Just intonation in the Renaissance



Footnotes:

1 [00:30] See our episode about temperaments in the Renaissance.

2 [00:36] For example, keyboard instruments have exclusively fixed pitches while wind instruments or non fretted string instruments (like the violin for example) are rather flexible.The most flexible instrument however is the human voice; it has complete freedom to tune each note as he wishes.

3 [01:11] See Zarlino, Istitutioni, Seconda parta, Cap. 45 (p. 157).

4 [01:29] See Giovanni Battista Benedetti, Diversarum speculationum mathematicarum, et physicarum liber (Torino: Bevilacqua, 1585) (google books). Later on, Vincenzo Galilei discussed the problem in his Discorso intorno all'opera di messe Gioseffo Zarlino da Chioggia... (Firenze: Marescotti, 1589) (imslp)

5 [02:40] See our episode about temperaments in the Renaissance.

6 [06:37] The famous song Doulce memoire by Pierre Sandrin for example, goes down one comma during its A part. This part is repeated, so by the beginning of the B part we are already 2 commas below the starting level. Furthermore, some sections that seem simple, like this example below from Michaelangelo Rossi’s madrigal “Per non mi dir”, migrate 3 commas down - which is about a semitone(!). The piece in total has a migration range of 11 commas(!) 
https://sites.google.com/site/earlymusicsources/home/more/youtube/justintonation/rossi-with-numbers-and-tonez.jpg?attredirects=0
7 [07:20] See the music example below: a) the solution presented in the video with the “too big semitone” on the Alto part. b) additional solution that have the bass C0 D0 C0. This makes the alto part move normally (with only one comma difference between the two notes: C0 B-1 C0). However, if this is so, then the top voice must take the big semitone of E-1 F+1. Thus, the problem was not solved, it was just transferred to another place. Also, in solution (b) the Alto is not pure with the other voices on the first bit of the second bar (see dashed vertical line).

8 [09:11] Examples for modern writings: Easley Blackwood, The structure of recognizable diatonic tunings (Princeton University Press,  1985); Ross W. Duffin, Just Intonation in Renaissance Theory and Practice [link]; Jonathan Walker, Intonational Injustice: A Defense of Just Intonation in the Performance of Renaissance Polyphony [link]; Yves Ouvrard, Jean-Pierre Vidal, & Olivier Bettens, ZARLINO: Just Intonation in Renaissance A Cappella Singing [link].

9 [10:09] Zarlino, Istitutioni, parte seconda, p. 136, cap. 45. In another place (Sopplementi musicali 1588 p. 151) he writes that when singing with instruments the singers will sing according to the instruments, although they are tempered.

10 [10:57] Vicentino describes in his treatise L’antica musica ridotta alla prattica moderna (Rome 1555, fol. 104 [imslp]) a system that is based on an extended meantone temperament that contains a row of fifths ranging from g flat until e sharp. To each of those notes, there is a pure fifths and a pure minor third on a second manual (the fifths and minor thirds are both smaller than pure by a quarter comma in a normal meantone temperament). With this system, it’s possible to have only pure consonances in a vertical sense, but melodic corrections of ¼ comma. This appears to be an almost ideal illusion of Just Intonation, without the problem of pitch migration and comma shifts.

11 [11:08] Nicola Vicentino, advertisement flyer (Venice, 1561) [link].

12 [12:54] Hercole Bottrigari, Il Desiderio, 1594, p.46. The quote in the video is shortened, see here the full quote in Italian: “Et tanto il desiderio loro di perfettare questa sorte d’Aromina con la unione, che oltre che per ciò ne volessero spessissime volte gli avvertimenti convonevoli dal Maestro di capella del Duomo musico in quei tempi molto principale, non raffinavano giamai di amonirsi? amorevolmente trà loro de’ proprij diffetti, et finalmente con lo stare stretti, con le persone insieme, et involti quasi ascosi ne’ manti cosi al buio dimostravano, che si come essi stavano quanto piu potevano con corpi uniti; de quali ne havrebbono certo voluto poter fare un solo; parimente bramavanno di far à loro potere una vera unione delle loro voci respettivamente; onde ne havesse à nascere un’armonia, dirò, quasi celeste.”