As you point out, alternate temperaments on the lute have inherent weaknesses, such as disparities for notes on the same fret that would be better raised as opposed to lowered from equal temperament, and so if one chooses the sharper or flatter variant it applies across every course of the instrument. And you're quite right to cite Galilei, as there was not unanimity on these matters (nor in keyboard tuning). I think what it amounts to is that in such alternate temperaments, the lutenist would have to be careful to leave out certain notes, or try different solutions for fretting accompanying chords (which may not be possible or easily playable). I don’t suppose you know if there are surviving tablatures for lute accompaniment of Gesualdo’s madrigals?
I think it’s absolutely clear from Vicentino, in addition to other theorists both before and after him, that enharmonic spellings of notes (e.g. F# / G♭) do not represent the same pitch class, which stands in direct conflict with 12-tone equal temperament; the chromatic madrigals of Gesualdo, Vicentino, Monteverdi and others associated with the Ferrarese school arguably should be sung justly, following the syntonic harmonic ratios where possible. As late as Leopold Mozart you find advocacy for enharmonic notes being different in pitch, so that the system of intonation is open, not closed.
The practice of the time up to about 1400 seems to have favoured strictly Pythagorean tuning, so that a note such as F# would be higher in pitch than the corresponding G♭, and which might have some bearing on the fact that major or minor thirds were not viewed as consonances that could be tolerated in final chords. By the start of the renaissance however, the lower harmonic position of the major third (5/4) seems to have come into play as it was especially noticed that tuning just fifths successively flatter (B♭, E♭, A♭, D♭, G♭, etc.) resulted in much sweeter consonances when they were utilised as the third, rather than going the other way through the sharps: the bug that the Pythagorean comma is about the same size as the syntonic comma actually turned out to be something of a feature. Split keys on keyboards catering for the flat/sharp comma difference began to appear. Rather than awkwardly respelling a D major triad as D-G♭-A, the Renaissance practice described by Vicentino went with simplicity (and the harmonic series) so that a note such as F# is flatter than its near-relative G♭, thus defining the sizes of chromatic and diatonic semitones quite differently from the Pythagorean method in addition to reversing the difference in magnitude.
From memory (i.e. reading Vicentino’s book about ten years ago) I think he actually was aware of the phenomenon of comma drift in unaccompanied music, though he didn’t seem to have a solution other than implementing an expanded equal-temperament scheme, which entailed 31 notes to the octave and a mean tone rather than the two major (9/8) and minor (10/9) whole tones, and thus his archicembalo could accompany his music without drift. The compromise on the size of the whole tone is much more to my personal taste than having impure thirds, but as described above Vicentino needed extra keys for his keyboard to deal with the blunted fifths (and seconds/ninths).
Anyway, this apparent distraction is returning fairly closely to the original questions posed a year and a half ago by the first poster above, so I propose to take these in turn:
My understanding is that singers always tune to the purest just intervals, and to me it seems as if this would erase the sense of key-color inherent in say meantone keyboard music. Is this true?
It is true that (good) singers will try to tune to the just intervals, which would seem to deny the sense of key-colour, since a justly-tuned chord of the same spacing should possess the same ratios at whatever pitch class. However, singers do not always sing pure intervals, mainly to recover pitch memory. As rwendell pointed out, the triadic progression I–IV–ii–V–I, if sung in syntonic just intonation with notes in each chord being held as pivots, would result in a pitch drop, because at some point an interval that should be sung as a major whole tone would be needed to be sung as a minor whole tone. In practice, good singers who retain the pitch of the original chord of I will very slightly compromise the tuning somewhere to avoid the pitch from falling. In doing so, this does tend to re-inject something of the flavour of different chords within a harmonic structure possessing noticeable colours.
Have choral (a capella) composers historically enjoyed the same sense of key equality that many composers in the 20th c. take for granted due to the prominence of 12-TET?
I think it’s clear that many composers of the pre-19th century period were well aware of the colours of different keys and exploited them, although the further back you look, you begin looking at the various modes rather than the simple major/minor dualism – and each of the modes obviously had their own “characters”. So your question seems meaningless historically, in that “key equality” is a badly defined concept except from a 20th/21st century perspective where 12-tone equal temperament is in almost total dominance.
Furthermore, if choirs always strive to tune intervals justly, what are the purest intervals?
The ones that are never challenged are the perfects: octave (2/1), fifth (3/2), and fourth (4/3). As I described above, for all the other intervals (seconds, thirds, sixths, and sevenths), singers are presented with a variety of options owing to following the circle of fifths on the one hand, or the natural harmonic series on the other.
For instance, if an exercise was performed in which the basses held a solid low C and the sopranos moved chromatically from a C up the octave (slowly, carefully tuning each note), what would be the intervallic relationship between each note and the pedal C?
You’ll get different answers depending on which system you’re following. Also, when you say “the sopranos moved chromatically”, it is also the case that enharmonically-spelled noted are different from one another, and have different associated intervallic ratios, so first of all you’d have to decide whether your sopranos were going to ascend from C to C# or from C to D♭, as well as deciding whether they were following strict Pythagorean tuning or harmonic syntonon tuning = four possible choices.
I assume 9/8 for D, 5/4 for E, 4/3 for F, 3/2 for G, but what of the other notes?
Choosing 5/4 for E explicitly favours the natural harmonic series over the Pythagorean 81/64 ratio, which is a syntonic comma sharper. It’s safe to assume that in keeping with that choice, the 5/3 ratio for A against C is probably the safer bet; 16/9 for B♭, and 16/15 for B.
For instance, how would Bb be tuned? Would it be 7/4, 16/9, or 9/5?
Good question. Classically, the seventh harmonic (along with the 11th, 13th, and 14th among the first four octaves) was regarded as an out-of-tune note in the harmonic series, so the harmonic seventh (7/4) would not be the “natural” choice, if you can excuse the pun. Nevertheless, some composers were happy to exploit it (there’s a wonderful moment in Beethoven 8 where Ludwig has all his horns and trumpets play the harmonic seventh, fortissimo!) as circumstances demanded. In your example of sopranos ascending against a pedal, the 16/9 tuning a major whole tone below the octave is the most likely choice. The 9/5 is the “octave minus a minor whole tone” version of that interval (the syntonic alternative), and unlikely to be favoured for the same reason that ascending from C to D via a 10/9 whole tone would be viewed as being a comma flat of the true note: going from G (3/2) to A (5/3) has already used the minor whole tone in that part of the scale. If on the other hand, the ascent from G had been a major whole tone, arriving at the Pythagorean 27/16 version of A, then the choice of a Pythagorean or syntonic semitone would result in 16/9 and 9/5 respectively.
Or am I mistaken about this whole business of singers always ideally tuning to a set of pure intervals in any given situation?
Just remember that an ideal is a standard of perfection that no choir ever meets in practice, though some obviously come closer than others! The art of singing isn’t confined to exact frequency ratios, and when making choral music issues of blend, rhythm, dynamics, tempo, vowel timbre, and vocal colour also compete for a director’s attention as variables similarly under his or her control as well as the singers’ own internal regulation. Thus the sort of theoretical discussion we have here is in strict isolation from all the other pressures that will be presented in rehearsal of a specific piece of music, though the singers and director should naturally be aware of them and promote or instill as necessary.
I've read that the Hilliard Ensemble insists on Pythagorean tuning for medieval music, but this seems to contradict my understanding that a capella singers of all periods essentially strive to have the same purity of intervals. Am I wrong?
I think you’ve slightly misinterpreted the issue, in that from the medieval to the renaissance period, there does seem to have been a shift in opinion on how to achieve the best tuning. I for one am in agreement with Paul Hilliard as viewing Pythagorean tuning to be generally better (i.e. historically appropriate) for the medieval repertoire, and syntonic tuning to be musically better for the a cappella
music of the renaissance, given that the ideal of a choir singing without any form of instrumental accompaniment was actually a lot rarer than one would expect from performance practice nowadays, outside of the very best choirs.
This is a “horses for courses” type of argument. In a cappella
music from the 19th century onwards, the necessity for equal temperament was pushed forward in order to be able to accomodate modulations to distant keys. However, this provides no excuse for a choir reading a work by Brahms, Schönberg, or Schnittke, say, to sing a triad with badly-tuned thirds. In these cases, you should tune to the just ratios vertically, even if this involves fudging intervals horizontally here and there when reading harmonically ambiguous accidentals.
Would it be appropriate to sing a M3 as something other than 5/4 (maybe several cents sharp or flat) in certain circumstances even in a capella singing?
Intervals being a few cents out here and there is unavoidable with human singers, especially in faster-moving music: we’re neither computers nor precisely modulated sine wave generators. If a slightly modified, impure interval is demanded by a certain context, then yes, it’s appropriate.