tuning: overthinking inharmonicity

This is an expanded version of a comment I posted earlier this week on the hpschd-l. Anyone who wants to get into historical harpsichord tuning and never come out again ought to search the archives of that list. There’s several lifetimes worth of tuning wisdom and tuning folly to be found there. In this post, I discuss one randomly picked topic from tuning lore: the so-called inharmonicity of strings and what to do with it.

A very short popularized version of the theory says that in stiff or thick (or both) strings the partials are out of tune; thick, stiff strings act acoustically as a rod. Imagine a 1950s staircase and its iron handrails. The rods that attach them to the ground say (when one tried harping on them, which one wasn’t supposed to do) plink/plank/plaing/plunk/plong, but mostly “ploink”; the “oi” factor in this ploink indicates that something – in terms of a pure sound – isn’t as it ought to be. When tuning very thick foreshortened bass strings or the treble strings in a modern piano, we are facing palpable manifestations of this inharmonicity. Even in the strings of early keyboard instruments there is theoretically some inharmonicity. This wisdom works wonders for the fantasy of some insiders.

1) Stretching the treble. In modern piano tuning, the treble octaves are supposed to be stretched, i.e. tuned somewhat too wide, in order to cope with the string’s inharmonicity. However, this octave-stretching varies mightily, even when we only consider the absolute top level of performance of professional modern piano tuners. If you have access to the following recordings, you can compare how the pianos are tuned:
— Rudolf Serkin’s 1984 recording of Reger’s Bach variatons Op. 81, CBS (the end of the fugue is a good testing point)
— Vladimir Ashkenazy’s 1996-98 recording of Shostakovich’s 24 Preludes & Fugues Op. 87, Decca (the second Prelude and Fugue as an example)
— Claudio Arrau’s mid-1980’s Liszt CD that was issued in 1990, with his second recording of the sonata in B minor and other pieces, Philips (listen a bit into “La Chapelle de Guillaume Tell”)
— Murray Perahia’s 1991 Franck and Liszt CD, CBS (the beginning of the Rhapsodie espagnole)
Two of these are examples of extremely stretched trebles (Perahia, Arrau); in the two other ones the stretch is imperceptible for the listener. All examples lie more or less within the scope of ‘what is done today’; nevertheless they seem to vary strongly in the level of deliberateness with which the octave stretch was applied.

In view of the fact that modern equal temperament makes use of very large major thirds anyway, I believe that treble stretch should be applied with a light hand even, or especially, in modern pianos. Listen again to the Major thirds in the spread-out arpeggios at the beginning of Perahia’s Rhapsodie espaniole: in the upper treble, they are painfully and entirely unnecessarily out of true. (All this has obviously nothing to do with artistry and interpretation. I’m criticizing the piano tuner here).

One would think that all this has nothing to do with historical instruments with their relatively thin and soft strings. However, once after I tuned a Viennese 5-octave fortepiano for a performance, someone came with an electronic tuner  (call it a wise-guy-gizmo) and curiously checked what I had done. As it turned out, (and if we can believe the accuracy of that sort of apparatus, which some don’t), I had stretched the octaves slightly, without knowing, or hearing it.

An analysis of this experience should lead to an abrupt conclusion of this chapter: inharmonicity makes one involuntarily compensate in certain ways during tuning. This is something totally different from having to do these things in order to meet the requirements of inharmonicity. In short, inharmonicity does something to us, but we shouldn’t worry too much about doing anything to inharmonicity. Anyone who spends his time doing more than trying to get the octaves subjectively in tune will likely go over the edge and ruin the tuning for the ears of others.

2) Can we actually hear inharmonicity in historical instruments? I have my doubts. True, it always seemed to me that the inharmonicity of certain fortepiano trebles is difficult to cope with. Inharmonicity in a harpsichord, on the other hand, is not something that has ever caused me any trouble – except, I thought, in the most abysmal specimens. But if I re-think this experience, I must admit that I am talking about false strings, which is not the same as inherently inharmonious strings. False strings (strings with kinks, rust, strings that are guided by loose bridge pins, too thin strings that are excited too strongly and so on) are impossible to get fully in tune, so much is true.

3) In conclusion, I believe that the matter of inharmonicity often gets too much attention in harpsichord and early piano tuning discussions. At a recent early keyboard meeting in Edinburgh, there was a doctoral student who discussed the “problem” of tuning the so-called Vis-à-vis by J.A. Stein (1777),  a large harpsichord-fortepiano combination instrument, at length and to no avail with a top specialist in the field. This specialist had tuned the recently restored instrument several times, and said that it had been no trouble at all, in spite of the fact that the fortepiano part of the instruments has thicker strings and a shorter scaling (consequently, the fortepiano strings have, in theory, a higher degree of inharmonicity than the harpsichord strings). The man kept asking “but how is this possible considering the difference in inharmonicity between the parts of the instrument?”

The answer is, similar to my reasoning above that, in tuning, everything is possible as long as it is possible, whatever the theory may be. Successful tuning is about whether our ears can cope with a situation or not, and not about whether the theory allows us to cope or not. This realization should help to reduce about half of the woo-woo out there, and leave us more time to keep our 4′ registers in tune, which can be a real pain.


6 Responses to “tuning: overthinking inharmonicity”

  1. Thomas D Says:

    What people don’t realise (and I didn’t either before working out the maths) is that inharmonicity of the ‘stiffness’ type *doesn’t* necessarily lead to any lack of coincidence between partials in whatever interval.

    In fact, if the inharmonicity coefficient ‘B’ increases smoothly with pitch (as the square of the frequency) then, theoretically, the partials of different notes are stretched just the right amount so that you *can* tune pure intervals with every partial nicely coincident, as if there wasn’t any inharmonicity at all. Of course the frequencies will be ‘stretched’ but no additional beats will appear.

    This ‘theoretically perfect’ behaviour of inharmonicity is obtained if 1) every string is equally thick and of the same material and 2) the scaling is exactly Pythagorean; i.e. length doubling at each octave.

    Problems arise when there are too-large deviations from either condition. This happens not in the treble (unless the scaling gets too long), but in the bass of every harpsichord except perhaps a long Italian. Surely you’ve noticed bass notes having some ‘fudginess’ in their sound such that they can be edged up or down a bit and still make some kind of reasonable octave. This is (I believe) due to their partials being *more* inharmonic than would be required to match with the tenor notes.

    Stretching the treble beyond the call of duty is mainly psychological, I would say.

  2. skowroneck Says:

    Thanks, this explains a lot.

  3. Steve Clay Says:

    Non-piano tuner here trying to understand this concept. I get the cause of inharmonicity, but am not clear about “stretching”. Is this just tuning the upper keys progressively sharper than they “should” be? E.g. a series of As might be 440hz, 881hz, 1764hz, etc. I would understand that this would make the fundamentals of the upper registers in tune with the harmonics of the middle, but wouldn’t this adversely affect intonation with other instruments in those high registers?

  4. skowroneck Says:

    Yes, your idea of stretching is as far as I’m concerned correct (although I haven’t got any information about amounts since I don’t check my results with electronic tuners).
    Talking modern pianos: since the stretch appears most clearly in the higher octaves, where everybody else has to fish for the proper pitch as well, the issue of other-instrument-compatibility is normally a bit covered-up by other concerns. But you do hear such problems often enough anyway. At the end of Rachmaninoff’s third piano concerto (those high-soaring chords), for example. Or in the opening of Tchaikovsky’s first concerto, on the top chords in every group of three.
    But, there are enough issues In the middle register to distract from all this. The normal problems are a possible equal-temperament incompatibility of other instruments, and the harmonics-incompatibility with the piano of some instruments. To illustrate the latter, there is a curious passage in Schumann’s Introduction and Concert allegro Op. 134 in the coda where the sum-it-all-up melody of the piano is doubled in unison by the trumpet. No matter what the trumpet soloist does, it always sounds slightly off. Clarinet is another one of these instruments.
    Short answer: stretch in modern pianos is (or should be) little enough not to be noticeable in combination with other instruments. But sometimes it actually is.

  5. Scott Portzline Says:

    I am very interested in this debate (discussion) and agree with the premise that too many people are impressed by the math, or impressed by their new discovery of inharmonicity. Hence, they “overthink it” as stated by the author of this page. That plays out in tuners giving more “weight” to the “math argument” than the please the “ear argument.”

    May I provide an example of this debate through the use of architecture. Building designers have realized that when standing at ground level and down the street, and looking toward the far upper corner of their building, it appears to be sticking out further than it should. It is a perception problem. To relieve that “condition of the eye,” designers added parapets (a decorative extension) to diminish this optical illusion. (Notice this next time in a city)

    Furthermore, the ear has aural illusions. Paying too much attention to the math would not recognize what happens to the “ear.” Just like, if the architects had adhered strictly to math, there would be no parapet and the eye would not be as pleased. If tuners pay too much attention to the math, the ear can become annoyed.

    Therefore, since we are talking about human perception, I like a good tuner who can please his and my ear. And, that certainly varies because of perception, for the tuner and the listener.

    I think too many people are impressed by the math when we are really trying to please the ear – not a calculation.
    Scott Portzline

  6. Joel Lidstrom Says:

    This is an old thread and may not still be viable. But I’ll comment, and see if anyone is still there and interested.

    I am a concert tech having spent forty years “stretching” octaves. The first thing I’ll say is that Mr. Portzline is right; this is all about “pleasing the ear” and not about calculations. Calculations may objectify—and hence explain—what it is that pleases the ear. But once objectified, such knowledge is but a small aid in learning how to “stretch” the scale. It takes extraordinary ability, well-honed; a dogged determination; and a level of humility as one struggles with the slight difference between well-stretched and over-stretched.

    Why do we stretch octaves? Because

    1. any given string on the piano produces sharp overtones to one degree or another
    2. this sharpness increases the higher we go in the overtone series
    3. so-called classical music is played over a spread of three-octaves (and more)
    4. the wider the compass, the more noticeable the flatness of un-stretched octaves when playing with hands spread wide.

    How does an under-stretched piano perform? A concert instrument with too little stretch has no “singing” quality, no sustain, no “flight to eternity”. A beautifully stretched piano seems to expand forever…

    How do we stretch? If we tune octaves “somewhat too wide”, we’re in for trouble. And here is the crux of why the best concert tuners are so good: Octaves must be stretched to exactly match the temperament AND end up so that each note of the piano has a fundamental that is nearly identical with the eighth harmonic (seventh overtone) produced three octaves below. I say “nearly”, because the preferable stretch may be to match the sixth harmonic two-octaves-and-fifth below, depending on the piano and where in the treble one is tuning. (Of course this stretch is effected throughout the piano, but it is in the treble where it is most evident to the ear.) There is no direct means of tuning a note to match the harmonics of a note three octaves below! This takes years of learning how to control the stretch of every note in every octave so that retrospectively you discover that yes, you did achieve your goal, or no you did not. This may be why Mr. Skowroneck complains about tunings he hears from ostensibly fine tuners.

    That said, we have to be careful not to make the mistake of blaming octave stretch for ugliness the result of a mediocre temperament, or a stretch that doesn’t uniformly replicate the temperament at the desired harmonic level, or brittle voicing that accentuates upper partials.

    As for pianos playing with other instruments, what are you going to do? We can’t tune octaves flat for accompanying clarinets or pipe organs, or tune them sharp to accompany saxophonists. We can only strive for excellence of octave stretch; for under-stretching produces a closed-in and poorly sustained sound, and over-stretching produces wildness not a little painful to the ear. The best is always beautifully stretched octaves, and instrumentalists who listen and wonderfully play in tune…with the piano!

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