Here the arithmetic does something the ear cannot: it proves a negative. De institutione musica II.30–31 shows that the whole tone, the ratio 9:8, cannot be divided into two equal semitones. If the halves were equal, "duo dimidia iuncta unum necessario efficerent tonum", and the terms would stand in a superparticular proportion that 9:8 does not admit. What remains are two unequal parts — the limma (256:243) and the apotome — and the number 243 that dominates these pages is the denominator of the smaller. This is the mathematical fact from which the whole later history of temperament descends: every tuning compromise from the medieval comma to equal temperament exists because this proof is true. It is also the clearest case of Boethius's own hierarchy in action. The proposition is inaudible; no ear distinguishes limma from apotome reliably. It is known by ratio alone, which is precisely why he takes reason and not hearing as the judge.
Boethius, De institutione arithmetica libri duo · De institutione musica libri quinque, ed. Gottfried Friedlein (Leipzig: Teubner, 1867) — read from page images, with the critical apparatus separated from the text
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INST. MUS. II, 30. 2603 proportioni, integri esse semitonii probaretur, duo di- midia iuncta unum necessario efficerent tonum. . Nunc | autem cum non sit extremorum terminorum sesquioctava proportio, manifestum est haec duo spatia proprie tono- rum dimidia non videri. Quicquid enim cuiuscunque est 5 dimidium, id si duplicetur, illud efficit, cuius dicitur esse Jimidium. Si vero illud inplere non possit, geminata particula minus est parte dimidia, si vero superfluat ac supervadat, plus est parte dimidia. Praeterea probabun- tur autem LXV. DXXXVI. non facere sesquioctavam pro- 10 portionem, si LVIII. XLVIIII. unitatibus comparentur, si octava pars LVIIII. XLVIIII. eisdem secundum eas, quae in arithmeticis dictae sunt regulas aggeratur. Quae quo- niam in integris numeris non consistit, idcirco eandem octavam partem relinquimus lectorum diligentiae compu- 15 tandam. Liquet igitur eam proportionem, quae in -CCLVI-. - et -CCXLIII- est constituta, non esse integrum dimidium toni. Quocirca id, quod vere semitonium nuncupatur, pars toni minor est quam dimidia. De maiore parte toni, in quibus minimis numeris constet. 20 XXX. Reliqua igitur pars, quae maior est, apotome nun- cupatur a Graecis, a nobis vero potest vocari decisio. Id enim natura fert, ut, quotiens aliquid secatur, ita ut non aequis partibus dividatur, quanto minor pars dimidio minor est, tanto maior pars eademque auctior dimidium vincat. ?5 Quantum igitur semitonium minus integro dimidio toni minus est, tantum apotome toni integrum superat dimidium. Et quoniam docuimus semitonium in -CCLVI- 1 duo om. i. || dimidia] semitonia o. 9 sit] sint f. 11 unitatibus om. g, k. 1*7 est] sit f. 18 toni om. g, k. 20 In quibus numeris maior pars toni sit, h, k. 21 -XXVI- h. [| apetomae g, h; apotoma i, k. 23 quoties g, k. à 24 pars om. o. — 25 eademque auctior] eadem quae est auctior 09; eadem auctior est f,i, k; eadem auctior g, h. .26 minus est integro i. 21 minus om, g. || integram i; integri et supra versum il um o. 28 dimidium vincit i; dimidium
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