1 Even the Italian origin of the ballata is no longer certain. Aurelio Roncaglia has proposed that the form was linked to the much earlier Arabo-Andalusian poetic genre of zajal, which has a similar formal structure. See his ‘Laisat estar lo gazel (contributo alla discussione sul rapporto fra lo zagial e la ritmica romanza)’, Cultura neolatina, 9 (1949), 67–99; ‘Di una tradizione lirica pretrovatoresca in lingua volgare’, Cultura neolatina, 11 (1951), 213–50; and ‘Sequenza adamiana e strofa zagialesca’, in La sequenza medievale. Atti del convegno internazionale. Milano, 7–8 aprile, 1984, ed. Agostino Ziino (Lucca, 1992), 141–54. The extensive bibliography on this topic is summarised in Heijkoop, Henk and Zwartjes, Otto, Muwaššaḥ, zajal, kharja: Bibliography of Strophic Poetry and Music from al-Andalus and Their Influence in East and West (Leiden, 2004)Google Scholar.

2 D'Agostino, Gianluca, ‘On the Ballata Form(s) of Fifteenth-Century Italy: A Case of Historical Misunderstanding’, in ‘ Et facciam dolçi canti’: Studi in onore di Agostino Ziino in occasione del suo 65° compleanno, ed. Antolini, Bianca Maria, Gialdroni, Teresa M. and Pugliese, Annunziato (Lucca, 2003), 1: 295–330 Google Scholar, at 296.

3 For the dating of this source, see Rijk, E. Abramov-van, ‘Evidence for a Revised Dating of the Anonymous Fourteenth-Century Italian Treatise Capitulum de vocibus applicatis verbis ’, Plainsong and Medieval Music, 16 (2007), 19–30 CrossRefGoogle Scholar, where the evidence is presented in the form of a citation that clearly permits dating the Capitulum later than 1332, definitely linking it to da Tempo's Summa.

4 Quoted in Burkard, Thorsten and Huck, Oliver, ‘Voces applicatae verbis: Ein musicologischer und poetologischer Traktat aus dem 14. Jahrhundert’, Acta Musicologica, 74 (2002), 1–34 CrossRefGoogle Scholar, at 14.

5 da Tempo, Antonio, Summa artis rithimici vulgaris dictaminis, ed. Andrews, Robert, Collezione di opere inedite o rare 136 (Bologna, 1977), 49 Google Scholar. The translations are mine unless otherwise indicated.

6 Such a style of performance surfaces in the description of a dance with singing in the Diaffonus, a Latin poem by Giovanni del Virgilio written in 1315–16. Here a young man began by singing the ritornello (‘Ornatae Juveni quae me sine jure peremptat / Murmure multorum funde, Cupido, preces’), which was then repeated three times by the group of young men and women, alternating with the strophes performed by the leader. See Marrocco, W. Thomas, ‘The Ballata. A Metaphoric Form’, Acta Musicologica, 31 (1959), 32–7CrossRefGoogle Scholar, at 33–4; and Bisanti, Armando, ‘Suggestioni classiche, mediolatine e romanze nel Diaffonus di Giovanni del Virgilio e ser Nuccio da Tolentino’, Schede Medievali, 46 (2008), 119–68Google Scholar.

7 de’ Prodenzani, Simone, Sollazzo, ed. Milani, Matteo, Biblioteca romanica: Studi e testi (Alessandria, 2004)Google Scholar. In fact, the metric form of Sollazzo's verses adheres more closely to the barzelletta than to the ballata. They are written in octonario (eight-syllable) verse, as was characteristic of the genre in the Quattrocento, instead of hendecasyllables and/or settenari, typical of the Trecento ballatas. Howard Mayer Brown discussed the question of how these ballatas might have been sung and danced: ‘To judge from Il Saporetto, Sollazzo sang the poems alone while the guests danced. Prodenzani neither mentions instruments, nor introduces the possibility that the dancers sang a refrain.’ ‘A Lost Trecento Dance Repertory Rediscovered’, in Studien zur italienischen Musikgeschichte, ed. Friedrich Lippmann, Analecta musicologica 30/1 (Laaber, 1998), 1–13, at 4.

8 Regarding the hierarchical value of contemporary poetic forms, Dante proposed, in his De vulgari eloquentia, that the poetic genres in which sublime ideas are treated must have no ritornellos. Thus, the noble canzone has no ritornello, whereas the ballata, as a genre with ritornello, is less elevated: ‘Dicimus ergo quod cantio . . . est equalium stantiarum sine responsorio ad unam sententiam tragica coniugatio’ (2.VIII.8) (I say, then, that the canzone is a linked series of equal stanze in the tragic style without a refrain and for a single train of thought.) Alighieri, Dante, De vulgari eloquentia, in Dante in Hell: The ‘De vulgari eloquentia ’, ed. Welliver, Warman (Ravenna, 1981)Google Scholar, 115.

9 I leave aside the complex topic of laude, religious sung poetry. Although most often shaped in ballata form, they are long polystrophic compositions, designed for a group singing during processions and other ceremonies, in which some kind of action or movement was supposed.

10 Jennings, Lauren McGuire, Senza Vestimenta: The Literary Tradition of Trecento Song (Farnham, UK, and Burlington, VT, 2014)Google Scholar, 22: ‘Six hundred and thirty seven secular songs with vernacular texts from the Italian ars nova survive today: 414 ballate, 196 madrigals, and 28 cacce.’

11 D'Agostino, ‘On the Ballata Form(s)’, 298.

12 Fischer, Kurt von and D'Agostino, Gianluca, ‘Ballata’, The New Grove Dictionary of Music and Musicians, 2nd edn (London, 2001), 2: 563–5Google Scholar, at 564.

13 Ziino, Agostino, ‘Rime per musica e danza’, in Storia della letteratura italiana: il Trecento (Rome, 1995), 2: 455–530 Google Scholar, at 474.

14 Capovilla, Guido, ‘Note sulla tecnica della ballata trecentesca’, in L'Ars Nova italiana del Trecento IV (Certaldo: Centro di Studi, 1978), 107–47Google Scholar, at 125–6.

15 See Rijk, Elena Abramov-van, Parlar cantando: The Practice of Reciting Verses in Italy from 1300 to 1600 (Bern, 2009), 81–7Google Scholar. For a discussion of this and other aspects of the ballata, see Gianluca D'Agostino, ‘On the Ballata Form(s)’, 279.

16 This can be understood as ‘singing without dancing’ or as ‘solo singing’. Burkard and Huck, ‘Voces applicatae verbis’, 11.

17 Da Tempo, Summa, 52.

18 Gallo, Franco Alberto, Ballata, in Handworterbuch der musikalischen Terminologie, ed. Eggebrecht, H. (Wiesbaden, 1981), 1: 1–7 Google Scholar, at 3. See also Baumann, Dorothea, ‘Ballata’, in Die Musik in Geschichte und Gegenwart, Sachteil, new rev. edn (Kassel, 1994), 1: 1157–63Google Scholar, at 1158.

19 Bembo, Pietro, Prose della volgar lingua, ed. Dionisotti, Carlo (Milan, 1989)Google Scholar, 153.

20 See Abramov-van Rijk, Parlar cantando, 59–63.

21 Ibid., 49.

22 Like madrigals, ballatas can be courtly, lyric, amorous, narrative, pastoral, realistic, satiric, moralising and philosophic. See Ziino, ‘Rime per musica e danza’, 487.

23 Ziino, ‘Rime per musica e danza’, 474–5.

24 Komponieren in Italien um 1400. Studien zu dreistimmig überlieferten Liedsätzen von Andrea und Paolo da Firenze, Bartolino da Padova, Antonio Zacara da Teramo und Johannes Ciconia, Musica Mensurabilis 6 (Hildesheim, 2012), 163.

25 Ibid.

26 Da Tempo, Summa, 49.

27 Florence: Biblioteca Medicea-Laurenziana, Palatino 87 (Squarcialupi Codex), fol. 156. Cf. Corsi, Giuseppe, Poesie musicali del Trecento (Bologna, 1970), 136 Google Scholar.

28 Folkerts, Menso, Euclid in Medieval Europe (Winnipeg, 1989)Google Scholar.

29 See Crombie, Alistair C., Augustine to Galileo, vol. 1: Science in the Middle Ages, 5th to 13th Centuries (London, 1979), 25–6Google Scholar.

30 Schüling, Hermann, Die Geschichte der axiomatischen Methode im 16. und beginnenden 17. Jahrhundert: Wandlung der Wissenschaftsauffassung (Hildesheim, 1969)Google Scholar, 21; see in particular ch. 3, ‘Theorie und Anwendung der euklidischen Methode in der Scholastik’, 20–3.

31 Benjamin, Francis S. Jr and Toomer, G.J., Campanus of Novara and Medieval Planetary Theory (Madison, WI, 1971)Google Scholar.

32 Villani, Giovanni, Nuova Cronica, ed. Porta, Giovanni, 3 vols. (Parma, 1991)Google Scholar, 3: 198 (Bk. 12, ch. 94, More about the greatness, status, and magnificence of the community of Florence.) I garzoni che stavano ad aprendere l'abbaco e algorisimo in VI scuole da M in MCC. E quelli che stavano ad aprendere gramatica e loica in IIII grandi scuole da DL in DC. 12, XCIV: Ancora della grandezza e stato e magnificenza del comune di Firenze.

33 ‘In his treatise on Optics Euclid states that the ratio between the apparent size of objects depends on the optical angle within which they are contained, while Alhazen, reformulating this axiom, states that we can be certain of the true size of an object only if, in addition to the angle, we also know the distance between the eye and the object measured along the central axis of the visual pyramid. In terms of trigonometry it is precisely these two factors, angle and distance, which are used to measure the dimensions of an object.’ Camerota, Filippo, ‘Teaching Euclid in a Practical Context: Linear Perspective and Practical Geometry’, Science and Education, 15 (2006): 323–34CrossRefGoogle Scholar, at 325. To recall, Landini's father was the renowned Florentine painter Jacopo da Casentino.

34 Verdon, Timothy, ‘Alza la voce con forza: L'iconografia del Campanile e l'annuncio cristiano’, in Alla riscoperta di Piazza del Duomo in Firenze, 7 vols. (Florence, 1994), 3: 85–115 Google Scholar.

35 ‘Preter hoc ad laudis sue cumulum accedit quod gramaticam atque dyaleticam plene didicerit.’ Villani, Filippo, De origine civitatis Florentie et de eiusdem famosis civibus, ed. Tanturli, Giuliano (Padua, 1997)Google Scholar, 410.

36 ‘Fioriva ancora in quel tempo Francesco degli Organi, musico e pratico . . . elli con ogni artista e filosofo gìo disputando non tanto della sua musica, ma in tutte l'arti liberali, perché di tutte quelle in buona parte erudito si n'era’ (bk. III). Prato, Giovanni Gherardi da, Il Paradiso degli Alberti, ed. Lanza, Antonio (Rome, 1975)Google Scholar, 165.

37 Ibid., 164. Today Grazia de’ Castellani is known for his philosophical works and comments on Dante, but, as Paolo Orvieto notes, though his mathematical works are now lost, he is mentioned with much admiration in numerous contemporary treatises on abacus. S.v. ‘Castellani, Grazia’, in Dizionario Biografico degli Italiani (1978), www.treccani.it/enciclopedia/grazia-castellani_%28Dizionario-Biografico%29/ (accessed 27 April 2016).

38 Rijk, Elena Abramov-van, ‘Who was Francesco Landini's antagonist in his defense of Ockham?’, Philomusica-online, 14 (2015), 1–24 Google Scholar, at 12–17.

39 Villani, De origine civitatis Florentie, 410.

40 Stinson, J.J., ‘Francesco Landini and the French Connexion’, Australian Journal of French Studies, 21 (1984), 266–80CrossRefGoogle Scholar, at 278, lists six modern editions of Landini's poem in defense of Ockham. The first is that in Il Paradiso degli Alberti: ritrovi e ragionamenti del 1389, romanzo di Giovanni da Prato dal codice autografo e anonimo della Riccardiana, ed. Alessandro Wesselofsky, 2 vols. (Bologna, 1867), 2: 296–301. For recent editions, see Lanza, Antonio, Polemiche e berte letterarie nella Firenze del primo Rinascimento, 1375–1449 (Rome, 1971), 233–8Google Scholar; and Michael Long, ‘Musical Tastes in Fourteenth-Century Italy: Notational Styles, Scholarly Traditions, and Historical Circumstances’, Ph.D. diss., Princeton University (1981), 219–22, with an English translation on pp. 136–41. Recently I proposed a new candidate for Landini's objections in my article ‘Who was Francesco Landini's antagonist’.

41 Long, ‘Musical Tastes in Fourteenth-Century Italy’, 134. See also Stinson, ‘Francesco Landini and the French Connexion’, 271.

42 This ballata ‘condensa nel breve giro d'una ballata monostrofica la Weltschauung d'una invettiva “in laudem loyce Ocham” lunga centottanta esametri latini’. ‘Per un'analisi delle strutture compositive nella musica di Francesco Landini: il caso della ballata Contemplar le gran cose (3¹)’, in ‘Col dolce suon che da te piove’: Studi su Francesco Landini e la musica del suo tempo. In memoria di Nino Pirrotta, ed. Antonio Delfino and Maria Teresa Rosa Barezzani (Florence, 1999), 259–321, at 262.

43 The fundamental work on Ockham's logic is Moody, Ernest A., The Logic of William of Ockham (New York, 1935), 252–80Google Scholar. See also Weinberg, Julius R., ‘Ockham's Theory of Scientific Method’, in Ockham, Descartes and Hume. Self-knowledge, Substance and Causality (Madison, WI, 1977), 22–32 Google Scholar. For a useful summary of Ockham's methods of demonstration, see Miller, Jon, ‘Spinoza and the A Priori ’, Canadian Journal of Philosophy, 34 (2004), 555–90CrossRefGoogle Scholar.

44 Miller, ‘Spinoza and the A Priori’, 556.

45 Ibid., 556–7.

46 See Weinberg, ‘Ockham's Theory of Scientific Method’.

47 Burkard and Huck, ‘Voces applicatae verbis’, 14.

48 Alighieri, Dante, Opere minori, 2 vols. (Milan and Naples, 1984) 1: 815–23Google Scholar. See also Bellomo, Saverio, Filologia e critica dantesca (Brescia, 2008), 152–5Google Scholar.

49 This type of rhyming is described by Antonio da Tempo in his Summa artis rythmici vulgaris dictaminis as ‘equivocus’, that is, the identical words with different meaning (Da Tempo, Summa, 87–90). See also Memelsdorff, Pedro, ‘ Equivocus: per una nuova lettura del rapporto testo-musica nel Trecento italiano’, in L'Ars Nova italiana del Trecento, VII: ‘Dolci e nuove note’, ed. Zimei, Francesco (Lucca, 2009), 143–87Google Scholar. For the correct interpretation of the ‘equivoci’ in the Detto d'amore, see the relevant entries in the Enciclopedia Dantesca (Rome, 1970).

50 The query whether it was a priori or a posteriori seems not to be so relevant for the ballata form since the reasoning in it is not rigorously syllogistic.

51 In this regard, the label poesia per musica, usually understood as the poetic texts of scant range and quality set to music, almost loses its sense when related to texts such as the ballata, Abbonda di virtù.

52 Baumann, Dorothea, Die dreistimmige italienische Lied-Satztechnik im Trecento (Baden-Baden, 1979)Google Scholar.

53 Vela, Maria Caraci, ‘Le intonazioni polifoniche de La fiera testa che d'uman si ciba: problemi di contestualizzazione e di esegesi’, in Musica e poesia nel Trecento italiano. Verso una nuova edizione critica dell'Ars nova (Florence, 2015), 93–141 Google Scholar. Of course, there were a number of tricks to avoid them, such as pauses (as in bar 4), syncopation and ‘delayed’ parallel motion (bar 3) (ibid., 102–5).

54 Baumann, Dorothea, ‘Bartolino da Padova’, in Die Musik in Geschichte und Gegenwart. Personenteil, 2nd edn (Kassel etc., 1999) 2: 406–9Google Scholar, and Fischer, Kurt von and D'Agostino, Gianluca, ‘Bartolino da Padova’, in The New Grove Dictionary of Music and Musicians, 2nd edn (London, 2001), 2: 820–2Google Scholar.

55 Corsi, Poesie musicali del Trecento, 255.

56 Translated by Giovanni Carsaniga, with some slight changes. www.lib.latrobe.edu.au/MMDB/composer/H0145019.HTM.

57 On the full spectrum of variants of Trecento ballata structure, see Pagnotta, Linda, Repertorio metrico della ballata italiana, secoli XIII e XIV (Milan and Naples, 1995)Google Scholar.

58 Ziino, Agostino, ‘Testi laudistici musicati da Palestrina’, in Atti del convegno di studi palestriniani, 28 settembre–2 ottobre 1975, ed. Luisi, Francesco (Palestrina, 1977), 381–408 Google Scholar; idem, ‘La ballata in musica dalla frottola al madrigale: campioni per una ricerca’, in La letteratura, la rappresentazione, la musica al tempo e nei luoghi di Giorgione, ed. Michelangelo Muraro (Roma, 1987), 259–72; idem, ‘Minima palestriniana: Giovanni Pierluigi da Palestrina e la ballata’, in Note su note: Rassegna di contributi musicologici curata e realizzata dalla cattedra di Storia della musica del Dipartimento di Scienze storiche, antropologiche e geografiche dell'Università degli studi di Catania, 2 (Catania, 1994), 221–72.

59 Newcomb, Anthony, ‘The Ballata and the “Free” Madrigal in the Second Half of the Sixteenth Century’, Journal of the American Musicological Society, 63 (2010), 427–97CrossRefGoogle Scholar, at 429.

60 Cited in Ziino, ‘Minima palestriniana: Giovanni Pierluigi da Palestrina e la ballata’, 225.

61 Franco Piperno is sure that ‘autori e fruitori del tempo percepissero senza difficoltà come ballata un testo che si discostasse dallo schema trecentesco solo per la rinunzia al predetto ritorno rimico’ (‘Ballate in musica, madrigali “a ballata” e gli ariosi di Antonio Barrè: predilezioni metriche e formali del madrigale a Roma a metà Cinquecento’, in ‘Et facciam dolçi canti’: studi in onore di Agostino Ziino in occasione del suo 65 compleanno, ed. Bianca Maria Antolini and Teresa M. Gialdroni (Lucca, 2003), 1: 459–89, at 462.

62 Spinoza, Benedict de, Ethics and the Improvement of the Understanding, trans. R.H.M. Elwes (Amherst, NY, 1989)Google Scholar, 263.

63 Quoted from http://aleph0.clarku.edu/~djoyce/elements/bookI/propI6.html. At present, this theorem is demonstrated differently, through drawing the altitude from the vertex angle of the given isosceles triangle and proving the congruency of the two resulting rectangular triangles. However, when Euclid was dealing with theorem 6 of Book 1, he was quite limited in the tools available for demonstration, consisting of a number of definitions, principles and axioms exposed in the introduction, and of the five previously demonstrated questions. That is why his demonstration appears to be fairly contrived as compared to the modern one. It was the nineteenth-century German mathematician David Hilbert who rearranged the axiomatic system. This, on the one hand, made possible more coherent and elegant demonstrations on the earlier stage of the studies, but, on the other hand, the gradual logical proceeding of Euclid's argumentations became much less evident for the student.