… washing the soul from the body that comes from the depth of the sea … (u sjećanje na Lou Reeda)

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”Tata, što si radio dok je bio rat?” Slušao Velvet Underground, više-manje. Nije da se ponosim. Nije da se sramim. Bilo je kako je bilo.

Uglavnom, jučer je umro Lou Reed, pjevač tužnih pjesama.

Jay Kennedy o muzičkoj strukturi Platonovih dijaloga

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Prije nešto više od tri godine jedno je otkriće vezano uz Platonove dijaloge dospjelo u masovne medije, npr. u Guardian i Večernji list (novinar VL je nažalost bio pobrkao tekst priopćenja sveučilišta i neke općenite ”izreke” koje kruže po internetu kao Platonove, a bile su navedene ispod priopćenja). Radi se o otkriću Jaya Kennedyja sa sveučilišta u Manchesteru da je Platon brojao riječi svojih dijaloga (to se naziva ”stihometrija”), i pritom primjenjivao jedan matematički obrazac vezan uz pitagorejsko shvaćanje muzika, a koji je su-određivao sadržaj pojedinih dijelova dijaloga.

Otkriće je objavljeno 2010. u časopisu Apeiron, u ovome članku: Plato’s Forms, Pythagorean Mathematics, and Stichometry.

Pročitah ga ponovo, evo nekoliko zanimljvih izvadaka:

The practice of counting the number of syllables in a line or the number of lines in a stanza was already routine in archaic poetry. Vitruvius, without giving his source, reports a tradition that ‘Pythagoreans’ and some comic playwrights mathematically organised longer works. … [T]he practice was already common during Plato’s lifetime. Callimachus’ catalogue, compiled about a century after Plato’s death, recorded the stichometric totals for each of the scrolls in the library of Alexandria. Diogenes Laertius’ report that Aristotle’s writings amounted to 445,270 lines may have derived from the Alexandrian catalogues.In this context, any authors with Pythagorean inclinations could avail themselves of stichometric counts to organise their works. …

This is apparently the first report of computer-based, stichometric investigations of Plato’s dialogues. … Although the data described below reveals some unexpected features of the dialogues, it is in retrospect natural that Plato would have given his works mathematical form. The dialogues reflect the revolution in mathematics that affected several of the arts and sciences during the fifth century, and mathematics is thought to have been important in the early Academy. Plato’s dialogues, of course, generally champion the importance of mathematics for philosophy and education. Embedding mathematical forms in their surface narratives also accords with the dialogue’s core philosophical conception of ‘forms beneath appearances.’ …

Some dialogues, like the Menexenus, the Symposium, and the Phaedrus, contain set speeches clearly demarcated from the surrounding text. The lengths of some of these speeches provide evidence that the composition of each dialogue was stichometrically organised.

In the Menexenus, for example, Socrates’ long speech lasts ten-twelfths of the length of the entire dialogue to within a fraction of a percent.

In the Symposium, Pausanias’ speech, Eryximachus’ speech (including the repartee over Aristophanes’ hiccups), and Aristophanes’ speech are each about one-twelfth of the dialogue. Socrates’ long speech, including his conversations with Agathon and Diotima, occupies three-twelfths or one quarter of the entire dialogue. Alcibiades’ speech lasts about two-twelfths of the dialogue.

duljina govora u Simpoziju

These length measurements suggest that an interval of one-twelfth of the dialogue plays a fundamental role. The relative location of the speeches within the Symposium provides an other form of evidence for the importance of this unit. The beginning of Pausanias’ speech is aligned with the point two-twelfths of the way through the dialogue, the beginning of Eryximachus’ speech (with hiccups) with the three-twelfths point, and the beginning of Aristophanes’ speech with the four-twelfths point. The climactic, rhetorical fireworks in praise of Eros that conclude Agathon’s speech occur at sixtwelfths, the centre of the dialogue. This scale of one to twelve plays a role even within the longer speeches. For example, the highlights of Diotima’s speech, her talk of intimate contact with Beauty and her description, at the top of her ‘ladder’, of transcendent Beauty as the form of the One, are aligned with points eight- and nine-twelfths of the way through the dialogue, and thus are also separated by an interval of one-twelfth.

In the Phaedrus, Socrates’ second speech is three times as long as his first speech to within a fraction of a percent. The first speech is somewhat longer than one-twelfth of the dialogue and the second is somewhat longer than three-twelfths. The beginning of the second speech occurs shortly before the four-twelfth point and the end is aligned with the seven-twelfth point.

The structure of arguments within individual dialogues is often organised around this scale of twelfths. Many examples could be given. In the Phaedo, Socrates concludes his argument for immortality from cyclic generation at the third twelfth; immediately thereafter he begins the argument from recollection which concludes at four-twelfths. In the Euthyphro, the first definition of holiness is at three-twelfths and the second definition is at four twelfths. In the Apology, Socrates begins his investigation of the oracle’s claim that he is ‘wisest’ at the two-twelfths point and concludes it at three twelfths. …

Using a figure of thirty-five letters per hexameter line, calculations of the total number of lines in the dialogues produce, with about one or two percent accuracy, impressively round numbers involving multiples of the number twelve:

• The Apology is 1200 lines, or 100 per twelfth.

• The Protagoras, Cratylus, Philebus, and the Symposium are each 2400 lines, or 200 per twelfth.

• The Gorgias is 3600 lines, or 300 per twelfth.

• The Republic is 12,000 lines, or 1000 per twelfth.

• the Laws is 14,400 lines, or 1200 lines per twelfth.

In sum, the lengths of speeches, the position of speeches within the dialogues, the location of significant turns in the arguments, and the absolute lengths of the dialogues all provide evidence for an underlying stichometric organisation and, in particular, for the importance of a twelve-part structure.

This evidence for a common twelve-part stichometric structure within individual dialogues suggests that they be read side-by-side, in order to compare their structures. Despite the different subjects of the dialogues, such com parisons reveal a surprising number of parallel passages, i.e., passages with similar content at the same relative locations in different dialogues. Many examples could be given. Here one, clear example is considered in a range of dialogues, both early and late. …

The Republic’s discussion of philosopher-kings and the form of the ideal just man occurs at the centre of the dialogue. Comparisons between the dialogues shows that passages describing the divine wisdom and justice of the ideal philosopher often recur near the centre. These terms also, of course, occur elsewhere in the dialogues, and that raises the chance that the following parallels are a coincidence. The immediate argument here against this possibility is simply the specificity, similarity, and precise locations of the passages:

Republic (50.0-50.5p): Socrates seeks justice and the just man who ‘participates’ in it, invokes Zeus, and first mentions the philosopher-kings who will lead (hegemoneuo) the city.

Phaedrus (49.5-50.3p): the followers of Zeus, the god of justice, seek a beloved with a ‘philosophical’ nature, who is a leader (hegemonikos), and ‘participates’ in the nature of god. The followers of Hera, on the other hand, seek a beloved with a ‘kingly’ nature.

Symposium (49.4-50.0p): Agathon praises Eros for being ‘the best and most beautiful leader’ (hegemon) and for being a ‘spectacle to the wise and admirable to the gods’ (including Zeus), and Socrates, perhaps for Plato an ideal philosopher and embodiment of Eros, jocularly claims to be a prophet (generally, a kind of divine knowledge).

Apart from the explicit repetition of forms of ‘hegemon,’ these three passages share a number of elements: Zeus and justice, the philosopher’s relation to divinity, and the notion of ruling or leading.

The Cratylus is useful for investigating parallels between the dialogues. Its series of etymologies is not organised in detail by any over-arching argument or narrative; the locations of the various terms analysed there, which typically appear only once, is generally determined by the underlying network of parallels between the dialogues. Here, for example, our leitmotiv occurs at the centre and nowhere else:

Cratylus (47.7-51.3p): the etymologies of wisdom, knowledge, the good, justice, Zeus, and of nous which rules itself and orders all things.

Some dialogues show this ideal philosopher in action at their centre, and repeat the cluster ‘philosophy, justice, and god’:

Apology (49.1-50.7p): Socrates claims to be wiser because he knows nothing — except that injustice is wrong for man and god, he will not give up philosophy, and he will obey the god.

Euthydemus (48.6-49.9p): one must philosophise, knowledge is more valuable than gold, and knowledge makes one immortal.

Euthyphro (48-50p): the gods dispute about justice, Socrates seeks to become wiser by being taught what the gods believe is correct (i.e., just), and will sing the praises of wisdom.

Gorgias (49.1-50.1p): Socrates asks about the nature of wisdom, behaves like an ideal philosopher by admitting his ignorance and seeking correction, and doubts whether justice is the stronger ruling over the weaker.

Finally, the Timaeus interrupts a long passage on natural philosophy at the centre of the dialogue with a paragraph of Pythagorean theology. Since justice is sometimes for Plato a kind of harmony, this passage would itself constitute an example of just and divine rule:

Timaeus (49.4-49.5p): Necessity willingly or unwillingly obeys God, who harmonises everything in the universe according to precise proportions.

Careful study of the parallels between the dialogues leads to another feature of their shared stichometric structure. Side-by-side comparisons of passages at the same relative locations shows that concepts with negative valuations within the dialogues, like disease, dishonesty, Hades, the body, difference, and negation, tend to cluster in definite ranges and at a definite locations, such as around and between the points ten and eleven twelfths of the way through the dialogues. Similarly, positive concepts, like the forms, virtue, the gods, goodness, justice, and the soul, tend to occur in distinct and equally definite ranges. These tendencies are never absolute, but the mixture of concepts in these ranges is clearly dominated either by more negative or by more positive concepts…

Za ovo Kennedy daje niz primjera u članku, ali možda slika govori najjasnije (prva se odnosi na Simpozij, druga na Fedon):

harmonični i disharmonični intervali u Simpoziju

harmonični i disharmonični intervali u Fedonu

Ima još zanimljivih stvari u članku, a i na Kennedyjevom blogu, o čemu u budućim zapisima. Za kraj, zaključak članka:

There are now several kinds of evidence that Plato’s dialogues have a stichometric structure: the lengths of speeches, the alignment of some speeches and key concepts with the twelfths, the parallel passages, and the parallel negative and positive ranges. The musical interpretation of these features is natural and coherent: a twelve-note scale with harmonic and dissonant ranges underlies the surface narrative of the dialogues. The evidence and its interpretation fit the historical context: stichometry was a common practice and applied to Plato’s dialogues, allegory was widely debated, the new mathematics was promoted by Plato and the Academy, the numeric representation of musical scales and harmonic theory were well-known, Plato’s correspondents, colleagues, and followers associated him with Pythagoreanism, and the Neo-Pythagoreans made the scale of twelve, regularly spaced notes part of their studies of the metaphysics allegedly hidden in the dialogues. … Though the evidence reported here will need to be verified and debated, it does clarify, in a surprising way, Aristotle’s once puzzling view that Plato was a Pythagorean.

matematika i muzika (Šikić i Šćekić)

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Kupih novo izdanje (Profil, 2013.) knjige Zvonimira Šikića i Zorana Šćekića Matematika i muzika. Malo sam guglao povodom toga pa pronađoh da Šikić ima blog (!), i na njemu prvo poglavlje te knjige (u dva zapisa):

Pitagora i matematička harmonija

Harmonija svijeta

Treba reći da to početno poglavlje nije reprezentativno za sadržaj knjige, koji je puno više stručan nego u tom poglavlju. Ja sam naučio podosta iz nje, a nisam još sve pročitao (ali nisam znao ništa o muzici, tako da…) Glavni problem je kako se ono što našem sluhu zvuči skladno odnosi spram matematičkih omjera mjerljivih veličina (duljina žica instrumenata i sl., odnosno frekvencija). O tome u ulomku na novom izdanju bloga:

matematika i muzika? (ulomak iz Z. Šikić/Z. Šćekić, Matematika i muzika)

Evo i muzike Zorana Šćekića:

Works in just intonation

Inače, zabavno mi je da engleski nazivi za dvije vrste ugađanja (just intonation i equal temperament) čuvaju Platonovo razlikovanje između pravednogjednakog. 🙂