft. In the two uppermost octaves, the key B exists in two variants. One, called hard or sharp, functions as "mi" in a hard hexachord, and corresponds to a B natural. The other, called soft or flat, functions as "fa" in a soft hexachord and corresponds to a B flat ⁴¹. Although it carries strings corresponding to these "altered notes," the Guidonian Hand remains a strictly diatonic structure.

Arising from the extension of each of the tetrachords into hexachords, the latter provide many more coincident pitches than the former. The common strings, which were occasional in the Greek system, are seen to become numerous in the Guidonian Hand. Six keys are common to three separate hexachords; eight keys to two. Only eight keys are specific to just one hexachord. Thus, redundancy is established as a principle.

Contrary to present-day usage in the Latin countries, the syllables ut, re, mi, fa, sol and la, here called voices, did not correspond to absolute pitch values in a scale, but only to one of six degrees of a hexachord. Their absolute pitch value depended on the position of the particular hexachord within the Guidonian Hand. In Renaissance times, if one wished to identify a degree of the Hand completely, it was the custom to follow the key letter with the voice or voices with which it could be implemented, beginning with the lowest hexachord. For example the note currently described as a G in some countries and a sol in others would be called G sol-re-ut ⁴².

This is then the system which served as a basis for musical education for nearly six centuries. Strictly instilled into generations of choristers, and deeply ingrained in the minds and instincts of singers over this long period, the Guidonian Hand formed a universal and durable diatonic framework, thanks to which singers could find their way around the musical scale with the support of solmisation ⁴³.

In practice, the singer who deciphered a melody using solmisation used in the first place the voices of the hexachord which covered the range of the melody in question. If it went outside the limits of a single hexachord (which was most often the case), the singer made a mutation (in Latin mutatio), that is to say he changed hexachord. For one voice of a current hexachord he substituted the voice of a new hexachord found at the same pitch level. This exploited the inherent redundancy of the system, and in effect moved along the horizontal lines of our diagram (Figure 2.9).

One advantage of the hexachord over the modern scale is that it does not contain the tritone (in the major scale between the fourth and seventh degree), nor other augmented or diminished perfect consonances. Thus if the singer is progressing within the confines of a given hexachord, he has no need to worry about avoiding, for example, an augmented fourth or a diminished fifth. Only if he makes a mutation does he have to address the problem. In the frame of a musical practice subject to the dogma of avoidance of such intervals, this characteristic of the system has great importance.

Another great advantage of the Guidonian Hand and of solmisation is that all the semitones have the same standing - they are sung mi-fa. From above and below, this central interval provides a landmark for the tones ut-re, re-mi, fa-sol and sol-la of each hexachord. Since the eight ecclesiastical modes have, in canonical form, their finals on D, E, F, and G, (which are the re, mi, fa and sol of the natural hexachord) it is possible, given any melodic line written in a particular mode, to transpose it by a fourth, a fifth, or an octave without modifying in any way the voices on which it is sung. Note that, transposed by a fourth, D-E-F gives G-A-B flat in the anglophone system, and ré-mi-fa gives sol-la-si bémol in the francophone system, but re-mi-fa remains re-mi-fa in solmisation. Furthermore, in whichever hexachord one sings, a chant in first mode will have a final on re, a chant in fifth mode on fa, and so on. One can see in this characteristic the main justification for the system - an adaptation of the great system of the Greeks with the objective of making it compatible with the ecclesiastical modes.

The seven hexachords of the Guidonian Hand form what it is convenient to call musica recta. It is possible, for example to avoid tritones, to add one or more further hexachords to the system, either in the direction "soft," i.e. going up by fourths or down by fifths on the right of our diagram; or in the direction "hard," i.e. going up by fifths or down by fourths on the left of our diagram. These virtual hexachords, not part of the Guidonian Hand, belong to the so-called musica ficta. Each additional hexachord in the "soft" direction adds a "fa" to the system, which corresponds, in modern terms, to a flat in the key signature. Each additional hexachord in the "hard" direction adds a "mi," which corresponds in modern terms to an additional sharp.

Stage four: Voyage to terra incognita

To summarise: We have followed the development of the Pythagorean tetrachord into a hexachord, as well as, in parallel, the development of the sistema massimo of the diatono into what we could call a Pythagorean Hand, which constituted the diatonic framework universally accepted in the Middle Ages and the Renaissance. In parallel, we have seen that, as a principle of intonation, the diatono did not satisfy Renaissance ears. It offended by its thirds, which were far from being pure. We can say also that Zarlino favoured the sintono, and we have information on the tetrachord and the sistema massimo constructed following this intonation principle. We are therefore now in a position to synthesise all of this into a Syntonic Hand, composed of hexachords inspired by the sintono of Ptolemy.

We are at the final stage of a logical process which should lead to a diatonic framework satisfying Zarlino's ideals. It may be that, for reasons now difficult to discover, he himself never made this step. Nevertheless he has carefully signposted the way. Anyone who had, in the Renaissance, a practical interest in just intonation could not do other than be led, as we are, to take this last step.

To transform a tetrachord with sintono tuning into a syntonic hexachord must involve extending it downwards by two degrees of a tone. As there exists, in this tuning scheme, a major tone and a minor tone, it is uncertain which combination to adopt. However we see immediately that in order to obtain a pure sixth ut-la we must add a pure major third to the fourth mi-la (the initial tetrachord), and thus a major tone plus a minor tone. On the other hand, if we are to maintain all the thirds of the hexachord (ut-mi, re-fa, mi-sol and fa-la) as pure, following the logic of the sintono, the only solution is to place the minor tone (t) between ut and re, and the major tone (T) between re and mi. We thus obtain a symmetrical structure in which the two extreme tones are minor, and the intermediate tones are major, enclosing the central semitone (S).

	Scale degrees	Ratios	Cents
{	La	5:3	884.36
{	t
{	Sol	3:2	701.96
The tetrachord	T
{	Fa	4:3	498.04
{	s
{	Mi	5:4	386.31
	T
	Re	10:9	182.4
	t
	Ut	1	0

Figure 2.10: Possible arrangement of a syntonic hexachord

This hexachord is not absolutely perfect. The fourth re-sol is too large by a comma. Whatever one does to avoid this problem, it is not possible to do better.

It is probable that Zarlino ⁴⁴ would have chosen to place the major tone between ut and re, and the minor tone between re and mi, and that he would have preferred the sequence T-t-S-T-t over our choice (t-T-S-T-t), this primarily for reasons associated with numerical elegance. This is certainly a possible option, but not our solution. As we shall see later, the risk of this choice, even if negligible from a theoretical point of view, has a certain practical importance.

It simply remains to build up a Syntonic Hand by combining, as necessary, seven hexachords (Figure 2.11). We find again, not to our surprise, the comma which appeared in the sistema massimo with sintono tuning, between the conjoint and the disjoint tetrachords. In the new Hand, it occurs for the key "d" between the sol of a hard hexachord and the re of a natural hexachord, this latter coincident in pitch with the la of a soft hexachord. But this is not all. The extension of the tetrachords into hexachords has given other near coincidences between minor tones and major tones, and thus mismatches of a comma. There are four in all for the whole of the Hand, and they are all situated, from left to right (i.e. from hard to soft) between a sol and a re.

A Syntonic Hand of this type forms a logical culmination to the theoretical sequence due to Zarlino, to which it simply adds a last link. Had it ever been actually and explicitly formalised, at that time or following Zarlino? Could it have had an implicit influence on the intonational habits of the period? Alternatively is it, in the same way as Helmholtz's "natural scale," a pure feat of imagination? We shall probably never know. Whatever the case, it is certainly worth the trouble to investigate, as apart from any historical interest, it is in itself full of information on the way diatonic structures work.

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