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I understand that the mathematical frequency ratios between certain intervals will correlate with the overtones that are present when a note sounds in a fundamental frequency. Certain predictable overtones will occur naturally at frequencies that are integer multiples of the fundamental note.

Sound waves produced by musical instruments (including the voice) translate into corresponding vibrations in the basilar membrane and other mechanical components of the inner ear. These vibrations produce their own overtones that work the same way as a plucked string on a musical instrument - producing certain predictable overtones in our auditory system that are integer multiples of the fundamental.

A perfect fifth in just or pure tuning based on the harmonic overtone series would obviously be more naturally consonant because the vibrations set off in the inner ear would flow together and not beat against one another. So theoretically, tuning to the perfect integer multipliers which match the harmonic overtone series would yield a more pleasing harmonious blending of notes than say equal temperament tuning of modern instruments.

I know that equal temperament tuning is a compromise to allow fixed tuning instruments such as a piano or guitar (with intervals determined by fret spacing - bending of notes notwithstanding) to be played in any key without being terribly out of tune in any one key (only slightly out of tune in every key). But tempering the tuning literally means stretching or shrinking the intervals such that they are no longer truly consonant and some dissonance is interjected by necessity.

I will acknowledge that it is likely that as we grow up listening to "out of tune" equal temperament tuned music - we develop a tolerance for and acceptance of it (some may argue even a preference for it but that is highly debatable). However, it seems to me that perfect consonance in keeping with the pure harmonic overtones, would naturally and instinctively be more pleasing to our ears without the need for (and in spite of) conditioning.

Given that the human voice is capable of microtonal adjustments to an infinite degree and not limited by the fixed tuning imparted upon an instrument such as a piano - I am wondering if singers who harmonize with the lead vocalist subconsciously adapt and instead of singing a equal tempered third or fifth actually sing something closer to a justly intoned harmonically natural interval.

If this does occur, it might help explain why some singers sound so incredible harmonizing together. I have been privileged to hear some performances where the vocal harmonies actually gave me shivers up and down my spine and seemed to touch my soul with their sensual richness.

Have there been any studies (or is there compelling evidence) which attempt to ascertain if trained and capable singers who sing harmony - instinctively gravitate towards pure intonation verses singing the notes they have learned to associate with equal temperament tuned instruments? In other words if they are singing a fifth above, would the instinctive control center of their brain override their musical training and adaptation and lead them towards singing a perfect natural fifth or would they sing something closer to a equal tempered fifth that a piano would play?

If some do and some do not, have any studies attempted to determine if there is a clear listener preference for pure harmonies vs. harmonies based on adapted and learned tempered tuning?

My personal hypothesis is that pure and natural harmonies that approximate the frequencies found in the harmonic overtone series would provide a more enjoyable listening experience unless said harmonies clashed directly with equal tempered tuned accompanying instruments. In many cases the vocals (including harmony) are not sung in unison with chords or corresponding melody notes. But it still may work better in a cappella singing (such as a trio or quartet).

But I will defer to the more educated music theorist and trained practitioners in the SE community for a more accurate assessment.

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    Tuning denotes a static property, which in my opinion does not apply to singing and fret-less string instruments (as long as no open strings are involved). One would typically deviate all the time, e.g. making major thirds a bit larger, leading-tones higher and so on. So I expect in the best case a probabilistic answer of limited value. – guidot Jan 26 '16 at 8:06
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    Pretty sure that violins will stray from 12tet into pure intonation, especially when there's no reference to 12tet. – Tim Jan 26 '16 at 8:21
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    A singer who is singing through Auto-Tune, which is almost everyone these days, is not equivalent to a fretless instrument at all, and their minor thirds will always be the same minor third. – Simon White Jan 26 '16 at 8:25
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    I think it is virtually impossible to not sing pure/just intervals when harmonizing, at least as long as you listen more to the other singer/singers than any accompanying (ET) instrument. It is simply a matter of gravitating towards the beatless interval. – Johannes Jan 26 '16 at 9:09
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    This is one of the big things that makes autotune sound noticeably bad. The intervals in harmonies are not "humanized". There's also formant shifting, but even minor tuning adjustments to "perfectly" fit ET can take away from a harmony part that has a lot more flavor to it. – Todd Wilcox Jan 26 '16 at 13:12
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I wish I could point you to some scientific studies; I cannot. But I can speak on the basis of a lifetime of my being a semi-professional traditional choral singer and soloist who has a university music school degree in singing. I have extensive experience with a cappella choral singing, with singing accompanied by piano and organ and orchestra, and even with singing with Baroque and Renaissance groups whose instrumental accompaniment uses different historical temperaments other than the modern 12-tone equal temperament.

It's pretty simple. If a group of singers sing a cappella, they sing in pure, just intonation which is adjusted constantly throughout the piece, measure-by-measure and chord-by-chord. That is why a professional a cappella choir will sound wonderfully in tune at every given point in a long piece of music, yet at the conclusion of the piece they might discover that they are quite a few cents away from the base pitch or key that they started on. This is because all the just intonation intervals don't add up to whole numbers.

If, on the other hand, a choir or even a vocal soloist sings with a piano, organ, guitar or harp, they automatically adjust their intonation to the 12-tone equal temperament (or other historical temperament) provided by the accompaniment.

However, to my perpetual astonishment, I have learned from experience that most singers, even some highly-accomplished professional ones, are not even intellectually aware of the distinction between just intonation and 12-tone equal temperament, and are not consciously aware of making adjustments between one scheme and the other. None of us singers do any of the mathematical calculations based on physics or musicology. It's just something we all learn to do informally and by ear.

Human singers are not taught to think of their voices in terms of the physics of the tension applied to vibrating strings or the tuned resonance of open or closed pipes. They just sing pitches and use their ears and their brains to observe and adjust to what is in tune and what is not, given the current context of the music and of the instruments being used to accompany that particular piece.

I will also observe, however, that everything that I've said above about singing in choirs also applies not to singing groups but to purely instrumental ensembles such as string quartets and string orchestras, small brass ensembles, and other instrumental ensembles that do not involve any keyboard instruments or instruments with fixed temperament -- when they play amongst themselves, they always modify their intonation toward pure just intervals. When the ensemble plays with a piano, they modify their pitches to line up with the intonation of that piano. That is, if they are very good musicians.

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    When I was in a pop-punk band singing harmonies along with guitar and bass, I noticed that we were constantly adjusting very slightly to try to find the best sound between matching each other and matching the instruments (which are not perfectly equal tempered in the first place!) One song we did had a long harmony held at the end of the song when all the music has stopped. We would adjust our harmony (without even thinking about it) as the guitar chords died away and end up with a just interval at the very end. I think with enough singing it becomes natural and no thought is required. – Todd Wilcox Jan 26 '16 at 13:10
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    @ToddWilcox interesting observation. I have noticed the same thing. Sometimes my band will sing a chorus a cappella and the singers sort of adjust to one another (but not sure if it's to pure intervals) until it sounds great. We have occasionally hit a magic spot on the last you... ..... in Amie by Pure Prairie League. – Rockin Cowboy Jan 26 '16 at 15:26
  • That makes sense Wheat. I used the words "subconsciously" and "instinctively" in my question on the supposition that what you believe may in fact be correct. I have a daughter who was a a gifted singer in Chorus from Elementary school through College and she could sight read music on paper and sing the written notes. She was graded during auditions for All State Chorus and apparently was able to sing what the examiner expected. I just don't know if she was singing the notes in the frequency that the piano reproduced or keeping the intervals between notes close to just intonation. – Rockin Cowboy Jan 26 '16 at 15:35
  • @RockinCowboy, I passed that very same audition for that very same group myself, in 1978. – user1044 Jan 26 '16 at 18:02
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    An excellent and very interesting answer. +1. I wonder, in your second para., if it's just that there's a tendency to wander, even with experienced singers, in a cappella, or is it actually due to the just intonation intervals? Any studies? – Tim Jan 27 '16 at 8:36
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You're not the first person to ask this question. A study done by Lottermoser and Meyer in 1960 found that professional choirs tend to sing the major thirds sharp and the minor thirds flat, thus veering toward Pythagorean rather than just intonation.

An octave or fifth can be tuned easily using the common overtones of the two notes to be sung. Thus, with competent choirs, these intervals tend to be just or nearly just (an equal-tempered fifth is only 2 cents away from just).

With thirds, by contrast, the common overtones are higher and weaker, so the tuning is much more susceptible to cultural conditioning. Western music since the Baroque features the dramatic contrast of major and minor, and it makes sense for the major and minor third over a common root to be farther apart than the 71-cent just interval. String players tend to play leading tones (which are a certain subset of major thirds) sharp, so that they tend tensely toward their tonics. Such practices are easily taken up by singers today.

By contrast, keyboard instruments in the Renaissance were most often tuned to some variant of meantone temperament, with low major thirds and high minor thirds, an aesthetic consistent with a performance practice where the difference between G and G# was inconsistently notated and often left to the singer. The same aesthetic persists among players of traditional Arabic music, who have been known to complain about the high major third of the Western piano. Since Arabic music is primarily monophonic with stepwise melodies, it seems that nearer equality of the seconds may be the driving factor there, not justness of the thirds (especially since some Arabic scales feature microtonal seconds and thirds with no just basis). In short, the human ear is capable of accepting a wide range of tunings of thirds.

  • Thanks for providing an answer that explores other considerations including historical context and non Western music. Also thanks for the reference to the "study". Plus 1 – Rockin Cowboy Sep 1 '18 at 15:48
  • Oh my goodness. Absolute gem of an answer. This deserves more attention. +1 – user45266 Feb 11 at 6:41
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The top barbershop quartets very specifically use just intonation as well as vowel modification to maximize overtone ringing. And it's likely that other top a cappella groups veer in that direction, some by instinct and others deliberately.

But the average a cappella choir likely can't tell the difference between just and well tempered intonation. Lots of studies of pitch perception show that singers are given latitude by listeners for being "in tune." I've sung in, directed, and listened to thousands of a cappella groups and sadly being just "in tune" by any definition as a group is a notable accomplishment!

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An extract from an interview from Ross Barbour of the Four Freshmen...

"If my note was a major seventh, I could sing it on top of the note — sing it sharp, you might say, so it and the tonic note became a little less than a half-step apart. That's what makes it buzz in your ear.

If we wanted a dominant seventh to ring, we'd sing it on the bottom of the pitch — especially if the voice leading was going down through that dominant seventh.

A major third should be sung brightly on top of the pitch, and a minor third should hang on the bottom.

We were singing those notes not because they were writ­ten and the piano said the pitch was "there." We sang them because they harmonized. They made overtones in our ears.

And we didn't discover some great breakthrough in har­mony. Good barbershop singers do it all the time; in fact singers have been doing it since at least the year 1700."

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    Singing major thirds on top of the pitch, and minor thirds on the bottom, is an effective means of melodic expression- but it's even more out of tune with the harmonic series than equal temperament. If you listen to good barbershop quartets, you will hear that they pitch their major thirds lower than equal temperament. But you are right about the dominant seventh- putting it low puts it close to the real blue note, the 7/4 interval. – Scott Wallace Feb 16 '17 at 11:26
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I disagree with your premise that a mathematically perfect tuning would be somehow preferable and that singers would naturally gravitate towards that.

Singing is a learned behavior. We gravitate towards tunings and scales that are familiar from our experience. If I grow up singing along with a piano in common tuning, playing major and minor scales, and then I sing a cappella, I’m going to sing major and minor scales in common tuning.

Auto-Tune has been almost as popular as reverb for some time now, and we see examples of young singers who sound Auto-Tuned even without Auto-Tune. They grew up hearing singers sing only a handful of very strictly tuned notes on a very tight grid, and that is what they sing back to the world.

Singers don’t exist in isolation. The fact that pianos or guitars were tuned a certain way and/or play certain scales causes the singers to be tuned that way and sing those scales. It’s a question of culture, not math.

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    So you are saying that two singers harmonizing would not sing the pure (beatless) intervals but somehow manage to sing in 12TET? I also believe that the auto-tune "sound" is mostly due to the timbral change caused by the software when it corrects a note and this is what singers mimic, not the tuning it imposes. – Johannes Jan 26 '16 at 9:04
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    @SimonWhite, as a lifetime semi-professional choral singer and vocal soloist with a university degree in singing, I disagree with your premise. All singing can be infinitely microtonal in nature. Good singers can control microtonality well, and poor singers cannot. – user1044 Jan 26 '16 at 11:46
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    What a good singer does with regard to intonation varies with the contexts of different pieces of music that are being sung. A good singer can choose between just intonation and 12-tone equal temperament at will. The singers in an a cappella choir can choose to exist in the isolation of just intonation, and in the same concert they can sing accompanied by a piano and shift gears into 12-tone equal temperament. – user1044 Jan 26 '16 at 12:03
  • I was wondering if that was the case Simon. That's why I posted the question - to see if it's actually been studied using something like a spectrum analyzer applied to a recording of the performance or during the live performance. In the absence of such a scientific analysis, its really impossible to say that you are right or wrong and it's possible that some singers do exactly as you say while others (perhaps because they don't have the skill to replicate what they hear in the instruments) instinctively gravitate towards a pure interval vs. the pure note they hear on piano. – Rockin Cowboy Jan 26 '16 at 15:21
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    Following the logic of this answer: if I grow up singing along with a piano that has one note (lets say F# above middle C) which is out of tune then I am going to learn a completely different tuning for the scale of C major from that of the scale of G major. I seriously doubt that this would be the case, although I can't actually point you towards any evidence to back me up on this. – JimM Feb 16 '17 at 11:30

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