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Quote from: http://sethares.engr.wisc.edu/consemi.html Relating Tuning and Timbre: “Clearly the timbre of an instrument strongly affects what tuning and scale sound best on that instrument.”

Before reading this I’d never considered the quality ‘timbre’ as having any judgmental relationship to ‘tuning’. To me, timbre is merely the quality of the sound an instrument produces. Apply timbre to one note and what results is the sound of that one note as produced by the instrument generating the timbre'd sound in question.

I now understand that relating tuning and timbre and consonance can be a complicated subject involving complicated math (so the article confirms) and is theoretically probably best explored through electronic ‘instruments’ capable of generating various waveforms.

QUESTION: Citing ACOUSTIC instruments,—in other words can you provide examples of musical instruments whose timbre makes them less consonant relative to, say, 12-tone equal temperament?

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    I don't believe, using 12-ET, that an acoustic instrument will sound different due to its timbre, in tuning to a particular key. If that's what you mean. – Tim Sep 24 '16 at 18:02
  • music.stackexchange.com/questions/41223/… may be relevant. – topo morto Sep 24 '16 at 18:34
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Timbre describes the spectrum of a sound, most noticeably, the distribution and relative strength of it's overtones. Most acoustic instruments are approximations of a one-dimensional oscillating body, usually either a string or a column of air. Such a body will intrinsically vibrate with overtones that align with the harmonic series, leading to a harmonic timbre. 12-TET is a compromise system that attempts to approximate the consonances of such a harmonic series (the 2nd, 3rd, and 5th overtone being equivalent to the octave, perfect fifth, and major third), while also having the property of isomorphism (all the intervals are equally-spaced).

However, there are two cases of acoustic instruments that I can think of where its timbre is somewhat out of line with a harmonic series, a property called inharmonicity.

The first case is instruments that do not approximate a one-dimensional object as closely. These are typically pitched percussion instruments, especially bells, and to a lesser extent, marimbas, xylophones, tubular bells, etc. In this case, you have a 3-D body vibrating in multiple dimensions, with considerably more complex overtones. This is what leads to these instruments generally having more of a characteristic "clangy" sound. And even still, these instruments are generally carefully tuned to and played in something like 12-TET. At the other extreme, if such instruments are not tuned, you can get a massive wave of sound, such as with a gong or cymbal.

The other case is pianos, where the thickness of the string (relative to length) combined with its rigidity, cause the higher overtones to be slightly sharper than the ideal harmonic. Piano tuners have to take this subtle effect into account; in order to make the piano sound "in tune" with itself, the higher keys have to actually be tuned slightly higher, and the lower keys slightly lower relative to the same note in a more central octave. This technique is called stretch tuning. The inharmonicity that produces this effect is also more pronounced in smaller pianos (where the strings have to be thicker) which is why they generally have a less-pleasant, more "clangy" sound.

Here's a video about inharmonicity, in the context of a bell.

And here's a video about stretch tuning on a piano, which also points out that guitar makers have to take this effect into account when placing frets.

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I tend to play quite a bit of Bach on accordion and have some choices of registration. I can play either with cassotto (a resonance chambers dampening overtones) or without, and I can play using "tremolo" in various gradations. Tremolo is not really a true timbre: it consists on adding a slightly detuned note to every note, resulting in a beating tone. On my instrument, I can determine just how much of such a note I allow in: if I keep it small, the detuning and consequently the speed of the beating actually is also small (I don't have a good explanation for the involved physics actually).

The last tuning was done by a pretty skilled tuner. If you now play with a single reed in cassotto, you get a timbre that is predominantly reproducing the fundamental frequency and not much else. As a result, all intervals you play have the beatings corresponding to that of equal-tempered tuning. For pieces with changing interval progressions, this results in a distracting progression of beatings of various speeds.

Stuff is better when playing outside of the cassotto. Playing with full tremolo is masking the effects of temperament thoroughly but the result is ungainly to hear. With the current piece I am practising, I basically play the final chord and pull in as much of the tremolo reed until the result no longer is aggravating due to the beating of the temperament and has not become aggravating due to the beating of the tremolo yet. And that's the setting I use for the piece.

I also know, when playing the violin, that there are certain fixed-intonation instruments (particularly recorders) which almost always clash with the violin whenever you play an empty string (which means that you cannot correct pitch with your finger or mask the beatings by employing vibrato), unless you have a very expert player who is able to use embouchure and breath in order to adjust intonation himself.

Of course, a violin being tuned in pure fifths means that a violin itself will not perfectly match tempered instruments' notes by default (except for the A that is used for tuning, of course).

As a rule of thumb: the stronger the fundamental component of your timbre is, and the less disharmonicity the overtones exhibit (thicker strings tend to produce overtones that are higher pitched than the proper harmonics of the fundamental would be, something having a definite influence on piano tunings), the more distracting the beatings of equal-tempered intervals become without the possibility of manual correction.

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