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 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).


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.


Intonation II: “Gimme That Ol’ Time Religion”

Here’s a still from the TV broadcast of us playing Henza’s Requiem
in the Cathedral of Cologne, April 2009

Our latest concert presented quite a challenge! The Cathedral of Cologne has an evening temperature (this time of year) of 15.3˚ C (59.4˚ F). That was with spotlights and extra spots from the television crew.

As we bundled up to play Hans Werner Henza’s Requiem, we took great care in tuning. Actually, it wasn’t too bad, all things considered.

Here’s what we had to consider:

From one equal-tempered semitone to the next there is a 6% increase in frequency.

A 10˚C change in temperature is equal to a 2% change in pitch frequency. That’s a whole third of a semitone!
[From "The Musician's Guide to Acoustics" by Murray Campbell, p. 201]

The Cathedral air was not a whole 10 degrees below room temperature (22˚C) but it was enough to make things really tricky.

So what was it like playing in one of the world’s tallest cathedrals? The acoustics of the choir area (on the east end, behind the altar) are not bad. The choir of Cologne Cathedral, measured between the piers, holds the distinction of having the largest height to width ratio of any Medieval church, 3.6:1. [info from Wiki] The whole cathedral is so large and so high that it is almost like playing outdoors, you don’t get that “churchy” acoustic. The nave is 43.35 meters high (144.22 feet) – the 4th highest in the world. Only when you stop playing, do you hear a long, long, decay of the sound.

When I stepped into the interior of the Cathedral for the first time in the summer of 1995 I almost cried, it was that moving and impressive. I am used to tall buildings, I’ve even been up the Sears tower in Chicago, but being inside such a vast structure is awe-inspiring. Imagine the impression it made on the pre-modern psyche!

Back to our intonation question:
How to stay on top of these extreme situations, hot or cold?
*keep flexible by practicing note bends – both ways. I find this absolutely crucial in piccolo playing.
*know alternate fingerings
*know the tendencies of other instruments under these extremes (strings go sharp in the cold, not like us!)


Intonation III : the Spectre of Spectralism

Some days ago I got the score for G. F. Haas’ new work „ … wie Stille brannte das Licht“. (What is it with German-speaking composers and their titles with elipses?). It got me thinking about how different composers notate microtonality. I like what Haas has done, it is explicit in placing the note within a frame of reference.

The notated C quarter-sharp in bar 241 is the 11th partial of G, and the A-flat in bar 245 is the 21st partial of E-flat (along with the indication that you are in a perfect fifth with the clarinet). I like having this kind of information in the score, but if he had notated the exact deviation in cents, that would have been even more helpful. This is easy enough to find out in Wiki

That 11th partial should a C# 49 cents flat, and the 21st partial should be an A-flat 29 cents flat. Like I said, easy enough for find out, but it would have been nice if the composer had provided this information.

We haven’t had rehearsal yet, so I can’t say how this will sound or how easy this will be to hear. [ed. read my follow up at the bottom of this post]
So now the question arises: How does one practice this stuff?

First of all, I get my tuner. Trying the A-flat first, I make sure that our Eb’s are in tune. Then I play an A-flat ca. 29 cents flat. Then I keep that tone while putting the sound on to Eb and I hear a Bb combination tone. This is a clue that I am on the right track, or within the correct overtone spectrum. Your combination tone (it may be a true difference tone or not, depending on the timbre and register of instrument) should lie in close relationship to the fundamental – say an octave, fifth or major third (which is the 5th partial, you don’t want to go much farther than that). Put in simpler terms, it should be part of the major triad formed by the fundamental.

Now to try the C# against the G. Again I tune the G’s, then test my C3 so the needle goes about 49 cents flat, then put the sound on G. My difference tone is B natural this time, still close enough (a major third – the 5th partial) in relation to G to be correct.

Again, I have no idea how this will work in the context of the piece, or if it will be heard. But now I know, theoretically, how much adjustment is “correct”.

Whether this works in context or not, I love working with combination tones. Scientists are still not in agreement as to what they are – but it is a wonderful example of how our brains work – how they “fill in the blanks” of the overtone spectrum. I wonder if this is the same phenomenon that allows transistor radios to work? Only the upper partials are projected, the brain fills in the rest.

Also, I’ve noticed that I’m one of those people who can read things like this, hence my spelling problems, most likely. (Thanks to my like-minded Uncle T for this text):
fi yuo cna raed tihs, yuo hvae a sgtrane mnid too
Cna yuo raed tihs? Olny 55 plepoe out of 100 can.
i cdnuolt blveiee taht I cluod aulaclty uesdnatnrd waht I was rdanieg. The phaonmneal pweor of the hmuan mnid, aoccdrnig to a rscheearch at Cmabrigde Uinervti sy, it dseno’t mtaetr in waht oerdr the ltteres in a wrod are, the olny iproamtnt tihng is taht the frsit and lsat ltteer be in the rghit pclae. The rset can be a taotl mses and you can sitll raed it whotuit a pboerlm. Tihs is bcuseae the huamn mnid deos not raed ervey lteter by istlef, but the wrod as a wlohe. Azanmig huh? yaeh and I awlyas tghuhot slpeling was ipmorantt!

[Follow-up added May 5, 2009]
The passage in question was acually not so difficult to hear! You are a part of this “lump” of sounds that are related to the fundamental. It is tricky that he has two harmonic spectrums going at once: those of G and E-flat.

When considering dynamics in this piece we realized that there was a lot more going on than just playing loudly or softly. It helped to think of crescendo/decrescendo passages as adding/subtracting harmonics to your individual sound rather than just making the sound louder or softer. This made for a much more interesting color. Also, the very quiet passages must be played with focus and good attack. Even the quietest notes need some harmonics in the sound, none of this fluffy airy stuff! It just didn’t match the color system.

I do admit the 11th partial gave me the most trouble in this piece. Tricky to hear! It must be 49 cents flat. I must make some exercises.

Speaking of which, I asked the composer and my colleagues how one can study and practice this music. The answer is always the same, learn the overtone series, horizontally, note by note, by ear. How can one do this? Programming a synthesizer seems to be the most popular idea. However you do it, once the sounds are in your head, you have to find a way to play them (of course, in a comfortable range of your instrument. I’m thinking middle octave) by using a combination of alternate fingerings and lip bending. Another project for me!