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I am reading "Fundamentals of Musical Composition" by Arnold Schoenberg. Chapter XI, page 99, states:

Chromaticism is difficult partly because the natural semi-tones differ in size from the tempered (a fact which causes choirs to get off pitch)

I do not understand this sentence. What is the difference between natural and tempered semi-tones? What does this have to do with chromaticism, or choirs getting off pitch?

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Schoenberg is talking about the difference between just intonation (which he refers to as "natural semi-tones") and 12-tone equal temperament (which he refers to as "tempered" semi-tones). This is a complicated subject. You can find several long posts about this subject on this site, or you can find a lot of references elsewhere on the Internet.

The gist of the matter is that in the modern era, a piano has 12 notes in an octave, and each of them are exactly 100 cents apart. This is the 12-tone equal temperament system. It is is an artificial compromise; it is not natural and it is not in keeping with the physics and mathematics of sound.

Singers in an a capella choir or string players in a string quartet are not limited to 12 equally-spaced half-steps. Choirs can sing in just intonation, which is pure, natural intonation. With pure, natural intonation, the size of the half-steps (and the difference in the frequencies between two pitches in a chord) will vary, depending on the chords and intervals being sung at the particular moment.

Here is a chart I created in answer to a previous question on this site.

enter image description here

As you can see, in pure intonation, some half-steps should be larger than 100 cents, and some half-steps should be smaller.

When singers or string players play with a modern piano, however, everybody must constrain themselves to the 12 equally-spaced half-steps that the piano can play.

This may surprise you, but the bottom line is that if you and your choir sing exactly the same pitches found on the piano, you are in fact singing out-of-tune to some degree. A choir can only sing perfectly in tune if they are singing a capella and they do not automatically adjust each pitch to be the same as the pitches on the piano, which are artificial.

Further observations:

Schoenberg observes that if the music is highly chromatic, a choir will make so many adjustments of semi-tones to produce pure intonation that they will not be able to avoid drifting off of the base pitch or frequency of the key or tonic note in which they started the piece, and they will finish the piece on a different tonic pitch (meaning they will "get off pitch"). In decades of singing in a choir, I have observed this many times. Generally we choral singers don't regard it as a problem, as long as we were internally consistent throughout the piece. By this, we mean that while we were singing it, each chord in the music sounded in-tune and there were no obvious wrong notes.

Frequently in a rehearsal of an a capella piece, the accompanist will play a chord on the piano at the beginning to give each section their pitch. When the piece is completed, the accompanist might play the same chord to show the choir how far off the base pitch they have drifted in the course of singing the piece -- at which point the choir often groans in chagrin. Sometimes it can be as much or more than a whole step sharp or flat from the starting pitch. Yet I contend that this is not a disaster as long as the piece sounds internally consistent. However, in a good choir, all the members strive to keep as near to the original base pitch as possible throughout the piece, and to finish there as well.

As a singer with perfect pitch, when I am in this situation, particularly in performance, it presents a dilemma: if I perceive the other choral singers around me drifting away from the starting pitch, tonic and key (taking into account the modulations that may be written into the piece), do I fight against the other singers by keeping my pitches up, relative to the starting pitch and tonic and key, or do I "go with it" and follow the gradual shift in tonic that everyone else is making? In most cases, it's better to "go with it" rather than "fight it", because it produces a more internally consistent sound. While I have perfect pitch, I must concede that if a choir were only to sing the 12 pitches on the piano, that would not be a good choir, because that is not really what choirs are supposed to do.

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