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Singing and Intonation

Posted: 10 Apr 2009 15:57
by jh87
Hello. These may be pretty elementary questions but I'm having difficulty finding a straight answer.

Basically I'm wondering how an a capella ensemble tunes itself (or for that matter, any ensemble which is composed entirely of pitch-flexible instruments). The source of my curiosity stems from a recent interest in keyboard tunings. I think it's fairly clear that a large aspect of keys having their own 'colors' according to pre-20th c. musicians and theorists stems from the slightly different character of each key due to conventions of tuning. I'm mainly curious to know whether or not such key-colors can be extended to say the music of Gesualdo. 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? 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?

Furthermore, if choirs always strive to tune intervals justly, what are the purest intervals? 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? I assume 9/8 for D, 5/4 for E, 4/3 for F, 3/2 for G, but what of the other notes? For instance, how would Bb be tuned? Would it be 7/4, 16/9, or 9/5? Or am I mistaken about this whole business of singers always ideally tuning to a set of pure intervals in any given situation? 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? 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?

I would assume that in the case of singing with continuo that the singers would always tune to that. This seems to suggest to me that choice of continuo instruments (particularly harmony instrument) is very important to a successful realization of a piece. For instance, lute would force the singer into 12-TET while harpsichord may require meantone singing. I had always assumed it was fairly arbitrary what continuo instruments were used but I take it I was wrong. Confirm? Deny?

Thank you. I'm just baffled by this whole business of choral intonation.

Re: Singing and Intonation

Posted: 12 Apr 2009 08:44
by vaarky
Very intg Q.

I think it's even more complex in theory than you mention, because a continuo tuned to the original key won't be true to the chords and keys as they may shift during the song. For example, an E that's the major third to the original key of C would need to be tuned ever so slightly differently once the piece shifts to Am for a while and that "same" E is now acting as the fifth to the A. So a continuo that's slavishly tied to the intervals of the old C scale will sound a little too tempered to the very refined ear when the intervals now need to be true to an A minor. However, this level of nuance is really, truly lost on most individuals even most professional singers, even if they're people who are bothered by equal-tempered tuning.

My personal sense is that keys have their own disctinctive color. People notice that choirs "always go flat" in certain keys (e.g. in F but not G? I don't remember well). I wonder if this ties to (and is a less advanced form of?) the synesthesia that people with perfect pitch experience where the individual pitches have an individual tone color. The various modes have their own characteristic, too, and they used to associate feelings such as "fortitude" with particular modes.

Curious what others have to say too...

Re: Singing and Intonation

Posted: 28 Jul 2009 23:02
by cdu
What a topic! For unaccompanied singing in traditional harmony I think most answers are found by following the implications of the harmonic series and the implications of each diatonic tone and chord. If we divide the major scale into two tetrachords, we see that the re, me, and fa have an implied resolution down to do, while sol fa and si resolve up to do. That being the case, the largest whole step is between fa and sol; also, because re is so close to do, it is a quite small whole step down. And interestingly, when I was first training choirs, I finally realized that often we were having intonation problems on one of those two chords. I thought I heard a low third, but the problem was that the root, usually the bass part, was too high. That was a revelation. I know that I'm over-simpliflying, but the laws of physics are reliable.

I must say, the only theory book that I know that is BRILLIANT in explaining the implications of the harmonic series for every factor of every chord is Schoenberg's book on harmony. A must read. (Leaves Piston and others in the dust.) Until final resolution, every pitch is being pulled in one direction or another and holding against the pull. If the pitch is placed correctly in relation to those two tension-creating factors, good tuning results. Of course, timbre and volume come into play as well. For instance, since a third, major or minor, is so strong a factor, chords can get out of focus and don't 'sound' correctly if there is too much third. I think sometimes neophytes hear that something is wrong and guess that it is tuning, when it is sometimes balance.

For beginners, who want to hear pitches move in one direction or another, practice the sight-singing exercises in Hindemith's Elementary Training for Musicians. The first set of excercises uses only whole steps and three pitches: G, A, and B. Each exercise tonicizes a different note. Sing the excercise with an F tonic; then, sing one with an B tonic. You will hear the A move higher.

I know that the original poster was talking about various tuning systems, but one needs to begin somewhere. I give these remarks as, perhaps, a help to someone just getting started. Once one has a good understanding of traditional harmony in performance, it is not too difficult to figure out the implications of the various church modes, then move to studying the various tuning systems.

Re: Singing and Intonation

Posted: 05 Aug 2009 12:19
by pml
There is ample evidence to suggest that when singers are accompanied by an instrument that is tuned in a particular intonation, they will tend to tune to it rather than work at the purity of intervals, so a starting point is to work a cappella the entire time, with nothing more than the occasional pitch check against a reliable reference.

It's well known that there are two different sizes for the diatonic semitone as opposed to a chromatic semitone - and in Pythagorean tuning, the size difference is reversed compared to the intervallic size derived from the natural harmonic series (Pythagorean chromatic semitones are wider than equal temperament, whereas the natural harmonic series demands larger diatonic semitones). In just tuning with a harmonic major third rather than a Pythagorean one, there should also be two distinct sizes for the whole tone, rather than averaging out the size to a mean tone (hence that term in the tuning of keyboards!).

If Pythagorean tuning is not being observed, then the mediant should be a syntonic comma flat, so that the major third will be made up out of a major whole tone (9/8) and a minor whole tone (10/9) to achieve the 5/4 (=80/64) harmonic ratio rather than the Pythagorean 81/64. The harmonic minor third (6/5) is also a syntonic comma sharper (81/80) than the equivalent Pythagorean interval (32/27).

So the question is, which tones in a particular mode should be the larger or smaller whole tone? The answer in polyphonic music is that this will actually depend on what other notes are already being sounded. So to answer the original question, if there is a pedal note (e.g., C) in the bass being held at a steady pitch, whether the pitch of another part (e.g. A) takes one of the possible alternatives will depend on other notes. If an E (5/4) or F (4/3) is being sounded, the A will normally be a syntonic comma lower (5/3) to achieve a perfect fourth or just harmonic third; on the other hand if a major whole tone D (9/8) has already been established against the C (as the dominant of the dominant), then the A will tend to take the higher Pythagorean tuning (27/16) to avoid a ‘wolf’ fifth.

It is quite possible to write diatonic progressions that will involve "comma drift", where instead of a mixture of major and minor whole tones and diatonic semitones, one only has major (or minor) tones, which can cause absolute pitch to shift up or down a syntonic comma when sung in sequence melodically. cdu already touched on this in the relationship between three simple notes - do, re, and mi will work just as well - when one of these notes is by implication the "tonic" note. If do is the tonic then the just harmonic mi will be a comma flat when compared to mi in the case where re has become the "new" tonic.

In practice, one needs an choir with excellent tuning to perceive comma drift; what usually happens instead is that there is enough "rough" tuning - not exactly out of tune, but imperceptibly short of just intonation - to even out the instances when a major whole tone has been sung as minor (and vice versa). And an obvious performance issue is that is not always "flat" or "sharp" singing that can cause a reasonably skilled choir that is aiming for just tuning to unexpectedly drift sharp or flat.

Regards, Philip

Re: Singing and Intonation

Posted: 12 Jul 2010 05:10
by ashantycapre
pLEASANT day guyz!!
Don't you know that I really loved these forum.I loved these topic to be discussed.!!
I really loved to sing.I'm a choir member in our town.I do have a band when I was in college.I enjoy singing and this is my life,my nature and most esp.this is my biggest dream;to sing.I have a lot of Idle when it comes to singing.Local nor Hollywood and I admire those who sings "out of tune"because they have the potential in singing.I do believe that those people wants to sings but "out of tune" has a possibility to improved their talent.If given a chance to have voice lesson just to improved my intonation and other sharp conflict,I will go for it.Why not,I don't to be ashame in having tutorial for singing carrer.Who will know,I may not be in fame for now,but then time will come.My fate will lead me to bring in fame.And I will not neglect these oppurtonities.This is God's Gift. then, treasure it.
Thank you!!!

Re: Singing and Intonation

Posted: 14 Oct 2010 06:39
by rwendell
First, we have long had a strong interest in the issues your question involves from several perspectives, including just intonation, tuning theory, historical temperaments, and choral singing, and so we naturally have a strong interest in addressing your question. Qualifications include being a choral intonation specialist who has been professionally delivering workshops in choral intonation for several years. We also have decades of experience as a choral director and have a Master of Music in choral conducting. We would like to address your question on the basis of our experience both as choral director and as someone who has not only a background in tuning theory, but six modern temperament designs, mostly well temperaments and one hybrid between a modified meantone and a well temperament. These designs are included in the embedded software of Verituner, who many in the National Piano Technician's Guild consider to be the premiere electronic tuning device. We have received rave reviews for these contributions from some extremely well-qualified tuning professionals at major institutions.

First, except for unusually well-trained choral ensembles (like the elite group to which your refer, we assume), this experience indicates strongly that most choirs drift flat not because of comma drift, but because they have poorly trained ears. If you observe virtually any public event in which singing occurs, churches, for example, usually accompanied by fix-tuned instruments, the public will sing the songs in their entirety invariably flat to the accompaniment. (A notable exception, for whatever reason, is young, excitable females, who sometimes have a tendency to sing sharp, especially in performance situations. Please note also that these are broad generalizations.) Unfortunately, this habituates people generally to hearing sung pitches flat even when they are supposedly matching pitches in order to create unisons with accompanying instruments. This pervasive reality unconsciously biases even choral singers (and too often their directors) and so choirs generally require training very specifically aimed to correct this. Those rare choirs with very high standards for intonation are no accident or mere result of careful auditions. Their directors train them. Intonation was a vital, central part of the late Robert Shaw's warm-ups even for his professional singers.

Secondly, just intonation that is rigorously adhered to is impractical, precisely because of comma drift. A shift of 21.5 cents (syntonic comma) in a sequence of only four simple triads, for example, with at least one common pitch between successive chords (e.g., I, IV, II, V, I), is intolerably noticeable for any well-trained ear or that of the highly musical ear of anyone sufficiently gifted natively. That is over a fifth of an equal-tempered semitone and quite significant. Bluntly put, anyone who thinks otherwise is revealing gross incompetence with regard to intonation. Therefore, in choral singing, the ideal tuning is some kind of adaptive just intonation. There are many possible schemes for adaptive just intonation, and historically many of them have been actually proposed. However, few have actually ever been employed. These typically remain on theoretical drawing boards.

One possible reason for this is that most choral directors do not have a string background or anything other than singing and/or keyboard skills, neither of which requires a rigorous sense of tuning accompanied by continuous practical adaptations in real-time performance as we find in the finest string quartets, for example. Any listener sufficiently skilled in intonation will notice that the de facto standard among the world's finest string quartets consists of very purely tuned, that is, just harmonic structures that are grounded in an equal tempered template for the harmonic roots. Otherwise these quartets could not possibly play Schubert quartets with frequent enharmonic modulations and still maintain just tunings in their vertical, harmonic structures.

Experience also demonstrates clearly that this de facto standard is by far the most practical model for adaptive just intonation. Strictly melodic structures have relatively arbitrary tuning requirements, as clearly demonstrated by quarter tones and other variations from western standards in some ethnic musics. However, the minute we introduce harmony into the equation that changes radically. There are strong acoustic and psychoacoustic factors that bias any reasonably musical human ear toward harmonic intervals that are coherently related. Most western instruments produce harmonic timbres, which means they have overtone structures that conform to the harmonic series. The harmonic series consists of pitches related by integer (whole number) ratios. The human mind generally loves order and coherence, and therefore also musical intervals with common overtones in exact unison.

Further, tones sounded simultaneously produce combinatorial products, including difference tones that lie on either an implied fundamental or an implied harmonic of it. Implied here simply means that the simultaneously sounded pitches in question would be found in the harmonic series of an implied fundamental. So all these squeaky clean relationships produce a coherence effect that is extremely pleasing to any musically sensitive ear. The pitches involved in such a relationship strongly reinforce mutually instead of interfering with each other to produce the infamous "sour note". Highly accurate intonation also produces a choral blend that is vastly superior to even slight variations from pure tuning, even the slight ones we find in equal temperament.

This strongly indicates that the demand for precision in harmonic music resides in the vertical, harmonic structures and not in the sequential pitches of a melody or harmony part. Yet most singers in a typical choir only conceive of their part as a melodic sequence (not necessarily THE melody, but also alto, etc.). When they learn their parts, they conceive of it as a tune and typically pay little conscious attention to how it fits in the harmonic structures surrounding it. If a choir is stocked with reasonably musical ears, however, even without conscious attention to a precise harmonic fit the alarm will sound if the pitch is too far off. The result is such that most directors regard it as acceptable and the story ends there with only occasional corrections of relatively gross pitch errors.

However, the very best choirs are well-trained and do indeed tend to sing in just harmonies. But to be practical they must use some form of adaptive just intonation (AJI), even if it is not consciously explicit, but merely intuitive. My suggestion (upon which I base my training) is the naturally evolved form of AJI we find as the de facto standard in world-class string quartets. The advantage is that we can use indefinitely sustainable pitch sources from well-tuned organs or from electronic keyboard timbres with no tremolo. We use these only for the roots and fifths and leave out the thirds in our training approach. Trainees need stable, continuous feedback for their ears, and so acoustic pianos are woefully inadequate for this purpose.

In equal temperament the octaves are pure, the fifths are negligibly flat (1.955 cents) for choral training purposes, and such keyboards are readily available to those who sufficiently value the enormous aesthetic improvement in the beauty of choral blend and sonority even slightly purer tunings make in already well-tuned choirs. Too many choral directors are utterly unaware of the awesome aesthetic power very minor tweaks in intonation provide. Worse, even when they are exposed to such a performance in another choir, they utterly fail to recognize its source in intonation and falsely attribute the results to collective vocal quality, good, well-coordinated vowels, etc. The simple truth is that for 99% of the world's choirs, intonation is where the greatest return on investment resides if and only if the director knows how to address intonation intelligently and personally has, or develops, the ear to implement such training.

Since the demand for pitch precision arises from harmonic structure, this clearly implies that tuning discrepancies are less noticeable in horizontal, that is, melodic pitch sequences (including pitch sequences in harmony parts). When choirs sing a cappella, they lack the guidance equal temperament provides them in cleverly hiding under the rug the pitch discrepancies we call commas by spreading them equally among all keys. We suggest the answer to this lies in training them to listen consciously for a precise harmonic fit, just as fine string players must. However, this alone is insufficient since just tunings need an adaptive component to be practical.

We must also train them to intuitively "fudge" horizontally to accomplish the same end equal temperament serves in fixed-tuned instruments. This is vastly superior to the fudging of vertical structures with tempered tuning as absolutely required in fixed-tuned instruments. Now the discrepancies are cleverly hidden in the horizontal structure and ideally never appear in the harmonic structures, which can remain impeccably pure, and which have also disappeared from the radar, buried in very slight shifts in the horizontal pitch sequence. Equal temperament as a template for harmonic roots under pure vertical tunings is ideal in that in minimizes these discrepancies in the horizontal, sequential structure as effectively as it minimizes the unavoidable mistuning in the vertical harmonies, but here they are virtually undetectable.

However, this requires a strong and precise pitch memory to complement this training. Such a precise pitch memory cannot exist when the ability to discriminate fine differences in pitch is lacking. Years of using computer-generated pitches to develop this has been a tremendous fount of information concerning how it all works together. This is especially true in training so-called tone-deaf students to reliably discriminate the direction of pitch differences of only five cents and to accurately detect the "sweet spot" for their pitch in a harmonic structure with what we call "harmonic listening" or "harmonic pitch sensitivity". The most inspiring result of this research experience has been the discovery that there is no difference in kind between the relatively small pitch problems found among natively talented singers and those who cannot initially match a pitch at all, but sing a different note entirely, often as far as a third or fourth away. The difference in how it all works is not in the least in kind, but strictly a matter of degree. The experience of dealing with such enormously magnified examples of precisely the same problems with regard to their fundamental nature has allowed vastly greater effectiveness in improving intonation in those who are already musical.

For anyone who wishes to check out the results of this approach in a high renaissance piece performed by a choir we founded and directed in a small town of 9,600 in rural Iowa, surrounded by other towns even smaller (i.e., not a suburb) and in which intonation standards were catacysmically poor before this choir existed, you may click on the following link for a live, unedited recording (no post-production). Please note that although there are some very minor imperfections here and there, the purity of harmonic tuning is every high while the pitch at the end is precisely the same as at the beginning after over two minutes of a cappella singing, so comma drift was not a problem for this choir:

Admittedly, Bruckner would present a greater challenge, but this choir has sung Bruckner as well. A similar recording for that is unfortunately unavailable.

Re: Singing and Intonation

Posted: 19 Oct 2010 06:08
by pml
First, except for unusually well-trained choral ensembles (like the elite group to which your refer, we assume), this experience indicates strongly that most choirs drift flat not because of comma drift, but because they have poorly trained ears.
An aside: I don’t think you meant to give us two “first” paragraphs. Naturally, my comments about comma drift had already made explicit that I was speaking of exceptionally good choirs, and usually this doesn’t entail that they are open for simply anyone to join. You raise the valuable point that it is often the director who has the power to affect the intonation of the choir by a combination of careful selection of voices at the audition stage, in addition to an insistence upon and reinforcement of good tuning in rehearsal, and every genuinely skillful conductor I’ve worked with has utilised these approaches. In addition, these conductors usually have had innate, active perfect pitch, and were sensitive to addressing both pitch and intonation issues as they arose.

I’d question that an adaptive just intonation necessarily has to be “grounded in an equal tempered template for the harmonic roots”: surely this depends on the repertoire? As you pointed out, enharmonic modulations are a common feature of music from the 19th century onwards, but with the exception of a few experimental pieces by figures such as Gesualdo, Lassus, Byrd, etc aside, the common practice through most of the last millennium until that point did reasonably well without modulating to distant keys, and thus one of the classical temperaments (or even the new attempts at constructing well temperaments) would seem both more historically appropriate and genuinely simple in providing the “template” as you call it, or the underpinning for justly tuned harmonic structures? The idea in well temperaments, after all, was not to disperse “the pitch discrepancies we call commas by spreading them equally among all keys” – it was to improve the purity of the commonly used keys by pushing the wolf to the distant key centres where it would be invoked extremely infrequently – if at all.

You rightly point out that in using equal-tempered keyboard instruments, the root and the fifth alone should be played, and this can't be stressed enough: the moment you use such a keyboard for playing triads involving thirds and sixths to a choir, you are badly compromising just tuning (major seconds and minor sevenths are about half-way to being as bad). How then is such an instrument to be used for playing a horizontal melody, for example, so that a soprano can hear her part in isolation from its harmonic context? Equal tempered instruments have the same problems “intervallicly” (sic) as they do harmonically when one is trying to implement just intonation. Frankly, if you are going to use a keyboard with a choir then use one of the readily available electronic keyboards that are beginning to come out with large numbers of historical temperaments pre-programmed into them, and reserve equal temperament for when it is actually demanded by the repertoire.

[/soapbox] Regards, PML

Re: Singing and Intonation

Posted: 19 Oct 2010 12:14
by Cdalitz
the exception of a few experimental pieces by figures such as Gesualdo, Lassus, Byrd, etc aside
Concerning "chromatic" music, I have always believed that the term "chromatics" itself referred to the colorful effect of alternating minor semitones (e.g. F to F#) and major semitones (e.g. F# to G) in nonequal temperaments. If this is right, then it follows as a corollary that this music is pointless when sung in equal temperament. Another corollary would be that this music can hardly be accompanied with instruments like the lute (Gesualdo's instrument was the lute, AFAIK), on which equal temperament was common.


Re: Singing and Intonation

Posted: 19 Oct 2010 23:42
by pml
Hi Chris,
Another corollary would be that this music can hardly be accompanied with instruments like the lute (Gesualdo's instrument was the lute, AFAIK), on which equal temperament was common.
Fretted instruments like the lute and viol were not necessarily tied to equal temperament, because the frets consist of loops of gut tied around the neck, and could be shifted on the fingerboard to provide some minor modifications of tuning (which would apply “equally” to each course obviously). On modern instruments like guitars the frets are fixed in place, but is a mistake to think that it has always been so. ;-)

And where experiments in chromaticism such as the Prophetiæ Sibyllarum were something of an exceptional avant garde work for Lassus, in the case of Gesualdo I think most of his œuvre is intended to exploit colouristic effects, and a number of mannerist madrigal composers had similar inclinations, though none were quite as wild as Nicola Vicentino, who wrote experimental compositions in 31-notes-to-the-octave temperaments: his treatise on the subject (The ancient music adapted to modern practice) specifies the intervals found in the diatonic (tones, semitones), chromatic (major and minor semitones), and enharmonic (major and minor dieses*) genera, which makes it clear that Gesualdo’s compositions were usually in a mixture of the diatonic and chromatic.

The historical evidence raked over by Maria Rika Maniantes seems to suggest that Vicentino’s highly trained choir were able to produce some good effects in highly chromatic music (and Vicentino had an instrument constructed, the archicembalo, to support these chromatic adventures in just tuning), but that the enharmonic music using microtones was prone to disaster.

Cheers, Philip

* Vicentino’s system divides the whole tone in five and the major semitone into three, and the octave closure achieved is fairly close to mean tone temperament. Perfect fifths are however rather blunted by about six cents (~696¢), so Vicentino’s 36-notes-to-the-octave keyboard had an alternate tuning with some extra keys to give exact tunings for these.

Re: Singing and Intonation

Posted: 20 Oct 2010 06:15
by Cdalitz
Dear Philip,

thanks for your informed comment!
Fretted instruments like the lute and viol were not necessarily tied to equal temperament
Right. Even though Galilei (a leading figure from the Florentine Camerata) advocated equal temperament on the lute, some modern lutanists achieve excellent results with meantone temperaments. I wonder however whether either temperament could be used to accompany chromatic vocal music. In meantone temperament e.g., there is the problem (on a lute in G) that c sharp and b flat lie on the same fret (the same with f sharp and e flat).

Can we take your observations on Vicentino as a proof that chromatic music was not meant to be sung in equal temperament?


Re: Singing and Intonation

Posted: 21 Oct 2010 01:01
by pml
Dear Chris,

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.

Cheers PML

Re: Singing and Intonation

Posted: 13 Nov 2010 06:41
by rwendell
In answer to pml's posted replies to mine, it should be clear from your quote from my post, pml, that I understood from the outset that it was quite possible that the elite choir in the questioner's original post was not drifting because of poor training. I was just covering a broader field of possibilities for the sake of completeness. However, I have been a member of choirs that were supposed to be highly elite, meaning very experienced singers, strong readers, etc. but their ears were not so well trained. I have found here in the United States that there are many choir members who have strong backgrounds both in terms of music education and experience, but who possess ears that aren't so very accurate. This unfortunately includes a high percentage of choral directors in this country who may be wonderful organists and who instantly recognize a miss of a half step, but never have had to deal in their instrumental domains with errors smaller than that. They often even play on badly tuned keyboard instruments with no qualms whatsoever concerning the implicit training of their choir to song off pitch.

However, if we ignore these pitiful excuses for choral directors who very unfortunately seem to dominate in this country, I still recommend as a practical tool the "roots on an equal tempered template"approach with the thirds left out for training purposes. This is only used for training in a cappella contexts in my approach and I recommend it simply because it is hard enough to get directors to use a stable source such as an electronic keyboard with a sustained timbre without any tremolo or vibrato effect. Most just want to use whatever stupid, out-of-tune piano that decays right away and happens to be available in the practice environment. For serious intonation training, the source has to sustain indefinitely to provide constant, consistent feedback while training for accurate harmonies.

Why use an equal tempered template for the roots in older music sung with just harmonies? In my experience, picking with the differences between an adaptive just intonation based on an equal tempered template for the roots and historical tunings such as quarter-comma meantone is much less important than simply getting justly tuned vertical harmonic structures. That's hard enough without worrying about such historical niceties. After all, recordings from choirs all over the world are available to us all, and very few choirs ANYWHERE in the world sing so well in tune that this becomes an issue for me personally, and in my humble opinion for anyone else not stuck in some unusual ivory tower, at least here in the U.S. Here there is strong academic opposition to picking with intonation at all. Those of us who care about it much have to fight this every inch of the way because there is a strong academic bias in most U.S. music schools toward a kind of cultural relativism misapplied to music even in intonation, with a concomitant denial that intonation is anything other than an arbitrary cultural convention without any legitimate acoustic or psychoacoustic basis. The errors common in far too many highly reputed university chamber choirs, supposedly with rigorously auditioned membership, are far greater than the relatively subtle differences between a just and an equal tempered third, for example.

As to Vincentino, he actually figured out that 31-tone ET was an extremely accurate approximation to quarter-comma meantone and built instruments with two such 31-tone-per-octave keyboards tuned 5 and 3/8 cents apart to get perfectly pure triads, since the thirds are already just in quarter-comma meantone while the P5s are 5 and 3/8 cents flat. The second, slightly higher 31-tone keyboard on a double manual provided the option for perfectly tuned P5s. This is quite a contrast with the attitudes I've just described above. However, this kind of historical evidence for keenly musical ears many centuries ago falls on deaf (pitch-deaf?) ears here, for the most part.

Since relatively cheap, highly portable electronic keyboards are ubiquitous now, at least there is some small chance that some will actually see the value in using them to train their choirs. I fail to see the value in trying to duplicate the precise melodic contours that were likely used centuries ago, since the aesthetic product of pure harmonies even when based on equal tempered relationships among the roots is wonderful. When forced to defend my position on the importance of good intonation when most of the public can't tell the difference (and "especially if we can't" often seems tacitly implied in the question), I respond with what seems, to me at least, to be a strong argument. Even if most in the audience don't notice when good intonation is missing, they notice how much better the aesthetic product is when the intonation is superb. Why? Because the difference is physical. It's acoustic and psychoacoustic. The common harmonics and combinatorial products line up coherently to mutually reinforce and this makes a vast difference in the blend and the resulting, amazing beauty of sonority. That's what matters to me. The audience doesn't have to understand where that's coming from. I find even most musicians attribute the results to anything but intonation. They just totally miss the point. We don't have to care if the audience doesn't understand the technical sources of the musical beauty we create, but we must fully understand if we want it to be consistent.

Finally, if we don't fudge the vertical, harmonic structures as we must in fixed-tuned instruments, but rather tune them pure, we have to fudge horizontally. After all, that's what adaptive just intonation means, of course. So we fudge horizontally. This is much less offensive to a fine musical ear and is much more difficult to detect than pitch errors among simultaneously sounding pitches. Better still, basing justly tuned vertical harmonic structures on an equal tempered template for the roots MINIMIZES THE FUDGING we must apply horizontally. Playing chords in 12-tone ET with the thirds missing in training for high renaissance music, for example, makes the horizontal fudging much less obvious than it would be with some proposals I've seen for such period music. Further, I've never actually witnessed, live or otherwise, any choir that actually implemented these other schemes unless they were singing with instruments using period temperaments, and even in this case, such precise conformity is quite rare and of course the intonation is still tempered, even if more purely, and therefore cannot be just. If we used quarter-comma meantone on our keyboard, all our fifths would be flat, for example, since we don't have Vincentino's instrument to play on. (Most of his students couldn't play on it, by the way, but only he and a couple of his most gifted students.)

Perhaps just as importantly, our ears are used to horizontal sequences based on equal temperament, so it is easier to train modern ears to make the relatively slight adjustments needed for the thirds with this approach. It is not easy to avoid political flack from a few experienced choral singers, including some with advanced degrees, even in this context. So I consider trying to comply with some theoretically perfect adaptive scheme for just intonation that we assume might more precisely duplicate historical realities to be unjustifiable overkill. After all, what is the basic purpose? I think it's precisely tuned harmony that creates an incredibly beautiful choral sonority rather than conformity with what we surmise to be a totally authentic reproduction of past performance parameters. I don't believe we can ever know precisely what "authentic"means anyway. Historically informed performance for me means a practical reproduction at a comparable level of gorgeous choral sonority.

Re: Singing and Intonation

Posted: 14 Nov 2010 17:53
by rwendell
"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?"
- Chris

My response to this in training is that modern ears are used to equal tempered thirds, which are 13.7 cents sharp. (Stacking three pure thirds on C for example yields the pitches E, G#, and B#. B# is NOT C, but 41 cents flat to the octave C. Our modern notation still reflects this because ancient musicians were quite aware of these discrepancies, which exist because of the prime factors 2,3 and 5 involved in pure harmonic ratios, and the incompatibility of primes. (Primes have no common factors, so intervals containing them as prime factors cannot neatly nest inside each other as they appear to in equal temperament. Equal temperament divides this equally among the three thirds and adds it to them. 41/3 is 13.7.) Given these facts, I recommend that M3s never be flat to pure, since modern ears already tend to hear pure thirds as flat. The beautiful sonority that pure thirds produce when everything else is also tuned accurately certainly makes this slight "violence" to modern pitch memories of how big a major third is fully justifiable. However, in my training, M3s, if they are at all less than pure in a cappella singing, should be somewhere between pure and 13.7 cents sharp and never outside these bounds. The intervals most sensitive to mis-tuning are the unison, octave and perfect fifth. Major thirds can be as much as 6 cents sharp and still yield almost no acoustic cues that they are not pure to any but the most finely developed ear in terms of harmonic tuning sensitivity. However, the typical choir is full of pitch errors much greater than 6 cents. No one would like to see this change more than I.

An interesting exception to the above is blues singing. The typical attitude toward blues, often even among African-related ethnicities, is that blues are less precisely tuned than standard western harmony. However, I propose that the best blues singers have very precise intonation and the discrepancies with our western tuning standards arises from blues being based on 11-limit just intonation rather than our typical 5-limit JI (meaning the highest primes for these tunings are 11 and 5 respectively). Blues often use the septimal seventh, which is quite flat to either our purely tuned or equal tempered minor seventh. Also, the "blues note" on F# in the C blues scale, for example, is not really F#, but rather the 11th harmonic of C, which is only about 51 cents sharp to an F. The minor third above C is also often a septimal minor third, or that which exists between the sixth and seventh harmonics from an F. This produces a very low minor third as well, exactly a P5 below the septimal seventh.

Re: Singing and Intonation

Posted: 18 Nov 2010 05:21
by vaarky
Thank you for the very englightening posting. I've always had the impression that F# and Eb, when sung right by a blues singer, rang in a very precise way. It's interesting to hear how the mathematical relationships would account for this.

Re: Singing and Intonation

Posted: 27 Nov 2010 00:06
by rwendell
You're so very welcome! It's lovely to know that some intuit this without the mathematical understanding. In fact, that's the kind of ability that motivates looking for the rigorous explanation in the first place. The other attitude springs from clueless ears (i.e., not very musical).