Intonation Exercises

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Here is a compendium of intonation exercises I have written over the years. They require either two players or one player and a sound-generator such as a tuner or an app. (The exception is the “Exchange” exercise.)

These exercises are based on being able to discern and manipulate difference tones, and contain a basic introduction to just intonation.

If you distribute these exercises please give credit where it is due. Have fun!



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.