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I'd like to open myself the discussion that Oliver Gerlach promised in the Low B-flat discussion. It refers to Guido of Arezzo's mention of the usage of the diesis in proper and inappropriate positions in the scale.

A first question that I have concerns the manuscripts that reproduce this mention, and their date. Guido's passage about this, reproduced in GS II, pp. 10-11, has been omitted in the CSM edition by Smits Van Waesberghe, and is missing also in the recent traductions of the Micrologus. Oliver himself mentions in his "Der Oktōīchos zwischen Mittelalter und heutiger Tradition" that this text nicht in allen Abschriften des Micrologus überliefert wird. I would be interested to know in which manuscripts this passage is copied, whether their date gives any clue as to why it is not included in all manuscripts, etc.

The fact is that there is, I think, some incoherence in this passage, as if it had been written by more than one person, and/or at more than one time. The text first says that the subductiones (by which I understand the "lowering of the note", producing the diesis) should happen only on the trito (by which I understand the third note of any tetrachord), i.e. F or C. Guido then adds that it (the subductio?) should not be done unless on "the third and sixth", which migh mean the third and sixth notes of a scale beginning on A (i.e. C and F). But a few line later he describes the inteval from G to a as corresponding to the string-length ratio 8:9 (i.e. 204 cents) and that from a to the note between b♮ and c as 6:7 (267 cents). In other words, Guido first appears to describe the diesis as a half-flattened F or C, then as a half-sharpened E or B♮.

I am both utterly interested and utterly puzzled by this, and I'd welcome any comment that anyone would make. What puzzles me is that in modern Arabic theory, as I think to understand it, the neutral second appears between D and F (divided by half flat E) or between A and C (with half flat B), while I fail to see the point of half flattening C or F (or, for that matter, of half sharpening B or E), which would not result in a neutral second, but in a "diminished semitone"...

Tags: Al-Farabi, Boethius, Chrysanthos, Guido of Arezzo, WilliamofVolpiano

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Replies to This Discussion

Of course, the term of lowering a note is a very problematic point of view, since the term diesis is known as a higher alteration of a certain pitch (not only in Greek, even in French), meaning that "a diesis" is caused by an attraction to the very close pitch "b flat", or "c" or "F" in other cases. This kind of alteration is also meant by Guido (the mi is always attracted by fa in all hexachords).

I quote the passage according to Gerbert (TML) who usually did editions which are rarely topped by later ones:

Metitur autem hoc modo. Cum a G. ad finem feceris novem passus, reperisque a. tunc ab a. ad finem partire per septem, et in termino primae partis reperies primam diesim, inter [sqb]. et c.

The three Guidonian dieses and their position on the monochord

Mox secundus et tertius passus erunt vacui, quartus vero tertii diesis obtinebit locum, qui similiter erit inter [sqb][sqb] et cc.

Modo simili a d. passus fiant totidem ad finem, moxque secundae patebit locus, supradicto ordine, quae erit inter e. et f.

Tunc revertens ad primam diesin, divide ad finem per quatuor, et primus item passus terminabit inter e. et f. secundus inter [sqb][sqb], et cc. reliqui vacant.

Admonemus vero lectorem, ne existimet nos desipere, eoquod primo omisimus ista scribere. Nos enim paratos habebit, post finem operis ex istis respondere sibi; nunc ad cepta revertamur.

One can measure them this way. If you make from G [hexachordum durum] to the end of the monochord nine parts so that you have already found a [8:9], you can divide the space from a into seven [equal] parts, and towards the end of the first seventh [6:7] you will find the first diesis between b natural and c, the following second and third sections remain empty, but the fourth indicates the position of the third diesis, which is between natural bb and cc.

You do it the same way from d with the same number of parts until the end, and within the same mentioned order you will find the second diesis, which will be between e and f.

Now if you return to the first diesis [b diesis], divide [the chord] until the end through 4, and the first step [3:4] will finish between e and f, the second [1:2] natural bb and cc and the missing others.
But we warn the reader, we don't wish to be regarded as fools, otherwise, we would rather omitted to write this. You will have us prepared to answer after the end of this work, for now we will return to the plan.

I expect that these last phrases encouraged later copists to omit Guido's passus about the diesis. I never took time for it, but I think yours is a very good proposition for the Medieval Music Theory Project to make a list of those copies with and without the passus about the diesis. 

If I follow you correctly, you think that Guido's diesis is a "leading tone", that G A B C, for instance would be sung 0–8:9–16:21 (=8:9 * 6:7) –3:4 (i.e. 0-204-471-498 cents), with B more than a quarter tone higher than its Pythagorean value !

In Engelbert of Admont (GS I, 312), subductio clearly is a synonym for limma, i.e. the Pythagorean diatonic semitone, the "normal" interval between B and C or E and F, and the passage from Guido may be confused about that. In the Brussels Ms of the Micrologus (B, Br, II 784), one reads Est et diesis id est uocum certa subductio. quae quia nunquam admittitur. nisi cum quarta aut septima in quinta. atque prima producitur – "And there is the diesis, i.e. a certain subductio of the notes, which is never admitted, unless when the quarta or septima proceeds to the quinta and prima". This seems to refer to the 4th note going to the 5th or the 7th going to the 1st (= the 8th?) – if the scale starts from G, this might refer to C♯-D and F♯-G.

One problem with your reading is that it presupposes hexachords, while Guido in the Micrologus more often refers to tetrachords. When one reads quasdam faciunt subductiones in trito,  does that mean the 3d note of the hexachord (i.e. mi), or of the tetrachord (i.e. fa)?]. Does In nullo enim sono valet fieri, excepto tertio & sexto, refer to the 3d and 6th notes of the hexachord (mi and la), or or the scale (B and E starting from G, C and F starting from A)? The description of numeric ratios starting from G do not necessarily point to the hexacordum durum, as you write above.

The meaning of the passage that you quote above and of the numeric ratios described seems inescapable. But it does not really appear to make sense with the other passages of the same chapter X mentioning the diesis.

The more I think of it, the more I think that the problem is in the passage that you quote. Guido says that, measuring as he describes, reperies primam diesim, inter ♮. et c, "you will find the diesis between ♮ and c". Now, indeed, the interval between ♮ and c is a diesis, a subductio, a limma. The diesis is an interval, not a note. Guido hardly could mean that "you will find the note, called diesis, that is somewhere between ♮ and c". The diesis is the distance between ♮ and c, not some note in between...

Of course, it is a Guidonian explanation, tempted by the starting point G. It is not the only possible way to look at it, but for those who understand the difference between modern solfeggio based on 7 syllables and Guido's mutatio hexachordis, it is at least one possibility. I see no plausible reason to avoid this point of view.

What I would rather avoid is Riemann's term "leading note", because it has nothing to do with the very imaginative orientation in modern tonality.

The proportion 6:7 means that the ditonus, which is already as sharp that any Eastern cantor would already perceive it as "enharmonic", is about a third (of a tonus) higher than usual. Everybody who has ears can easily learn to intone such a simple proportion. It is not my business to judge those who always have to stick to their old habits, but if you skill the singers of your ensemble to understand their perception of pure intervals according to harmonics which come together, they really start to discover modality. If you can perceive easily 9:8 (the tonus), why not 8:7 or 7:6?

At least with an ison on D or G that should be possible. If you really do so, you will see that after some months you can anyway recognise it also without concentrating too much on proportions and their effect on overtones. What I am talking here is a very precise intonation which is the essence of doing modal music and it is naturally related to a vocal technique which does not avoid stable harmonics.

We are talking about something which is beyond the imagination of many musicians (depending on their school and its principles). But the way to modal music is always open, whenever a musician feels ready to go it.

Some remarks on the Greek terminology:

The interval 256:243, known as "semitonium" among Latin theorists, is called λεῖμμα in Greek. Since 204' cent minus 96' is not 96' as "semitonium" might suggest it to ignorants (or those who learnt to mistune everything equally which I really doubt how they manage this, because only a pianist might rely on that...), the resting 108' cents are called ἀποτομή.

During the 19th century Chrysanthos of Madytos even used the term ἡμίτονον, but he meant all kinds of intervals from the smallest microtone to the minor tone which was considerably larger than the Latin proportion identified with semitonium

Oliver, trying to understand Guido (even if he was wrong) is for me a real challenge, and an interesting one.The challenge is about understanding how theorists may have thought a thousand years ago. I am not sure that disparaging comments about equal temperament can help us understand.

I don't think that the term "leading tone" belongs to Riemann; it must be much older. I understand by it a tone that is attracted towards another and that, therefore, raises closer to it. It is well known that, in tonal music, violinists tend to sharpen the leading tones and to play rather narrow diatonic semitones (as in Pythagorean tuning). I have no idea of (and I am not particularly interested by) intonations that medieval singers may have prefered, but I do believe that the whole idea about musica ficta is of the kind of what later was called "leading tone".

I suppose that by "ditonus", you mean the Pythagorean one of 408 cents. I fail to understand in what sense this has to do with "enharmonic". I understand that the diatonic tetrachord was seen in the middle ages as the division of the perfect 4th (498 cents) in two tones (forming a ditone) and a limma, 204+204+90=498 cents, and the enharmonic tetrachord as consisting in the same ditone (408 cents) and two little intervals at times called dieses. In other words the ditone to me not particularly more enharmonic than diatonic.

The Pythagorean limma was often known as semitonium in Latin theory: it was considered the "true" semitone because it was the singable one – the diatonic semitone. The Pythagorean limma corresponds to 256:243, i.e. about 90 cents. The apotome is the rest, as its name indicate, i.e. the other part of the division of the tone, 114 cents (204 - 90).

Only ignorants would believe that the word semitone, or its Greek equivalent ἡμίτονον or ἡμίτονίον, could denote the division of the tone in two equal parts: semi- or hemi- indicate a division, no more. It is true that these terms connote the hope that the tone might be divided in two equal parts – a hope that can be documented throughout the history of theory from Greek Antiquity to the end of the Renaissance and later, but that became a reality only after the discovery of continous fractions in the late 16th century and of logarithms in the 17th century.

I don't see where your value of 96 cents comes from. The equal-temperament semitone by definition is 100 cents; the Pythagorean limma (256:243) is about 90 cents, the Pythagorean apotome (2187:2048) about 114 cents.

And nothing of this brings us any closer to understanding whether Guido, describing the diesis, meant the distance between, say, ♮ and c, or a quarter tone roughly half way (or two thirds of the way) between the two.

You are right, I remembered the cent value of a semitonium not correctly. Hence, λεῖμμα has 90' cents, and ἀποτομή 114', exactly as you wrote.

I think, it is very crucial what I wrote about modern tonality (therefore very useful to explain common misunderstandings and also a good luggage you might need for such a travel in time). Nevertheless, certain interpretations of yours are based on a very common confusion between chromatic or enharmonic as a "corrupted" term of Western music theory and "chromatic / enharmonic genus" as a modal term which was the only one known to Guido. I mean it is useful to distinct both. Cultural differences are not mentioned here in order to neglect them, but in order to respect them. Thus, they even might help us to understand.

Despite I really think it is useful to mention some experience in performance practice and this includes the perception that Pythagorean is nowadays defined as "hard diatonic" according to the chant manual of Georgios Konstantinou, but it never was according to Byzantine theory, where the "hard diatonic" division was only used as enharmonic triphonia and associated with phthora nana. It included a further attraction within an interval of the leimma, but such a microtone was not called leimma, but very vague as hemitonon.

My point to refer to Guido's hexachords was that the passus about the diesis even works within his very specific hexachord theory. For this very reason I am convinced it was not added later, but it was really written by Guido himself. Of course, we have to verify this with the given sources and their real age.

As far as I know, Guido never found any resonance among Greek authors. One of the possible reasons is that his hexachord concept only works under the condition that the tetrachords are divided by a ditonus and a leimma, this was not the condition of Boethius, because he only specified one tonus within the tetrachord, but not the other intervals. He still did refer various divisions which he found in Ptolemy's treatise.

If you try to understand the different perception which results from the simple choice, whether you divide the tetrachord just by two diatonic intervals (let's call them "tonus" and "semitonium") or by three intervals which are all different (these two options always existed within Greek theory through all the centuries), you will very easily understand the difference between a tetrachord system and its possibility of transposition, on the one hand, and Guido's translation of the same into his hexachord system and the mutatio as a faked transposition which simply has to illustrate the untransposed tone system, on the other hand. We often believe we do understand the tetrachord system, but the real challenge is to imagine it with tetrachords divided by three different intervals, and then we will easily find out, what the transposition of the whole system does really mean, especially if it should work temporarily.

What comes as a surprise is the very large difference between 7:6 and 9:8, it clearly rejects the common simplified concept, that the diesis fits twice within the semitonium. The rest of the fourth left by a ditonus sharpened with Guido's diesis is so small, that it would not even fill twice the interval of one semitonium.

"Subductio" could be indeed a Latin term chosen to translate the Greek term δίεσις quite literally, but I strongly advice against a confusion between λεῖμμα and δίεσις. Guido's measures exemplified on the monochord leave no doubt that a much smaller interval is the result of an alteration. I think that diesis means as well the resulting small interval towards the attracting degree of the mode as the difference in proportion caused by the alteration itself. This ambiguity was based on a common simplified imagination of a diesis with respect to a semitonium, so it was expected to be more or less the same interval (in Greek explanations we find very similar imaginations, that common intervals are somehow subdivided or even cut into two halves with one half added to the neigbour interval), but Guido's proportion clearly points at the possibility that such a use of the term might be somehow imprecise...

I also freely admit, it is very strange indeed, that he does neither measure nor mention (explicitely) the a diesis which we do definitely expect (whether we might think of tetrachords or hexachords).

Oliver, two points:

1) I have nothing against the idea in itself of considering hexachords instead of tetrachords. However, the third degree of the hexachord is mi (in solmization), while that of the tetrachord is fa. The question, then is whether Guido affects the diesis to mi or to fa.

2) Guido explicitly speaks of subductiones [in trito], quae dieses appellantur. There is no doubt that for him, and for Engelbert of Admont, "diesis" and "subductio" are synonyms, any ancient Greek conception notwithstanding.

Engelbert writes: tonus [...] dividitur in duas partes inaequales, scilicet in maiorem, quae vocatur Apotome, id est, superdivisa, et haec est semitonium maius; et in minorem quae vocatur diesis, id est, subductio a semitonio maiori; et haec est semitonium minus. It appears that for him, the diesis is the minor semitone, called subductio as the part of the tone that remains after the apotome has been taken. [This meaning of subductio, if I am not mistaken, would be a good translation of the Greek ἀποτομή, "what is cut away"; but never mind. This only shows the Latin misunderstanding of Greek terms.]

As you write, one would expect the diesis (the "quarter tone", it being understood that it is not exactly half a semitone) to be added to the semitone, to form a "neutral second" of three "quarter tones". This would be the case if the diesis was added to fa (i.e. to F or C), dividing the trihemitone E-G or B-D in two equal halves, E-F+-G and B-C+-D (where the + indicates the sharpening by a diesis, in this case a half sharp). Arabic theory does the same, but dividing the trihemitones D-F and A-C and lowering the note in between by a half flat.

For sure, the ratio 6:7 is a strange one, but why not? It produces an interval of about one tone and two thirds (267 cents), which might be described as "three quarter tones", a neutral second (considering once again that a quarter tone needs not be exactly half the semitone).After all, Al-Urmawi once described a neutral second formed by a semitone + a comma...

I am not puzzled by the irregular halves, but by the fact that Guido apparently does not use the diesis (or subductio) as a neutral second, but as further dividing the semitones B♮-C or E-F in quarter tones. Unless when he writes reperies primam diesim, inter ♮ and c, he means "between B♮ and C+", i.e. forming a neutral second between B♮ and C half-sharp. But that would mean measuring 6:7 from B, not A...

Thank you for quoting Engelbert, and I assure you I rather prefer talking about proportions than about quartertones which is very imprecise indeed (and no ear can listen to such a geometric way of symmetrical subdivision), the surprise was just, that Guido's diesis is so spicy!

Since I was talking about a certain imprecision concerning the diesis, I do not think that Engelbert's Latin is so bad or simply a misunderstanding. You have just to look for the right context he mentioned subductio, many authors have a very refined concept talking about the triphonic system (also known as the Lesser Perfect System), but only a very few readers today understand, what they are really talking about, because they do not even know it.

As I said to use the term semitonium as a name for very different sizes of small intervals corresponds to the Greek use of talking about hemitona, rather than to the more specific use, when semitonium was chosen to translate leimma. I think apotome as the difference between tonus and leimma is very specific and it is indeed larger than the leimma, but what was cut away in Guido's case (the alteration) is the much larger part of the leimma with respect to the rest interval which Engelbert might have called diesis. Nevertheless, he translated diesis as a "subductio" which is connected to a larger hemitonon (there is the preposition "from", a with ablativus, but it could mean a larger interval in relation to the one of the diesis itself). It seems to me that his description is quite precise.

Concerning the Arabic treatise, al-Farabi described two diatonic tetrachord divisions and their transpositions on the 'Ud keyboard: the so-called "Pythagorean" and a softer one defined after the ring finger fret of Zalzal (a famous 'Ud player of Baghdad). If you prefer to think in tetrachords the resulting division would be 9:8 x 12:11 x 88:83. But as I said, I expect that the hard diatonic division probably already sounded close to an enharmonic division for Greek ears, it makes Guido's diesis even appear spicier, if not over-spiced...

Oliver, I can assure you that I have some knowledge of the Lesser Perfect System, and of ancient Greek theory in general.

Guido's ratio of 6:7 correponds to an interval of about 2,7 semitones (to be more precise would involve deciding which semitone, but that is not important here). He says that the diesis is built as follows:

G (8:9) A (6:7) B2/3♯ [... C?]

with B2/3♯ falling somewhere between B♮ and C (and "subductio" somehow meaning "sharpening by 2/3 semitone"). It leaves with only about 1/3 of a semitone between B2/3♯ and C, and I find it extremely puzzling. I don't understand the purpose (musical or other) of such a division.

The ratios I have for al-Farabi's Zalzalian division are not exactly yours, which seem incorrect because they do not totalize to a perfect fourth of 4:3. You wrote 88:83, but I think you meant 88:81. This would produce the following tetrachord, again starting from G (I use the ratios for the string lengths, as did al-Farabi and Guido, i.e. the reverse of those for frequencies, but this changes nothing):

G (8:9) A (11:12) B½♭ (81:88) C

Now we might assume that Guido's diesis (6:7, about 8 quartertones of 1/3 semitone, i.e. 16/6 of a semitone) is roughly the same as al-Farabi's neutral third (8:9 x 11:12 = 22/27, about 5 quartertones of 1/2 semitone each, i.e. 15/6 of a semitone). But the difference is that Guido builds it on A, while al-Farabi would have built in on G.

The question, to me, reduces to this: is Guido confusedly describing something similar to al-Farabi's Zalzalian division, or is he speaking of something entirely different?

The Lesser Perfect System also had chromatic and enharmonic divisions and needed many different intervals.

Guido unlike Engelbert uses diesis, when he talks about diesis (he never tried to translate the term in itself), and semitonium, when he talks about leimma. Engelbert on the other hand is much closer to the Greek terminology and their flexible use of hemitonon (that is why I wrote not to confuse leimma with diesis). Neither Guido nor Engelbert, as far as I know, offer any evidence for the soft diatonic division, it is just al-Farabi who defined the chordal proportion of the ring finger fret "Zalzal" as 22:27.

The only exception is an 11th-century compilation known as alia musica, one of the compiled tonaries speaks about an intonation AIANOEOEANE which is rubrified as autentus deuterus. Since he/she mentions a low and an intense intonation of that modal degree which is supposed to be the final, I believe the low intonation was not according to the Boethian diagramme, but meant as the tonal centre of the mode, while the diesis treated the same degree as mobile. There was a Byzantine influence, because AIA was the name of tetartos, it was not devteros, so E was indeed a mobile degree and as basis note meant as a kind of mesos tetartos (I mean there was no pentachord between E plagios and b kyrios as in devteros). I doubt that Greek singers would ever use a diesis on the modal degree of the base note, but we cannot know for sure! Certain Latin cantors obviously did...

In any case I would not go so far to suggest, that Guido's diesis contradicted the Boethian diagramme, he just made a precise interval definition for the enharmonic division. That's all!

The difference between theory and practice was, that the enharmonic diesis was sometimes used within the diatonic genus (we have so many concrete descriptions which offer enough evidence about this practice), so precisely as he described it a singer would intone the ditonus considerably higher, she/he would hardly intone the ditonus followed by the diesis and then continue to FA, although this division was supposed to be the enharmonic genus according to the theory. Guido like Engelbert were clearly talking about a microtonal attraction, the ab with ablative was clearly opposed to the direction, which is always expressed in Latin with the preposition ad and the accusative of direction.

If you divide a tetrachord by one tonus, the Zalzal proportion looks very weird, because the other intervals seem so different: the rest is divided into 12:11 and 88:81 (not 88:83!), but it was indeed nearly a symmetric division of the rest. With their methods you could make the comparison with arithmetics:

108  96   88   81


   12 + 8 + 7

Chrysanthos did not like the sum of 27 for the fourth, because he wanted to divide it into four minor tones (instead of 5 halftones), so he corrupted the arithmetic number and changed it into 28, by adding one division more to the middle tone (12 + 9 + 7). Then there was a patriarchal synod in 1881 which defined (even more absurd) an equal tempered division into 12 + 10 + 8 (200' + 166' + 133' = 500'), but the original proportions known since al-Farabi meant that there was a very small difference between the middle and the minor tone...



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