The Mystery of
The High Third

Excerpted from “Lies My Music Teacher Told Me,” page 70
In case you are getting the impression that I think I have it all figured
out, let me assure you that I haven’t. Every book needs a bit of mystery.
Here’s ours. The numbers indicate that the fifth partial (scale step 3) is lower
than the tempered major third. Yet, I have observed that many pitch-sensitive
musicians (particularly string players and singers) seem to prefer a tuning
that is higher than the tempered third when the fifth of the chord is
also sounding. Evidently, there is more happening here than just
simple partials. Here's an opportunity for cutting-edge research.


Before you read more below about this fascinating topic, listen to this audio clip to find out whether you hear “something” move up or down in pitch. First listen for the overall effect. Then play it again and listen with more attention to individual pitches in the chord.

Click to listen - (Wave 782kb) (Mp3 74kb)

In this clip you will hear two pitches initially, then a third pitch enters. Finally, the initial two pitches drop out leaving the third pitch sounding alone. Do you hear the single pitch at the end of the clip change when the others drop out? If so, do you hear the pitch go up or down?

Click to listen - (Wave, 978kb) (Mp3, 92kb)

(Theses audio clips were created by Paul Erlich, a very knowledgeable musician
and theorist whom I met on the Yahoo “Tuning List,” where a number of
experienced tuning experts participated in a search for a
mathematical explanation for the “high third.”)

Please, before you read further, tell us by email what you heard. Are you a musical performer? If so, it will be helpful to know your performance medium (instrument, voice) and the number of years of experience. We’ll post the results (anonymously, of course) as they are received. It should at least be interesting, and perhaps informative.

Now, here’s the story.

I began to wonder about the mysterious “high third” quite a number of years ago while teaching my entry-level college choir to sing in tune. Once the group could sing a pure unison, I demonstrated the perception of basic acoustic fifths and thirds and then asked the singers to tune these over a sustained fundamental (root) pitch—G over C, then E over C, for example. The idea was to seek the “best” tuning by moving the pitch (G or E) up or down ever so slightly until it “locked” with the fundamental pitch (C).

Part of the lesson was to show that intervals tuned “by ear” are not the same intervals when played on a keyboard. Once a well-tuned interval was sounding vocally, our accompanist tapped the corresponding pitch on the keyboard for comparison. The contrast in pitch was clearly audible. The vocal intervals sounded clean and “in tune.” The piano intervals sounded somewhat “cloudy” and less focused.

As would be expected (according to the math of acoustics), the piano’s perfect fifth (G) was slightly lower than the singer’s perfect fifth and the piano’s major third (E) was clearly higher than the singer’s major third. The Harvard Dictionary of Music states that the few cents difference in the acoustic (singer’s) fifth and the tempered (piano’s) fifth is not noticeable to the human ear. Our young singers, who likely had never even heard of the Harvard Dictionary, had no trouble hearing the discrepancy.

(This exercise can be performed by most anyone using a bit of patience
and some careful listening, and I suggest you try it yourself. For help,
click here to bring up the appropriate exercises from Natural Ear
Training
in a new browser window. Be sure to move your
vocal pitch very slightly up or down to the “best”
tuning in order to experience acoustic tuning.)

Once the singers could securely tune the major third and perfect fifth separately to the sounding root (two simultaneous pitches, in each case), I asked them to sing all three pitches simultaneously, creating a C major chord. We first tuned the basses to a piano reference pitch (C below middle C). Then the tenors added a well-tuned fifth (G above the basses’ C). The sopranos and altos then added the third (E above middle C).

When everything sounded homogenous, I asked the accompanist to tap on the piano each of the sung pitches to check for discrepancy. The basses were still sounding a solid C. The tenors were still providing a fifth slightly above the piano’s G. However, the sopranos and altos were singing a third that was higher than the piano’s E, yet, the chord sounded pure and “in tune.”

Apparently, the presence of a well-tuned fifth had caused the third to be tuned much higher than when the third was tuned to the root alone—that is, without the fifth. This curious turn of events prompted me to consult outside help to find out whether there is an acoustical (mathematical) explanation for this phenomenon.

Consulting the experts

Fortunately, I was aware of Yahoo’s “Tuning List,” an on-line email conversation among the most experienced music/math experts anywhere. I described my “problem” to the microtonalists there, and to my delight many of them rolled up their theoretical sleeves and went to work. After about three months of exchanging ideas, crunching numbers and testing theories, no clear-cut solution appeared.

Some felt the effect was simply a learned stylistic preference. To me, this didn’t seem to jibe with the fact that my choir consisted almost entirely of novice singers. It also didn’t explain why the “learned preference” wasn’t sung when the fifth wasn’t sounding.

Others felt the effect was likely a psychoacoustic phenomenon. They suspected that the actual pitches being sung were in accord with acoustic norms, and that the ear created some sort of illusion of the third appearing higher. I thanked everyone for their help and left it at that. I had known for many years that experienced singers as well as string and wind players frequently performed and acknowledged the “high third.” In fact, I remember being startled when I discovered that the acoustic facts didn’t support the major third I grew up with. This seemed to me to support the idea that the high third was indeed a learned preference, but it still bothered me that I had consistently observed it being “preferred” by my novice singers.

About a year after my first Tuning List visit, I returned with new evidence. It occurred to me to record the tuning lesson at the beginning of a new semester and ask the tuning gurus to analyze electronically the pitches being sung. Here is the audio clip I posted for their consideration (a bit fuzzy sounding here, but I think you'll get the idea)..

Click to listen: (Wave, 3.36mb) or (Mp3, 316kb)

The basses get their pitch (C) from the piano (replaced here by a synth) at the start. The sopranos and altos then tune their third (E) to the bass root. The tenors then insert the fifth (G). Notice that the keyboard reference pitch first sounds higher than the sopranos and altos and then later sound lower than the sopranos and altos.

The idea was to find out whether the soprano/alto pitch actually moves higher after the tenors enter or simply appears higher to the ear. Electronic analysis showed that the pitch actually changed—from lower than piano pitch before the fifth entered to higher than piano pitch after the fifth entered.

That observation seemed to support the idea that the high-third experience is not simply a psychoacoustic phenomenon, but reflects some sort of physical/acoustic reality that my novice singers had sensed. For me, that evidence was welcome, since my own experience was very comfortable with the notion of a math-based high third. However, some of the best minds available were not able to discover what that explanation might be.

While the electronic analysis was proceeding, Paul Erlich (whose audio clips you listened to at the top of this page) took another tack. I had suggested that the “real” high third might likely be found by searching between the one-cent measurements usually considered adequate for microtonal work. I made this suggestion because I had not been able to find “my” high third while adjusting pitches in cents on my synthesizer.

Paul prepared a set of triads having various placements of thirds. I and other participants on the list were invited to select our preferred items from blind choices. Paul later disclosed the actual mathematical descriptions. My preferences were fairly consistent, but they did change somewhat from day to day. I began to think that either there was no actual high third or that it still did not appear here because none of these examples actually “nailed it” the way one does when tuning pitches in actual performance. For about a week, I elicited judgments from visitors to my office. Experienced ears tended to agree somewhat on certain “high-third” examples.

A breakthrough?

Then a very startling thing occurred. Paul posted a series of "high third" triads. When I heard the first triad, I felt my heart leap. "Paul has discovered the high third," I thought.

Click to listen: (Wave, 3.36 mg) or (Mp3, 316kb)

Another list member, Joseph Pehrson, also reported that he heard this triad as having a third higher than tempered-tuned. Others said they simply heard a just-tuned triad. Paul then revealed that the third in that triad was in fact an acoustically correct 4:5 ratio to the root. As you might imagine, this evoked some lively conversation.

Paul then posted an audio clip in which a just-tuned (acoustically-tuned) major triad was followed immediately by a tempered-tuned (keyboard) major triad. It’s the one you heard at the top of this discussion. When I first heard the clip (before Paul revealed its content), I sensed that the third of chord moved downward in the second tuning, as did some others. Most reported that they heard the third move upward in the second tuning. I found that when I focused on the pitch of the third, it could clearly be heard as ascending. I then shifted back and forth, first focusing on the chord as a whole and then focusing on the single pitch. My experience appeared to be consistent, hearing the high third only when I focused on the triad as a whole.

That experience clearly favors the idea that the high third is a psychoacoustic illusion. But what about the actual raising of the physical third found in my college choir recording? Do some of us actually physically move the pitch of the third because we are influenced by an illusion of what is consonant? Are singers and string/wind players more susceptible to this “illusion” than are keyboardists and other instrumentalists?

What do you think? Email us and we’ll post your ideas on our Sharing page.

Update: No conclusive trends have emerged so far. Evidently there is more mystery here than a simple survey can solve. Any grad students interested in a research project???

 

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