Do you use difference tones to tune your multiphonics?

24 Jul 2015

Jenny Gooder July, 2015

I really enjoy these videos, and am learning a lot. I have a question about tuning my G/E. I noticed that when you play, the E is lower for the multiphonic than it is for the NBC tune you played. I also noticed that you get a really strong difference tone "C" when you play the multiphonic. Is this on purpose? Do you use the difference tone to tune the interval? I'm really curious about this.

Bonnie McAlvin July, 2015

+Jenny Gooder Wow! What a great question. Thank you Jenny. This is a big can of worms, but I will try to be as clear as possible. I'll also think up a demonstration which clarifies the difference between equal temperament and just intonation, and include it in the next video.

The quick answer is yes, I use difference tones to tune, and I play in just intonation as often as I can when I play multiphonics. Having said that, the design of the flute (which was not built to play chords) sometimes makes it necessary to settle for something closer to equal temperament.

The more in-depth answer is that there are many 'styles' of tuning. Modern pianos are tuned in what we call equal temperament. This means that the half steps of the chromatic scale are all roughly equal. The distance between B and C is almost identical to the distance between C and C#. Electronic tuners are tuned to equal temperament as well, and as long as you're playing with piano, equal temperament does the job just fine. You can play the notes in tune with an electronic tuner, and it will sound mostly-good with a tuned piano.

However, equal temperament is a compromise that we make so that we can go from key to key freely. You might notice that the Mozart Concerti and Bach Sonatas that we all love are written in just a couple different keys. That's because in those days, keyboard players tuned in various 'mean-tone temperaments', which allowed the musicians to play in a couple different keys and sound really good, but not stray tremendously far from the tonic key. For instance, if the piece began in D major, it wouldn't stray too far from the keys that have only a few sharps. Modulating to the key of Ab major would really sound ridiculous, unless the harpsichord player retuned their instrument suddenly (lol, while they were playing). In order to minimize re-tunings, most music was written in the same few keys.

Do you notice that the further along we move in history, the more key changes happen within one piece or movement? This trend correlates with a trend toward using equal temperament. (But it's hard to say whether one trend caused the other, or vice-versa).

Equal temperament and mean-tone temperaments are both compromises to what we call just intonation. Just intonation strives to replicate ratios between notes that occur in nature. String players learn that when they divide their string exactly in half, they get an octave, which means the string vibrates twice as quickly, and the pitch sounds one octave higher. If you divide the same string into thirds, the two string lengths (the open string, and the same string touched at 2/3 of its distance) vibrate in a ratio of exactly 3:2. This produces a Perfect 5th above the octave. All of the intervals of the major triad are in these types of perfect ratios--not 3:2.01, or 3.01:2.11. Exactly 3:2, or 4:3, or 5:4--exact whole numbers. When we play in just intonation, we vibrate the air molecules in these perfect ratios, and it sounds more full, rich and harmonious.

However, just intonation has its own limitations. If we were divide a octave (say, the distance between A-440 and A-880) by 12 (there are 12 notes in our chromatic scale), we get an irrational number. If you don't remember what an irrational number is, it means that there is a crazy remainder. Check 440 divided by 12 on your calculator. You get something like 36.66666 repeating to infinity. How do we tune a piano string to infinity????

So we can't tune a piano according to just intonation...because we have this remainder that we just don't know what to do with. (This is called the 'comma' in piano tuning). Where people put that comma is what differentiates all the different tuning styles throughout western music's history.

The comma is not overly noticeable when playing with moveable-pitch instruments, such as other flutes, strings, clarinets, etc. So when I play duets I play in just intonation. When I play a chord in orchestra or with other moveable-pitch instruments, I play in just intonation. If I'm playing with piano, and the texture is rich, I aim for equal temperament, since this is the tuning that the piano uses. However, if there is a true duet between flute and one hand of the piano (two notes at a time total), I take advantage of the opportunity to use just intonation whenever possible. Off the top of my head, I seem to remember some moments in Copland Duo which allow this: the flute plays one note, and the piano plays one note.

Your question made me wonder if playing NBC was a useful thing at all for the tutorial...I definitely want people to play Parallel Transformations in tune, because it is a tonal piece; it relies on the chord progression for its effect. I felt that opening this can of works, and talking about just intonation vs. other types of intonation would be too much for 3-5 minute videos, so I resorted to the popular ear-training method of singing memorable melodies to find intervals. But I will think again about this, and fortnight 7 might take a different direction. I really, really appreciate the question, and I'm looking forward to thinking of ways to tame these worms so everyone can learn more!