I would like to learn more about the kind of sounds used in music that can make a scale discernable. Regular drums for instance do a bad job, even if they can be tuned. I've no idea if that's true really, but hopefully you get my point. From the little I've learned so far I think this is to do with harmonic content. Instruments that play notes produce a note that has 3rds & 5ths etc that naturally occur. Drums don't do that so much. But then a synth can be made to only produce 1 waveform and that works just fine. Is it perhaps to do with how close it gets to white noise? My goal is to explore sounds that can be used for discernable notes, but if I knew more about the subject i'd go down fewer blind alleys. Sorry for the dreadful explanation there! Just after some research pointers. Thanks
Well, sound is pretty much all about spectrum. And it so happens that there is a fundamental difference between typical unpitched percussion sounds and the typical sounds of melodic instruments (even when they are actuated by a hammer like the piano or xylophone): Unpitched sounds are usually the result of a 2D oscillator, while pitched sounds are practically 1D oscillators.
1D oscillators are things like strings, rods, air columns, or even thin strips of wood (xylophone). These typically have overtones with frequencies of the form n times fundamental frequency. Our ear is adapted to recognize such overtone series and will deduce all the overtones to belong to a single fundamental pitch. This is what you need for melodic instruments.
2D oscillators are things like drum skins, plates of wood, bells, cymbals, etc. All of these vibrate in a true 2D pattern, and the overtone sequence of these are very different from the n*f frequency pattern of 1D oscillators. Precisely because of the 2D shape. As such, our ear cannot find a common base pitch easily, and perceives them as more or less pitchless. As such, they are hard to use for melodic purposes, and usually require specialized crafting to produce beautiful melodic sounds. For instance, if you want a bell to sound with an easily recognizable tone, you need to be very careful about the precise shape of the bell, or it will have a very dirty sound.
Note that the 1D oscillator is an idealization. True strings do not have their overtones perfectly in tune: Due to the rigidity of the strings (most notable in the thick and short soprano strings of a piano) their overtones will be just a tad sharp. Sharp enough that you cannot tune a piano to the theoretical 12 tone equal temperament. Our ear still recognizes these almost correct spectra as 1D oscillator spectra, though.
For the purposes of this question, lets consider three broad categories of musical sounds:
- Unpitched sounds - examples include cymbals, bullroarer
- Semi-pitched sounds - examples include tom-tom drums, wood blocks
- Pitched sounds - examples include woodwinds, strings, brass, xylophone, glockenspiel, most non-percussion instruments
Maybe instead of categories we should consider it more like a spectrum. At one extreme end are sounds that create no sense of pitch at all, or very little sense. A crash cymbal does sound "high" and sizzly, but we would never ascribe a particular note to a crash cymbal. Some crash cymbals sound higher and some sound lower, but again, we can't hear a musical interval in a crash cymbal, and a crash cymbal never sounds out of tune with other instruments.
In the middle of the spectrum we have things like toms from a drum kit. These kinds of sounds clearly generate high, middle, and low sensations when we hear them, and we can hear somewhat of an actual pitch in many drums. Drum kit toms can and often are tuned to actual notes for the fundamental frequency. Still, a drummer can play "tuned" toms with a wide range of instruments and never sound out of tune with the band. That said, there are many examples of drums being tuned to fit a particular song, generally in recording sessions where each song can be catered to separately in between takes.
And at the opposite end from unpitched sounds, we have firmly pitched sounds. Violins, flutes, and trumpets are quite clearly pitched, as are trained singers. We can clearly hear notes and we notice right away when pitched instruments are out of tune.
You are right that the difference is due almost entirely to harmonic content. A sustained pure tone (containing one frequency of sound), especially in the middle and upper ranges of human hearing, generates a definite pitch sense in humans. The flute is probably the closest instrument to this (excluding a synthesizer). Sounds that include harmonic overtones above a fundamental pure tone also are firmly pitched sounds. Harmonic overtones are those that have frequencies that are integer multiples of the fundamental frequency. A more complex sound like a violin has harmonic overtones with decreasing intensity at 2, 3, 4, 5, 6, 7, etc. times the fundamental frequency. It's this mathematical relationship between the multiple frequencies that make the sound have a pitch.
At the other end of the spectrum are sounds where the different frequencies that make up the sound have no mathematical relationship. A cymbal crash is essentially a burst of shaped noise, which is a random assortment of frequencies. Shaped means that some of the frequencies of human hearing are not present (cymbals are generally random groups of higher frequencies - the lowest frequencies of human hearing aren't generated). The randomness and lack of harmonic relationship between the frequencies is what causes a cymbal to have no sense of pitch.
In the middle are sounds that have some frequencies that are harmonically related and other frequencies that don't have a harmonic relationship. Toms have some sense of pitch because we can hear and detect notes when we play them, but they also include other overtones (for some fascinating physical reasons which won't fit here) that are not harmonically related to the loudest frequencies, so the sense of pitch gets "blurred" to a greater or lesser degree, depending on the drum, its tuning, and how it is struck.
There are also musical sounds where the frequencies present are close to having a harmonic relationship, but are not quite on. If the frequencies present are very close to having a harmonic relationship, then we hear definite pitches. The piano is an example - the stiffness of the strings means that they don't quite make a perfect harmonic series when they vibrate. Pitched mallet percussion like xylophones, glockenspiels, bells, etc. also fall into this category (some bells have some inharmonic overtones like drums to, as well as near-harmonic tones and harmonic tones). These instruments fall in various places along the spectrum of pitched to unpitched.
Now, there is a way that an instrument that has largely harmonic overtones can sound as an unpitched instrument, and that is if the sound it generates is too short and/or if the pitch of the sound does not stay steady enough during the time it sounds. One comment on the question mentions synthesized drums, and this is a great example. A classic 808 kick drum sound is made by sweeping a sine wave (which would normally sound clearly pitched at mid and high frequencies, like the flute) very rapidly from a high frequency to a very low frequency. If we slowed down an 808 kick sound, we would hear something like a siren going from high to low. Also, often an 808 kick sound is fairly short, to better imitate real kick drums. The combination of the lack of a steady frequency combined with the short duration of the sound means our ears and brains cannot decode a sense of pitch from that kind of sound, even though the source of the sound is a pure sine wave. What's important is the makeup of the resulting sound, and the 808 kick does not have a final complete sound that presents our ears with a harmonic set of frequencies for long enough for our brains to decode a pitch.
One great counter example to the idea that duration of the sound is primary in pitch sense is cymbals. Many cymbals can ring out for several seconds. A tam tam (gong) can ring for over a minute. And yet we never get a sense of pitch, because the sound is essentially noise.
Additional note on synth sounds:
In the question, you wrote "But then a synth can be made to only produce 1 waveform and that works just fine." Note that the single waveforms produced by subtractive synths are made up of multiple frequencies that have a harmonic relationship. The triangle, sawtooth, square, and pulse waveform all generate a harmonic series of frequencies, so we hear them as pitches. A waveform is not a single frequency. The only waveform that is composed of a single frequency is the sine wave.
Synthesizers are great tools to experiment with the spectrum between pitched and unpitched sounds, especially if they contain a noise source and FM modulation capabilities. If you modulate the frequency of one (audio rate - not LFO) oscillator with another audio rate oscillator, you'll quickly be able to generate bell-like ringing sounds and tones that don't evoke as definite a pitch. If you add or use noise in the sound, that also decreases the sense of pitch. And if you modulate the frequency of an oscillator with a noise source, you can very quickly take a pitched sound and turn it into an unpitched sound.
"Regular drums for instance do a bad job, even if they can be tuned. I've no idea if that's true really, but hopefully you get my point."
You would need to define what a regular drum is.
I seems like you are interested in synthesis. I will try and help but I'm going to take a physics detour. I have actually developed software that generates synthesized sound from a variety of instruments and other sources.
You mention several things like:
"Instruments that play notes produce a note that has 3rds & 5ths etc that naturally occur. Drums don't do that so much."
"But then a synth can be made to only produce 1 waveform and that works just fine."
" Is it perhaps to do with how close it gets to white noise?"
First let me say that I think you are confusing "waveform" with "harmonics" but that's why you asked the question. A synth may produce "one waveform" but that does not mean that it contains a single note. What is in that waveform depends on how sophisticated the synthesis algorithm is. This can be implemented in software or hardware.
Let me also say that the difference between a drum and say a guitar is not white noise.
Now for some music related physics.
It is true that vibrating systems like a string fixed at two ends or standing waves in an air column contain a natural set of harmonics that are tuned to the vibrating system. The harmonic sequence is fn = n * f1, where f1 is the fundamental tone (that which the instrument is tuned to) and n = 1, 2, 3, 4, infinity. A guitar string tuned to A = 220Hz will also vibrate at 440Hz, 660Hz, etc. It is also true that a few of these harmonics match the 3rd and 5th degree of the major scale starting on the fundamental (if you assume Just tuning). But there are others. In fact, all the notes of major scale, perhaps even the chromatic scale, are "close to" tones contained within the harmonic sequence. The real issue is how do they get excited. The excitation of these harmonics is related to the attack of the instrument, attack = how the vibration is excited. When you pluck a guitar string the initial shape and velocity of the string elements determines the harmonic content of the waveform produced by the instrument. Once set in motion there are various mechanisms in the body of the instrument and the air around it that bleed away energy, this leads to decay of the sound. The rate at which energy decays depends on vibrational frequency, so higher harmonics will decay faster leaving the lower pitch notes at the end of some time. Your statement about drums in not altogether correct. Drum heads are vibrating membranes and just like strings, have a natural set of harmonics. However, for 2D systems the harmonics do not obey the simple relationship fn = n * f1. This means that the overall sound can contain some dissonance.
To synthesize a full waveform from an instrument you need to model the attack and to some degree the physics of the instrument to get the correct sustain and decay behavior across all harmonics. But once all the data is combined it will be a single waveform in time. In that sense there is no difference between a drum or sax in terms of synthesis. Each is governed by different physics and will therefore lead to different combined sounds but we can understand exactly how this is created. For people in my line of work being able to mimic reality from mathematical models is a main goal. But to build a good synthesizer one can also use sampled sound from instruments, apply signal processing to them and use that to better understand the fingerprint of various instruments. Models are very pristine and idealized and tend to deviate from real life, even though the laws of physics from which the model is built is pretty well grounded. The issue is in the complexity of the system. For an acoustic guitar the type of glue in the joints can affect timbre and models may not catch this (unless you want to model the physics of glue, which some do).
Percussive sounds also have a spectrum. When you whack something you create a spectrum of frequencies that are all based on the materials that were whacked and the physics of the whacking. While it may seem like noise it is not likely "white noise" and depending on the nature of the percussion and materials involved one could hear a choir of ringing tones over time. So it really isn't fair to describe them as not discernible as tones or notes. Here is an example that I've measured. I once had access to a very large tuning fork that was about 8 or 9 inches long and tuned to a frequency below 100Hz (thought I don't recall the exact value). Being a stiff rod these forks are governed by a different set of equations than a string and produce very out of tune harmonics (also called overtones). When harmonics mix together in most musical systems they blend in such a way that we hear the fundamental as the "note" being played and the harmonics as the "tone" or "timbre". Terms such as twangy or raspy, warm, etc. are all descriptions of the harmonic content of the note. In contrast for stiff rods and other systems one can literally hear multiple distinct notes from a single whack. This makes identifying "the note" a meaningless endeavor.
Based on your original question you might be interested in waveform synthesis or signal processing to dissect a signal to search for "notes", say in a surveillance device (;-)). Either way you would benefit from a deeper knowledge of wave physics in coupled vibrating systems. I offer a few references for your review depending on your back ground.
 Physics and the Sound of Music, Rigden (very beginner level text)
 The Physics of Musical Instruments, Fletcher and Rossing (there are 2 versions of this, one with no math and the other for engineers)
 On the Sensation of Tone, Helmholtz (rather advanced and the first text on the subject)
 Principles of Vibration and Sound, Fletcher and Rossing (very short but very dense mathematically)