Atonal Intonation in Light of Berio’s Sequenzas

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Luciano Berio‘s Sequenza no.1 for flute is one of my favorite pieces. I also love teaching it; it has so much to offer in terms of technique (especially articulation!), style and presentation. There is an unwritten book inside me about this work, but for now I would like to consider one aspect of the work that often gets ignored: Intonation.

This is one reason it is not my favorite piece to listen to. If you know me, you know I don’t have perfect pitch, or even flawless intonation. Nevertheless, neither you nor I can assume that a work without a traditional tonal center and without traditional harmonic relationships is devoid of centers and relationships entirely. I would argue that in this context, these matters require even more consideration. I’d like to address this generally and specifically, not as a how-to guide, but as food for thought in your own practicing.

In general, there are rules of thumb for atonal solo works. Here I quote Doris Geller’s “Praktische Intonationslehre“, page 117 (my translation):*

In free-tonal music there is also a hierarchy of intervals, the most important points of orientation being the prime intervals (octaves, fifths, fourths), especially when they form tones that draw attention to themselves. These could be, for example, long, held-out notes or notes that follow a rest.

Here she is referring to Debussy‘s Syrinx, and gives specific examples. However,  these words and her further advice to analyze goal notes, high points, low points, and melodic turning points can apply to all solo works. Edgar Varèse‘s Density 21.5 especially offers the same points of consideration.

Specifically for the Sequenza, I consider the soul of the work to be in the long, held-out notes. If you listen to the other Sequenzas of Berio,  you will hear this particular pattern of drawing the listener in. Often there are rapid, virtuosic passages punctuated by the stillness of a single note, where the quality of sound and the relationship to its environment are of utmost importance.

*In a previous entry, I write more about Doris Geller and the intonation of melodic intervals.



Inharmonicity of hearing

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In a previous post on flutonation I admitted my tendency to play melodic octaves too wide. Re-reading Doris Geller‘s super book Praktische Intonationslehre  I realize this is a universal phenomenon, which she describes as the “inharmonicity of hearing” or perhaps “inharmonicity of the ear”. (original: Inharmonizität des Gehörs).

Doris Geller

Doris Geller

Here is my paraphrased translation of what she has to say (original German below):

The ear seems to naturally favor tones whose harmonics are spread, as those of a piano are. [Read about this in my previous blog entry here.] This preference is most strongly expressed when we hear successive tones: we find jumps of octaves, fifths, and fourths ideal when they are slightly wider than  justly tuned.

And not only these intervals! In a small-scale study she shows the range and average of what students and teachers considered to be an ideal-sounding melodic, linear interval (as opposed to a chordal, vertical interval) relative to equal temperment:geller

The vertical lines for each interval show the range in which the test subjects found the interval to be ideal. The short horizontal dashes (which may look more like a dots in this picture) through these lines show the average. The 0 line is the interval at equal temperment. The intervals are referred to by number (8= Octave), “kl.” means minor, “gr.” means major.

This really shows how subjective listening can be.  3rds, 6ths, 2nds and 7ths can bear a fairly wide range of variation: they are often context-dependent on their environment. It is no surprise that, out of context, listeners will find their own comfort zone.

I paraphrase from page 84:

The largest diversions at 5.6 cents are quite small, but this can add up. If for example you play these wide fifths or fourths in succession, the third note will produce an octave 8 cents too wide. A whole tone scale with these “ideal” seconds would produce and octave 30 cents too high! However, in solo playing, our pitch memory keeps us from making these mistakes, since temporary overall pitch relationships and the relation to the tonic of the key keep us in line.

Hopefully, when playing an unaccompanied, tonal solo piece, your intonational “snap to grid” function is checked!

Since we get more than the lion’s share of melodic roles, and we have a fantastic body of unaccompanied literature, the expressive and tasteful use of intonation is an important tool for flutists. I am fascinated by the possibilities, and the different practices of vertical (chordal) and linear (melodic) intonation. You will probably be hearing more from me about this as I continue to re-read. It is a pity that Ms. Geller’s book has not been translated into English.


original German from page 83:

Das Gehör scheint von Natur aus Klänge zu bevorzugen, deren Teiltöne als etwas gespreizt liegen, änlich wie es beim Klavierton der Fall ist. … Noch stärker äußert siche diese Vorliebe aber bei nacheinander erklingenden Tönen. Wir empfinden einen Oktav-, Quint- oder Quartsprung erst dann als ideal, wenn er im Vergleich zur reinen Stimmung etwas zu weit ist.

page 84:

Die Abweichungen sind mit maximal 5,6c zwar gering, doch können sie sich bei Aneinanderreihung in einer Richtung erheblich aufsummieren. Wenn man z.B. Quinte und Quarte in ihrer erweiterten Form in einer Richtung hintereinander spielt, gelangt man beim dritten Ton zu einer um 8c erweiterten Oktave. Und würde man eine Ganztonleiter ausschließlich aus “Idealsekunden” spielen, so wäre die Oktave am Schluß um 30c zu hoch! …In der Einstimmigkeit bewahrt unsunser Tonhöhengedächtnis vor derartigen Intonationsfehlern, denn auch die zeitlich übergreifenden Tonbeziehungen sowie die Beziehungen zum Grundton der gerade herrschenden Tonart werden zur Intonationskontrolle mit herangezogen.


Just Intonation: Thirds and Sixths, an exercise

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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 in real life is when tuning minor thirds in minor chords (See Claudio’s comment below). Here, the equally-tempered minor third works better. Here’s why: 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, well, 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).