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If I consider the range of musical instruments, one thing that seems to vary is tonality. For instance, a snare drum clearly doesn't produce a tonal sound whereas a piano does.

How does one measure the degree of tonality of an instrument? How can I tell if one instrument is more tonal than another besides just the obvious contrasts between a drum and piano?

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In your question, you use the word "tonality", but I do not think it means what you think it means. Tonality refers to a certain harmonic vocabulary, especially involving the use of dominant-tonic chord relationships. You are asking about the tone quality, or timbre, of an instrument, and what could be refered to as its harmonicity. More frequently, we refer to the opposite concept: inharmonicity. To understand this concept, we have to take a detour into physics.

When an object produces a musical tone, it usually vibrates in various "modes". This means that the object produces numerous different frequencies at the same time, in varying strengths. This distribution of frequencies is known as a spectrum, and is what determines the timbre of an instrument. To illustrate, here are a few extreme cases of spectra:

  • A pure sine wave has a single peak at a single frequency.
  • White noise ("static") has no peaks, and has approximately equal amplitude across all spectra.

Most musical instruments, at least in western music, approximate a one-dimensional object: usually a length of string or a column of air. In an ideal one dimensional case, the object will resonate most strongly at frequencies that are integer multiples of some base fundamental frequency, and all other frequencies will dampen out. If the fundamental is f, the spectrum will peak strongly at f, 2f, 3f, 4f, etc..., and higher multiples of f will generally have smaller peaks. These peaks are called harmonics, and a spectrum that follows this pattern (or an instrument that produces such a spectrum) is said to be harmonic.

Two dimensional objects (such as drum heads) and three dimensional objects (such as bars, or bells) vibrate in more complex modes that don't necessarily produce a harmonic series. In the case of drum heads, there is only a vague sensation of relative pitch (e.g. kick vs. snare, or low toms vs. high toms). In metallophones and other similar pitched percussion, there is a perception of a distinct pitch, but the harmonics are still shifted of off their "ideal" harmonic locations, leading to a distinctive "clangy" sound. When created, such instruments are often designed in such a way to minimize this inharmonicity, by locating the mounting points at a place that dampens out certain inharmonic modes.

Even in string instruments, plucked and hammered stings (pizzicato, guitar, piano) will sound more inharmonic than bowed strings. In fact, over the course of a pianos range, this inharmonicity necessitates a technique known as "stretched tuning". The inharmonicity is worse in stiffer strings, and particularly affect the bass strings on an upright piano (which are shorter and stiffer than those on a grand piano), which explains why grand pianos are considered higher quality.

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I'm not qualified to give an in depth answer, but to me the simple requirement is the mathematical relationship between harmonics.

Eg if you have a note and then a range of harmonics at multiples of the fundamental it will sound tonic. Snare and other percussion instruments have many frequencies that are not related mathematically. These sound atonal.

  • +1 hmm. Yes, that's a start to an answer. I think more depth is needed, but definitely helpful. Thanks! Also, is there a difference between the terms toneless and atonal? – Stan Shunpike Sep 5 '15 at 0:26
  • Just to clarify, are you saying then a tone has to be comprised of integer multiples of the fundamental? I am a bit unclear about this. – Stan Shunpike Sep 5 '15 at 0:33
  • No, but simple mathematical relationships are a key factor. – Doktor Mayhem Sep 5 '15 at 0:35
  • @StanShunpike Dr. Mayhem is referring to the harmonics heard when a note is played on many instruments. Consciously you hear the main note (fundamental), but on almost a subliminal level you hear other notes ringing out simultaneously at frequencies that are multiples of the frequency of the fundamental note. Your brain hears what you perceive as the fundamental note. But the fundamental note is made up of many notes blended together. To see this you can play a harmonic on the 7th fret of a guitar and then lightly touch string at the 12 fret to cancel some of the harmonic frequencies. – Rockin Cowboy Sep 5 '15 at 14:57
  • @RockinCowboy the fundamental note is made up of many notes blended together... it might be better to say that "a note is made of many partials blended together". It is possible to have an audible note with a given frequency with no energy in the partial at that frequency - i.e. with no fundamental at all! – topo morto Dec 30 '15 at 0:12

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