A week or so ago, I had the opportunity to sit with Gene Puerling for a couple of hours to talk about his days with The Hi-Lo’s, The Singers Unlimited, and what he’s been doing since those days. I’ll be working on transcribing that and editing it for a column soon. If my talk with Don Shelton is any indication, this interview will also probably run to two or three installments, and it will take a while to get into print. But it’s coming.
In the mean time, here’s a piece I’ve prepared on the difference between tempered and just intonation. It may seem like a dry subject that doesn’t really relate to the beat of A Cappella Jazz, but intonation is pretty fundamental to any a cappella music – or any music at all, for that matter. If you want your dominant 7, sharp 9 and your major 7, sharp 11 to sound right, you’d better make sure that the root, third and fifth are all in their proper place. Otherwise, you’re pretty much sunk before you set sail.
Background:
After many years of meditation on the subject, trial and error, acoustic observation and even some mathematical analysis, I have come to the following basic conclusions. An even-tempered (well-tempered) major third as one would find on a freshly tuned piano is too large. The inverse, a tempered minor third, is too small. A fifth is not large enough, and a fourth is too large.
I’ll get into the math in a moment, but first, let’s look at the harmonic series, whence all “just” (natural) intervals and pitch relationships derive. We didn’t develop western harmony out of thin air. It comes from some basic physics relationships, combined with a desire to be able to play equally well in a variety of keys. If you eliminate that multiple keys requirement, you open a door to a lot of different musical styles, including most varieties of instrumental folk music from around the world, which are often limited to a limited number of key centers, based on the instruments used. You also open up the possibility of a number of different pitch relationship systems, but I won’t get into that here.
In the discussions that follow, “just pitch” is a technical term referring to the unmodified pitch as it would occur naturally in a harmonic series (relative to the fundamental in that series).
The Mathematics of the Harmonic Series:
Let’s start with an A 440. An octave below that is exactly half the number of vibrations per second, 220 Hz. An octave above A 440 would be twice its value, or 880 cycles per second.
To make the math a little simpler (and to bring it back into a more listenable range for most humans) let’s start at a lower A as our fundamental and climb the natural harmonic series. As you continue to add the frequency of the fundamental note (110 Hz, which is 110 cycles per second), you end up outlining the precise pitches of the harmonic series, as outlined in the tables below. In these tables, “Low” indicates that a pitch is below the labeled note, but isn’t so low that it can be labeled the next chromatic tone down.
 table 1
In the next octave, it becomes a little more interesting.
 table 2
The D found at 1210 could be considered a flat D# or a sharp D natural, and in the literature of baroque natural trumpet (an instrument that uses the natural harmonic series), this note is used for both tonalities, though there are compensating intonation tricks to bring it into a real D sharp or D natural position.
Note: Most of the trumpet literature from that era is in the key of D, so we’re talking about a G or G#. But in the second Brandenburg concerto by J.S. Bach (in the key of F Major), there are B flats / B naturals in the trumpet part, depending on the harmonies at any particular time.
In the tables above, the A’s, the E’s and the C#’s in the series are perfectly in tune in the key of A. The physics of this intonation produces subtones and additive tones. You can verify their in-tune-ness by the presence of these sympathetic notes. There is actually a technique I learned during college that used this phenomenon as a way to tell when the brass section was actually locking and in tune.
A truly in-tune major third or major 10th will produce a sympathetic tone of a fifth of the chord (displaced an octave or two above). A truly in-tune perfect fifth will produce a sympathetic tone of a third (similarly displaced above). A truly in-tune major sixth (which is found in the harmonic series as the fifth of a chord and the third above it) will produce a sympathetic tone of the relative root (displaced below the pair being played).
If you don’t believe the preceding couple of paragraphs, it’s worth the 15 or 20 minutes or so it will take for you to do some experimenting. Just don’t use a piano or other tempered instrument (including any electronic or MIDI instruments). And at least one of the tones has to be from a source that can be adjusted in microtones, such as the voice, a string instrument, or something similarly variable. I’ll get into this exercise a little later, though.
Tempering:
As you probably know already, Bach’s “well-tempered clavier” was not an instrument with an attitude adjustment. It was a keyboard that could play in all keys (equally, though minutely, out of tune). That’s why he wrote a prelude and a fugue in each key. It was a demonstration of what the new type of instrument could do, not because of the construction of it, but by the new method of intonation. On a keyboard that hadn’t been tempered, playing a piece in C#, or F#, for example, would be horribly out of tune, since they were generally optimized for specific keys – often D Major.
An even-tempered scale is commonly defined as being made up of even intervals such that a half step is equal to the ratio of the inverse of the twelfth power of two. An easier way to say it is that if you multiply a given frequency by 1.059463, you will arrive at the next tempered half step. Try it. It works.
At the end of this particular octave (A 880 through A 1760) as defined by this method, the octave has lost only .002 cycles per second. However, since each of the A’s on a piano are tuned relative to the 440, those .002 cycles are reset at each octave and don’t have a chance to accumulate into a margin of error that would be audible.
The following is a listing of the tempered frequencies of the notes of this octave.
 table 3
As you can see, the tempered C# (a major third from the root) is 1108.73 cycles per second. The naturally occurring C# is only 1100 cycles per second. 8.7 cycles per second may not seem like much, but if you detune two strings on a guitar by 8.7 cycles per second, you will hear a distinct beat pattern (that’s right, exactly 8.7 beats per second) which would be simply unacceptable. With voices, it just sounds like they’re out of tune.
The tempered E (perfect fifth from the root) is set at 1318.51 cycles per second. The naturally occurring E is 1320 cycles per second. The difference of 1.5 cycles per second is much less striking than the detuning of a tempered major third, but it is significant enough to matter. When you experience an in-tune open fifth, it rings in a way that a tempered fifth just doesn’t.
What does all of this mean in practice? If one needs to perform with tempered instruments (keyboards, mallet percussion, etc.), you have to be prepared to either be out of tune, or to adapt to the tempered scale. However, when working with instruments that are not bound by the fixed pitches of the tempered system (strings, brass, woodwinds, voices), this limitation can be overcome and the intonation can be much more satisfying.
The voice is certainly not bound by fixed pitches, so in a cappella music one may choose to have the chords ring in just intonation. I have found that once I got used to being able to play and sing in-tune, I was no longer happy performing in any situation where that was not the norm. In fact, my senior recital in trumpet performance had no keyboard except for a harpsichord that was part of the continuo for Bach’s Cantata #51 (playing along with a chamber orchestra). The rest was chamber music performed with other instruments that had the ability to play in tune. I’m not saying that we always did play in tune. I’m just saying that we had the opportunity to do so <grin>.
If one chooses to hear a chord “ring” in-tune, then the major third must be noticeably lower than it would occur on the piano, and the fifth of the chord must be just a touch higher. One can achieve a similar effect of an in-tune major third by raising the root a bit in order to shorten the distance to the third. In a choral setting, that also has the advantage of possibly recovering some “lost” pitch if the group tends to sag, so I often use that technique to gain both better intonation on specific chords, and better global intonation relative to the starting pitch.
Assuming no change of the fifth, it goes a little more out of tune when you raise the root like this, but the in-tune third usually has a stronger effect than the out of tune fifth (and whoever is singing the fifth often adjusts the right way unconsciously, anyway). I often lean that way since, as a bass singer, I usually have more control over where the root sits than anything else. And yes, I often tune chords where a major third was high by raising the root even more to match it.
By comparing the relationships between the naturally occurring root, third and fifth and their tempered counterparts, one can evaluate the relative detuning of all of the consonant intervals, along with the reason they’re out of tune in the tempered scale. These examples are relative to the key of A to be consistent with the above discussions. However, since the whole point of the tempered scale is to have everything uniformly out of tune, the conclusions are applicable to any other key in the tempered world. :
 table 4
Exercises To Hear The Difference:
To experience the phenomena described here, establish a static fundamental pitch and then sing the major third you would expect to hear above it. It’s best if the two notes are of similar tones (two voices, two unfretted strings, etc.).
Now take that major third and very slowly glissando up a half step and down a half step, and then as you come back up, stop when you can hear the sympathetic fifth sounding in your ear. Do it again. After a few tries, when you can lock into it consistently, you’ll probably find that it’s lower than you originally thought. If you have a keyboard handy (though it tends to spoil the feeling), you can play where the tempered pitch actually sits and compare it to what you’re singing. I usually find that this example astonishes those who are experiencing it for the first time.
You can do the same exercise with a fifth, a fourth, and either a major or minor sixth. The sympathetic tone to listen for changes depending on the interval you’re tuning, but the practice is basically the same. But start with a major third. It’s a more pronounced difference than for the 5th and the 4th, since in the tempered scale, it’s detuned more drastically than they are.
 table 5
Do you see a pattern here? All of the intervals and resulting tones describe a major triad in some inversion or another. A major tonality is simply the natural way of things from a physics point of view.
Some of these will appear in the octave or two above the interval you’re tuning (additive tones or harmonics). Some of them will appear in the octave or two below (subtractive or sub-tones). If I remember correctly, usually the thirds and fifths that appear are in the octaves above the interval being tuned. The missing roots often appear in the octaves below the interval being tuned. I’m not verifying those “above & below” locations as I write this section (I’m too sick right now with a cold to sing anything very well in tune or to hear the resulting tones, even if I did!), so don’t hold me to it.
Vibrato:
Vibrato makes it impossible to achieve just intonation. The best possible outcome when tuning two pitches that have vibrato is a reasonably close impression or approximation of being in tune. This can be complicated when one vibrates above the pitch and another vibrates below the pitch. That will simply never sound right. Well, to my ear, anyway. But the math and the physics aren’t subjective, so I’ll go out on a limb and say I’m simply correct on that point.
When there is a section of performers all using vibrato, the variances tend to cancel each other out and create a fluffy “zone” of pitch. Obvious examples are modern string sections and the typical classical choral ensemble.
However, if you listen to early music specialists on viols or a good group of chamber singers (Chanticleer or the Kings Singers are good examples), the predominant tone is straight with little or no vibrato. The simpler harmonies of early music when performed by these groups achieve a clarity that they wouldn’t have if they were performed with vibrato.
One example where vibrato is actually required (and a wide one, at that) is in performing traditional gospel music and spirituals. To sing that literature with a straight tone or even a modest vibrato just sounds wrong. An ensemble that really sounds authentic in that genre will almost always have quite a variety of ranges and depths to their vibratos. But as a whole, it creates a unique sound that we recognize as “right” in the style.
Large vs. Small Vocal Ensembles, and Maintaining A Solid Pitch Center:
An added variable in tuning vocal music is directly related to the size of the ensemble and the sensitivity of the members to keeping a stable pitch center. I have found that, in general, the smaller the ensemble, the more important it is to be able to eliminate vibrato from the sound.
Larger groups, less experienced singers, less accurate singers, and singers whose voices are tired or poorly maintained will tend to lose pitch over the course of an a cappella piece. Conductors often play tricks on the group to get them to maintain or salvage their pitch center. Some use visualizations, some try tempering the intervals so that descending intervals are smaller and ascending intervals are wider. But these are methods to fix something that’s broken.
In a chamber ensemble where everyone has the technique, the ability, and the vocal health to maintain a solid pitch center, I find it more interesting (and infinitely more rewarding) to aim for just intonation wherever possible. This is especially true in older or harmonically simpler music.
Vocal Jazz Intonation:
I’ve been asked from time to time how Clockwork tunes. We tend to start with tempered pitches (from a pitch pipe, not a keyboard), but when a piece is slow enough or becomes familiar enough to pay attention to finer intonation, we generally tend toward just intonation, though we don’t talk about it much in those terms. I think we all know to lower a major third and raise a fifth in a chord, but whether or not we all know about it intellectually, we know how to execute it. What’s unusual in a jazz group is that intervals like minor seconds and major sevenths start to become as consonant to the ear as a major third or a perfect fifth. But make no mistake about it, the upper partials (7ths, 9ths, 11ths) tend to find their way home much more easily when the lower partials (root, third and fifth) are properly in tune.
Instrumentalists and Just Intonation:
It is my experience that when playing with a good brass quintet or a good professional orchestra brass section, everyone is expected to know where their parts are sitting in a given chord and how to adjust their individual instruments to accomplish proper just intonation. Without doing that, the section sounds brash and loud. With solid “just” intonation up and down the chord, the sound is significantly stronger, yet it doesn’t sound like they are playing “loud” as much as they just sound amazingly strong and in-synch with each other.
The Chicago Symphony Orchestra’s brass section is famous for this effect. I can say from personal experience that hearing them play together on a piece like Wagner’s Lohengrin Overture, pounding out major chords with every tone adjusted to perfection is an experience unlike any other. I had played professionally and had studied trumpet for three or four years in college before I heard that, and it was so drastically different than any orchestral sound I’d ever heard before, it changed my whole view of what was and wasn’t possible.
I realize that this is a very long-winded explanation of a very specific and esoteric subject. I’ve gone into such depth more to put into writing some of the concepts that I’ve spend a lot of time thinking about over the years, and have never collected in this way before. I realize that I’m going against what may be the majority of current thought in at least the choral conducting world. But when you have the empirical evidence and experience, and then have the math to back it up, it’s not being cocky to claim you’re right. It’s just defending the truth. And spreading the word of truth has got to be a good thing.
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