Just Intonation: Thirds and Sixths, an exercise

I’d like to take the opportunity to write about the benefits of doing intonation exercises with 3rds and 6ths using just intonation.

  • To refine the ear. These are simple intervals, and the difference tone (or combination tone) is strong enough to easily adjust.
  • Flexibility. To make these adjustments, a flutist must be willing to make minute changes of the angle of the air by manipulating any three points: lips, jaw, or rotating the flute in or out.
  • Accuracy in tuning chords (vertical intonation). The theoretical knowledge that, from the bass note, major thirds are 14 cents flatter and minor thirds 16 cents sharper will cut out some of the fishing around for the right direction. (That’s thinking like a flutist. Objectively stated: major thirds are narrower, minor thirds are wider.)
  • Grasp of microtonality. Seriously. Take the second bar of the exercise in the link below. The G is first played as a just major third to an E-flat (=14 cents flat). Then the bass note changes and it becomes the just minor third to E-natural (=16 cents sharp). The difference you have traveled is 30 cents, almost a sixth-tone! You get a feel for these sixth tones, double that, you’ve got third tones and you’re off!

But why do these exercises? After all, I do not propose that thirds and sixths should always be tuned justly. There are many times when it makes sense to tune these intervals using equal temperment, such as when playing with any fixed pitch instrument. (I wish conductors would also take this seriously. How many times have you worked on intonation during a wind sectional rehearsal, when your ears will naturally drift to just intonation, only to have it completely different when you add the strings, harp, percussion or piano?) It also makes sense to play more temperately when you have the melodic line or when you want to make other expressive adjustments such as raising the leading tone.

Another place to avoid just intonation is when tuning minor thirds in minor chords (See Claudio’s comment below). Here, the equally-tempered minor third works better. Remember, if you tune an interval justly, the difference/combination tone you should hear will belong to (or complete) the implied major chord. For example, let’s take the minor chord:

A justly-played C and E-flat will give you a difference tone A-flat, because A-flat is the major chord that the interval C – E-flat implies. That sounds very nice! But add the G and it’s no longer nice because G and A-flat are causing dissonance. This may be why, historically, those beautiful medieval works in minor keys always ended on major chords. See what you can learn about Early Music by delving into the details of intonation! The practices were practical, not academic.

While playing this exercise it will also become apparent why, historically, notes with flats were generally played sharper and notes with sharps were generally played flatter.

Directions for playing with a tuner: during the fermatas, change the pitch of the tuner with the right hand while holding the flute (or piccolo) with the left hand only (use B-flat thumb for Bb and A#). Try not to interrupt playing during this process so you can make the adjustment as finely as possible.

Click here for the exercise (this is the same one that was previously on my website).


Microtonality: some basic tips

Microtonality is the use of intervals smaller than a half-step such as quarter tones, sixth tones, eighth tones and so on.

There are several standard ways in which microtones are used (which may be interconnected):
• As part of a “just” intonation scheme (based on pure intervals instead of equal [keyboard] temperment). Used in a simple way (also known as mean-tone tuning), one plays perfectly in tune in a given key, less so in related keys and totally “out there” in remote keys. So far I have come across no better explanation and description for just intonation than David Doty’s Just Intonation Primer. Or click here for an on-line explanation of just intonation.
• As part of a “spectral” scheme where the notes are not tuned according to equal [keyboard] temperment, but according to the intonation of upper partials of a given overtone sequence. This is related to just intonation (and again David Doty’s book explains this wonderfully), however in a “just”intonation environment there may be a tonality implied, whereas in “spectral” music a tonality is rarely implied (although there may be a tonal center). Here is an example of how the overtone sequence on our low C is naturally (purely) tuned. The deviation from equal temperment is measured in cents. (Cents are measured by dividing an equal tempered half tone into 100 units. These are marked on most tuners to indicate the degree to which one is sharp or flat.)
◦ 1st harmonic (fundamental): C (no change)
◦ 2nd harmonic C (no change)
◦ 3rd harmonic G: (2 cents sharp)
◦ 4th harmonic C: (no change)
◦ 5th harmonic E: (14 cents flat)
◦ 6th harmonic G: (2 cents sharp)
◦ 7th harmonic Bb (31 cents flat)
◦ 8th harmonic C (no change)
◦ 9th harmonic D (4 cents sharp)
For the deviation up to the 31st harmonic, see Wikipedia’s entry on the Harmonic Spectrum.
For tips on how to get these partials in tune without having to resort to watching the cent meter on your tuner (i.e. by ear) read my entry on spectralism.

• Sometimes a composer may invent a tuning system, then it is up to you to determine: a) why/how the composer uses microtones and b) how you should approach them. These questions will help to determine whether or not the composer has included microtones as special “effect”; i.e., should the “de-tuned” notes be given special colors to contrast the “normal” notes, or does the composer want consistency of timbre? This is an important factor in determining fingering (if not already prescribed by the composer).

Once the interpretive questions have been addressed, there comes the time to actually play them. There are several fingering charts available such as Matts Möller’s quarter-tone chart. They are good resources but think of them as starting points. My personal advice is to know as many fingerings as possible for a particular note. Be flexible in the choice of fingering because there are several factors to consider when making your choice:
• the speed of the gesture
• the intervallic relationship to its neighboring notes
• the dynamic and tone color
Although each of these is an important consideration, it is crucial to know your end tempo and always have it in mind. I have made the mistake of carefully going through a score and writing in all the “correct” fingerings for microtones, only to have to change them later as I got the piece up to tempo. Practice the notes in tempo, if only two or three at a time to get a feel for this.

Sometimes the solution can be simpler than you think. Turning the flute in or out to “de-tune” a note can work just as well as a really complicated fingering. For example: on a standard flute there is no stable fingering for F 3/4 sharp. If you need a loud, stable tone just play F# and lip up. If you need a quiet clear tone, finger G and lip down. (Of course if you want that hollow, bamboo sound, use low B and half hole the F-key [index finger right hand].)

A note on different flute models: I play on a quarter-tone Kingma System (produced by Osten-Brannen). This is a really great system, and I can recommend it for anyone who wants to play a lot of contemporary music. These flutes are also suitable for all repertoire. However, every flute has the capacity to play microtonally (as we all know, sometimes inadvertently!). Please don’t be discouraged from playing the modern repertoire if you have a standard flute, even if you only have a student model with closed holes. There is still repertoire that can be played on student models such as the flute solos by Karlheinz Stockhausen. See my repertoire list for more suggestions.


Intonation IV: Our Partner in Crime

By Partner in Crime I mean our most common collaborator in the traditional repertoire: the piano (or harpsichord). Here is some information I’ve gleaned from reading Fundamentals of Musical Acoustics by Arthur H. Benade (standing left in photo, playing homemade flute). To know the acoustical properties of the piano is to be able to deal with its intonational quirks. The burden of this lies with us, the flutists. The piano is not able to adjust to our intonational quirks.

Pianos and harpsichords have what are called inharmonic partials as opposed to a flute’s regular harmonic partials.

A flute sound will be comprised of its fundamental and its more weakly-heard upper partials: an octave (2 x the frequency of the fundamental), 12th (3 x the frequency of the fundamental), octave again (4 x the frequency), and so on. Everything is all laid out and predictable. What an orderly instrument!

Pianos and harpsichords have inharmonic partials due to string stiffness and effects of the sound board. Their partials are spread, that is, the first partial will be slightly more than 2 x the fundamental. This is important for us to know: given the same note, the upper partials of a piano have a higher frequency (are sharper) than the flute’s. Just how sharp these partials are will depend on the quality of piano: an expensive concert grand will have less deviation – an upright with a too-short sound board will have more deviation.

Here is an example of a typical deviation for a decent instrument, based on the frequency of C4=261.63hz
Fundamental Flute:261.63 Piano:261.63
2nd partial Flute: 523.26 Piano:523.51
3rd partial Flute:784.89 Piano:785.91
etc. You can see that the higher you go, the farther apart the frequencies will be between the two instruments.

The piano’s “spread” inharmonic spectrum explains why its attack sounds sharper (higher in pitch) than its immediate decay. Initially a great number of partials are excited, including the higher ones that display spread inharmonicity. As the sound dies away, we are left with the lower few partials, which are more or less “normal”, that is, less inharmonic.

What else does this inharmonicity mean for us flutists?

I need to back up and explain another acoustical lesson from Benade. Sometimes, there is a difference between matching a pitch between alternately presented sounds and simultaneous sounds (p. 268). Why? Because sometimes the ear is matching overall pitch rather than frequency. What?

Overall pitch is what the ears interpret, given any sound with a harmonic spectrum. We infer the fundamental from that sound. That fundamental, which we infer from its upper partials, may not actually match the frequency of the fundamental. (Really! It’s a case of the mind inferring its own reality. Maybe like an optical illusion?)

How does that affect a flutist who plays the same note in alteration with a piano? Given the example above for C=261.63 hz, and “assuming the first six partials [only the first 3 listed here] to be equally important in determining the pitch, one finds that the normal [flute] tone must have its pitch raised about 4 cents…if the two are to agree when presented alternately.” (p. 318)

He gives a further example on page 323:

“Suppose for example that a flutist plays a mezzo-forte G4, maintaining it accurately in tune with the G4 produced by a single harpsichord string (whose inharmonicity is very similar to that of the strings we have been discussing all along). The flute is sounded steadily, and the corresponding harpsichord key is struck repetitively at the rate of about 2 per second, so that the tone is restored quickly after each dying away. (…)If, however, the harpsichordist sounds one more note after the flutist has shut off his well-tuned tone, this last note sounds a trifle sharp to our ears…”

Well, I beg to differ… the harpsichordist will not sound sharp: the flutist will sound flat!

There are a couple of other things that are interesting to know about the piano.
On page 319 Benade describes what he calls “the piano tuner’s octave”. Octaves on the piano are not completely pure, it seems. They are also tuned a few cents too wide. Funny, I would have thought that would be the one interval that could have a 2/1 ratio (the upper note having exactly 2 x the frequency of the lower). But, if you think about that spread inharmonicity, it does make sense.

Furthermore, not even unisons are pure on pianos! (p. 334) This is what blew me away. Most piano notes are produced by multiple strings, which may have as many as 1 – 8 cents difference between them. This is deliberately done to enhance the decay time.

Given the shimmering effect of de-tuned unisons, together with the spread inharmonicity of each individual string, the piano creates a vibrant, pulsing sound. This is one reason I think it doesn’t make sense to play non-vibrato with piano. Of course, there are musical contexts where non-vibrato is appropriate. However, as a general rule, I think the flute should go with the piano’s flow of vibrations!

Photo credit: AIP Emilio Segre Visual Archives