Tuning and temperaments in the Renaissance - Part I:

Tuning and temperaments in the Renaissance - Part II:

Main sources

Arnolt Schlick, Spiegel der Orgelmacher und Organisten (1511, Mainz) [Google books, see chapter 8]

Pietro Aron, Il Toscanello (1523, Venice) [imslp, see chapter XLI]

Costanzo Antegnati, L'arte Organica (1608, Brescia) [Google books / Italian transcription]

Michael Praetorius, Syntagma Musicum (1619, Wolfenbüttel) [imslp, see Vol. II pp. 149-158]

Footnotes

1 [part II, 0:50]: When a consonant interval (unison, octave, fifth, fourth, major third) is very close to pure, you can hear the phenomena of beating. It’s like a vibrato of volume (amplitude), getting slower the closer the interval is to it’s pure form. The beating disappears in the moment the interval is perfectly pure. Depending on the type of instrument and on environmental factors, the beating of intervals can be heard easily or not at all. The speed of the beating is a reliable indicator for the size of an interval and can be used to temper consonances by ear with very high precision. Example: narrowing the fifths between d and a by ¼ syntonic comma results in a beating speed of three times per second. However, the beating speed depends on the absolute pitch of the interval. A fifth of the same size, but one octave lower, D-A, beats only 1.5 times per second. In the context of our investigation, the phenomena was mentioned only by Schlick and Praetorius, but they used it only to define if an interval is either pure or impure, never to objectively measure the precise size of intervals.

2 [part II, 4:48]: Rein in German is commonly translated as "pure". The description of an interval as "pure" is nowadays understood as an interval that is physically pure and not beating. However it is not clear from Praetorius' tuning instructions whether the thirds should be absolutely pure and not beating since his terminology is at times inconsistent.

3 [part II, 5:07]: On such micro levels the human ear cannot perceive an absolute size of an interval, let alone describe it. Mostly, it can note if an interval is either pure or impure. Musicians with good ears can also note if the interval is smaller than pure or bigger than pure. It is not possible however to identify by how much it is smaller or bigger - these are just many of shades of gray that humans cannot perceive objectively, and therefore they cannot be communicated through words in a book. [Concerning Schwebung/beating see footnote 1]

4 [part II, 7:35]: The ideal third is never described simply as pure, but always in a more indirect way. This is in contrast to the sources' descriptions of octaves for example: [Aron:] 'perfectly united'/'sempre bene unito', or [Antegnati:] 'appearing as one unified sound'/'due corde [...] paiono una sola'.

5 [part II, 8:09] To interpret the Renaissance sources as a description of that consists of heavy normalization and simplification: namely, to interpret their description of thirds as pure thirds, and to assume that all the fifths are tempered by the same amount.

6 [part II, 8:17]: The first theoretical sources that actually described ¼ comma meantone were only at the end of the 17th century, most prominently Christiaan Huygens [Christiaan Huygens, Lettre touchant le cycle harmonique, 1691, edited in Martinus Nijhoff (ed.), Oeuvres complètes de Christiaan Huygens, Tome Vingtième, Musique et Mathématique], and this is only thanks to the invention of logarithms. Logarithms were discovered by John Napier in 1614 (see his Mirifici Logarithmorum Canonis Descriptio...), but used for the theoretical description of temperaments only later.

7 [part II, 8:26]: In modern times, the most influential scholar who interpreted the Renaissance sources as descriptions of ¼ comma meantone is James Barbour [James Barbour, Tuning and Temperament. A Historical Survey, East Lansing 1953].