Helen Bledsoe

Category: intonation

  • Sixth-Tone Exercise Audio

    This is a very simple exercise for playing sixth-tones The audio file has undulations on three notes: (I picked these because they appear in a piece I am coaching.)

    • Bb3 – Bb3 sixth-tone lower
    • Db4 – Db4 sixth-tone lower
    • C3 – C3 sixth-tone higher

    All you have to do is practice at first playing with the undulations, then against them so when the audio plays Bb – you try to play Bb sixth-tone lower, and check yourself as the tone moves. Here is a crude display, you can imagine the audio as the black line, your sound as the blue line.

    Tuning is A=441

    Here is the audio file

  • Intonation Exercises

    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!

    CLICK HERE FOR INTONATION EXERCISES PDF

  • Atonal Intonation in Light of Berio’s Sequenzas

    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

    Inharmonicity of hearing

    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

    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:
    G
    E-flat
    C

    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

  • Intonation I : Flutonation

    Practicing intonation, I’ve noticed a few funny quirks of mine which I know are shared by many other flutists, so I think it is worthwhile confessing and hanging them out to dry.
    (By the way, barring any live musicians I can scare up to do my nerdy exercises, my partner in intonation practice is my Korg OT-12. It’s a bit chunky and pricey, but it offers decent range of sound output. It is also recommended by orchestral piccoloists for its good registration of high pitches.)

    But first I need to get another pet peeve off my chest: flutonation. It’s the natural intonation of the flute. I often hear it when a flutist is playing a solo piece without accompaniment, esp. a contemporary (atonal) piece that they think doesn’t need to be in tune. Oh boy….eyes rolling…..
    I admit that I have flutonation in spades: C# too sharp, low and mid Eb too flat, but the high Eb too sharp. This is why I hate hearing it in other players 🙂

    Now for those other quirks:
    When tuning unisons, I notice I tend to tune just a few cents sharp. I realized why after awhile: when perfectly in tune with my OT-12, the sound of the OT-12 disappears completely! The harmonic structures are so interlocked that they are indistinguishable. If I’m a little sharp though, I can still hear the tuner. And in my quest to always listen, to play so I can hear my partner (even if mechanical), I play so that both can be heard. Funny, huh? A case where overdoing one aspect can mess you up in another area. Sort of opposite the way pitch rises in orchestra – where you play sharp so you can hear yourself.

    Another thing is octaves. I can tune vertical octaves without any silliness, but melodic (horizontal) octaves are another story. They are almost always too wide. I don’t know why, maybe I have played the flute too long and have a severe case of octave flutonation. Then there is picctonation. I hear octaves on the piccolo too narrow. That’s probably because I like to play with the cork rather close to the embouchure hole – but still, I should know better. I really have to re-train my ears with a fixed pitch instrument or my korg. I’ve developed some exercises for octaves with my korg that involve listening, not looking at the blinking lights.

    In general, I’ve got several exercises for tuning with a tuner that involve listening to combination tones, complete with explanations. They are no longer publicly on my site because I am considering publication, but- if someone is really interested I can send them pdf.