I am no mathematician. I never got farther than sophomore geometry in high school, and can’t remember what I learned about fractions in elementary school. And to paraphrase Ben Jonson on Shakespeare, I have no Latin and less Greek. Combine these deficiencies and you will understand why I have always been baffled –– not to say repelled –– by the vast literature dealing with calculations of micro-intervals in connection with keyboard temperament. Nature’s inscrutable wave theory tempted some unfortunate civilizations into filling up the gaps between her pure intervals in order to create scales, made up of steps on these ladders, comprised of whole and half-tones, clumped together by the ancient Greeks into tetrachords for their lyres and kitharas. Asia and Africa wisely avoided this trap and settled for five notes to an octave which could all be tuned pure. A ancient Greek named Pythagoras, however, decided that tetrachords and their component intervals needed to be measured in exact mathematical units, classified in various modes, and glued together according to diverse systems.
That way lay madness. He and other Greek authorities were immortalized by the late-Roman theorist Boethius, an advisor to the Ostrogoth King Theoderic the Great. Boethius collected as much of Greek music theory as he could in his De musica. This fundamental treatise continued to be copied throughout the Middle Ages and really took off when printing commenced. It brims with head-spinning terms like proslambanomene and hypate hypaton; the mysterious twins Nete and Mese (the latter always distracting me with visions of Lebanese appetizers) abound; each note has a different Greek name, and intervals all have Greek names as well. I have never mastered any of them other than diapente (fifth) and diatesseron (fourth), and I may be wrong about those. As a harpsichord tuner I have to know a little about commas, but don’t get anywhere near me with algorithms and cents.
The Renaissance took up this material with a vengeance. The possibilities for continued interval calculations and use of abstruse Greek terminology were irresistible, resulting in endless, impenetrable books and violent controversies about a subject which had almost no practical application. Music’s official status as a science rather than an art contributed to this farrago of obscurities.
I recently made the acquaintance of the man who probably knows more about this field than anyone else on the planet. One of the loveliest and most mentally agile guys I ever met, 84-year-old (as of this writing) Charles Atkinson has an important position at the university near where I live. I attended a couple of his lectures about the evolution of ancient music theory, which, due to my ignorance, had me utterly lost after their introductory paragraphs. But I found myself fascinated by Charles’ erudition and evident passion for his subject. Like Charles, I adore the art and history of the Middle Ages, even if I can’t relate to its theocentric philosophies or its music theory. And we bonded over our past careers –– his as a clarinetist, mine as a fortepiano player. Mozart’s “Kegelstatt Trio” is a topic for endless discussion.
***
The other day I finally began reading a set of books I had bought mostly for their pretty yellow covers. Back in the halcyon days of the Early Music Revival, people bought great numbers of facsimiles of scores and treatises. There were six main publishers: Broude, Garland, Gregg, Minkoff (usually known as Rip-off), SPES and Fuzeau*. None survive except as remainders; the market has collapsed, partly in parallel with the collapse of classical music in general and high-quality Early Music in particular, partly because everything is now online. When Fuzeau went out of business I snapped up a number of titles on sale, including four volumes of theoretical treatises from 16th-century France.
I have since been admiring their backs, the color of an unripe peach, from a respectful distance; but with my Scarlatti edition on hold at Bärenreiter I waded into them last week. I was brought up short by the first treatise in Vol. III, the Solitaire Second, ou Prose de la Musique (Lyon, 1555) by Pontus de Tyard, a minor Burgundian nobleman who was a member of the illustrious Pléiade of poets grouped around Ronsard. He was also a mathematician and music lover. The treatise at hand combines these attributes in the manner of a master of classical French letters, which is to say: with heartbreaking elegance and the involvement of a female. His meandering sentences are liberally punctuated with colons.
Tyard chooses the popular dialogue form. He embodies himself as “Solitaire”, a quiet scholar of advancing age, who is instructing “Pasithée” –– young and beautiful, va sans dire –– in music theory. Her name corresponds to “Pasithea”, one of the ancient Greek Graces, sometimes called a Muse, said to be all-knowing. Thus she takes on a double function: the human and distinctly erotic, and the quasi-divine. Indeed, for a young lady supposedly seeking knowledge, she is suspiciously perspicacious, understanding all immediately and asking the right questions to keep the conversation going, while occasionally teasing, chaffing or flirting with Solitaire, and provoking a bit of friendly male rivalry when her suitor “Curieus” unexpectedly appears and joins the conversation. There is considerable talk of the ecstasies produced by music, and conversely, of its necessity for a temperate society.
All this folderol only serves to make the meat of the thing a little more digestible, for here is the surprise: the 175-page book is a full discussion of Greek music theory, as passed down by Boethius and later writers (Tyard names Peletier, Gaffurius and Glareanus) which I think would have Charles Atkison’s full attention, and as far as I can tell, admiration. I was surprised to find the topic so thoroughly discussed in mid-16th-century France, printed in the latest Lyon italics, after two volumes of Guidonian hands and sol-fa in Gothic characters. Charming Pasithée soaks it all up without difficulty, while admitting (p. 69) that “Truly, these subtle considerations seem to me of such admirable industriousness that I do not find it strange that so few people are bold enough to plunge into this deep abyss.” I skimmed the subtle considerations in despair, looking for things to take note of.
***
This first thing to be established, given my abyss-mal ignorance, was Tyard’s own attitude to the mass of Greek music science. He shows himself rather ambivalent. After 80 pages expounding the three Greek genera, Pasithée asks him if there is not another? Solitaire reluctantly admits that there is a school of thought centering around Aristoxenus. This 4th-century-BC student of Aristotle is often incorrectly cited as advocating equality of intervals, whereas he only proposes two equal semitones to a whole tone once, as a thought experiment. What he does recommend is tuning by ear, rather than by numbers, and a generally more sensible attitude to such matters. Tyard seems to secretly harbor similar wishes, expressed for example on pp. 42-3:
“[The Pythagorians] give incomparably more credence to reason than to exterior sensations, although even the illustrious Pythagoras could not attain that degree of authority without being contradicted: for, whether they follow the doctrine of Aristoxenus or of whomever else, there also exists another sect of musicians called Harmoniques, who have more faith, and more firmly anchor their opinions, in the sensations of the ear than in reason: & of these two sects I know not which had the honor of being the earlier: and as to the truth of their opinions, the former please me with the subtlety of their speculations, the latter with their simple and evident proofs: but I am even more pleased by those who have followed a middle road with Ptolemy, giving reason a very honorable place, and yet by no means rejecting the merit due to physical sensations. Such that (if I am not deceived) judgements regarding music should be considered correct, and sane, when reason and the senses meet together without controversy.”
Solitaire then proceeds to hedge his bets, whereupon Pasithée justly remarks: “It seems to me that you are trying to confound and cast a shadow over this question, or that you wish to dissimulate your opinion.” Solitaire answers that precise calculations are important, and thinks that the ear, while important, is limited in its discernment; who can hear the difference between an apotome and a diese? He then trows up a smokescreen of Greek nomenclature and moves on to another systeme.
Tyard is thus somewhat trapped in the Boethian Establishment; he loves showing off his erudition, but has to admit its limits. At one point (p. 64) he goes to Pasithée’s espinette –– the generic name for all instruments of the harpsichord family in France at this period –– to satisfy her desire to know whether one can discern the difference between a major and a minor semitone:
“Take the trouble (I said to her as I got up) to come to your espinette, & I will show you. She approached the special trestle on which it usually stands, and confirmed that the upper key [feinte] between A & B can demonstrate such an evident discrepancy that the ear can hear it: since in truth it seems that the interval from A to the upper key, which I have called Trite synemmenon, is smaller than that which rises from the upper key to Paramese, or B-natural. After this proof (she says), Solitaire, let us return to our seat, & finish, if you please, the proportions of the tetrachord synemmenon, the which I have caused you to interrupt.”
(You see what I mean about Pasithée keeping the conversation going. And her espinette must have been extraordinarily well in tune.)
***
I of course pounced on the few remaining mentions of the espinette. Here is what Tyard himself called “a paradox of little profit” regarding ranges (p. 25):
“Let us observe how in some musical instruments such as the lyre, lute and guitar, the thick strings are in a higher position: & in some others, like the harp and espinette, they are lower.”
Then on p. 77, during a discussion of the effect of pure consonants when doubled: “In truth (she says), I have sometimes noticed this while playing the lute, & (although not so clearly, because of the liquidation or couverture of the other chords) on the espinette.” I have left two words in the original because I am not sure what she means. Best guess: the damping of a previous chord is imperfect on the espinette, a together with its continuing resonance, hinders perception of new resonances.
On p. 102 Tyard skirts the practical issue of temperament, the enemy of all theoretical systemes:
“And so the opinions of Aristoxenus never received enough approval for them to be accepted by those who succeeded him: on the contrary, they combatted him using many arguments which it is not necessary to repeat to you: given that the research involved is dear and difficult, and the fruit of it of little use…And truly, if actual usage & execution had accompanied their imaginary designs, they would have realized that music only builds on its true foundation when the senses and reason are joined in common consent: and that proofs [of such systems] are difficult, and that human curiosity [regarding them] will never be satisfied using voices [which cannot reliably render micro-intervals], but only by the disposition of [organ] pipes of solid material, & by strings, whose flexibility allows them to be tightened or loosened at the will of the person who handles them.”
***
The lute, still the instrument par excellence, gets more space in a discussion of Francesco Canova da Milano (1497-1543) beginning on p. 113. Curieus praises the power of music:
“In sum, music is the sovereign mistress that consoles the mourner, calms anger, restrains audacity, tempers desire, cures illness, solaces misery, comforts languor, and sweetens the pain of love. You could tell (said Pasithée) many ancient stories on this subject: but it would be difficult to find one to greater effect than that which was recently told to us by Monsieur [Jacques] de Vintimille [1512-82] (whose friendship is so dear to you, Solitaire, as shown by your frequent recollections), who, being at Milan, didn’t spare himself the trouble, according to his diligent nature where the virtues are concerned, of acquainting himself with all persons of merit, and was invited (given that such a person cannot remain unknown, wherever he may be) to a magnificent and sumptuous banquet, arranged for the most illustrious persons in the city, in a house of the same rank: where among the rare things assembled for the entertainment of the guests was found Francesco di [sic] Milano, the man who is held to have attained (if the thing be possible) the pinnacle of perfection in playing the lute. The tables having been moved aside, he went to one, &, as if to try the out the tuning, he began, sitting near the end of the table, to improvise [rechercher] a fantasie. He hadn’t played three notes when the guests broke off the conversations they had commenced among each other, and having constrained them to turn their faces in his direction, he continued with ravishing assiduity, and little by little through his divine manner of playing having caused the strings to die under his fingers, he transported all the listeners into such a tender melancholy –– so that one rested his head on his hand supported by his elbow; or that another, stretched out casually, his limbs in a posture of careless unconcern, jaw dropped and eyes more than half closed, seemingly glued to the strings & with his chin resting on his chest, veiled his face with the saddest taciturnity ever seen –– that they [all] remained deprived of sensation except that of hearing, as if their souls, having abandoned the seats of all the other senses, had retired to the edge of their ears, in order to more fully enjoy such a ravissante symphonie: and believe me (said Monsieur de Vintimille), we would still be in that state, had [Francesco] himself not (I know not how) revived the strings, and bit by bit reinvigorating his playing with sweet power, reinstated the soul and the senses to the place whence he had previously stripped them: not without leaving all of us equally astonished at having been lifted into some transport of divine frenzy.”
There follows a complex passage where the three interlocutors delicately probe their mutual sentiments as only classical French can, and Solitaire eventually confesses whence his greatest inspiration comes:
“[Melancholy] is his nature (says Curieus), but if I beguile him [si le s’en je] he is tractable enough to be led away from it towards contentment. And truly (Pasithée replied), if we wish to believe him he will make us as sad as he is: but Solitaire (turning towards me), tell us what testimony you can put forward [out of your own experience] regarding the efficacy of music. It is (I responded) that the sound of your voice, accompanied by your harp, or espinette, has ravished me.”
***
An interesting observation regarding compartmentalization of disciplines, lamentable then as now, occurs on p. 133:
“…Musicians are all, or for the most part, without knowledge of literature and poetry: just as the greater number of poets mistrust, &, if I dare say, do not understand music.”
Here is another passage on the effects of music (p. 153):
“Just as poetry joined with music can keenly move human emotions and passions, these can be pacified, as we discussed today, by the same means, and human understanding put back on the high road back to the tranquility of its origins: & the fragments of the soul, scattered and vagabonding here & there in disorder, be reassembled into a whole…And this is to be understood not only in regard to harmonious music of voices and instruments, but also concerning that harmony of virtues which guides the soul to its highest good of perfect tranquility, where, as if by an enrapturement, it is raised above all physical desires; like when the sounds of music in perfect tempo [parfaittement mesuree] pull the listeners away from all other thoughts, and block all senses except for hearing, rendering the others inert…”
My translation of parfaittement mesuree may be too literal, biased by my disgust with the free-tempo performances so prevalent today. Tyard may mean something broader, like “perfectly balanced”, or “perfectly proportioned”. But for now I will stick with “in perfect tempo”, because the steady measurement of passing time was in fact the backbone of music from the moment it became polyphonic, and then MENSURAL, in Western civilization. That was Pontus de Tyard’s world.
He closes his Solitaire Second with one more reference to the skills in music of his Muse/beloved, and to the espinette. A third day of discussions has been proposed by Solitaire, and the irritating Curieus begs an invitation:
“But (she says) je vous en prie, as I feel assured that we cannot be without the aid of your gentle spirit. The hour of departure approached, wherefore, Curieus taking his leave (after, among other gracious acts, she had shown her rare & admirable industrie in playing the espinette) & I having honored her with a humble reverence, leaving her we retired to my home.
AMOUR IMMORTELLE
FIN DU SECOND SOLITAIRE”
St. Nicholas Day, 2025
*My first published musicological work appeared in a short-lived facsimile imprint; see Article 11.
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