What are notes in music?

Factual Questions
1

Basic music question. Why is a note a certain frequency and not another?

Musical notes all seem to be quantised into certain frequencies, with their harmonics being the same note in other octaves. I understand that A is 440.000 Hz. Why is it not, say, 400 Hz?

Even across cultures, different scales use the same quantised frequencies. I understand, for example, that from the chromatic scale Indian music has a C (523.251 Hz), even though it doesn’t use D (587.330 Hz).

I think I might be missing something fundamental (he he) here. 440.000 Hz looks very precise so I suppose it was was chosen for A. If 400.000 Hz or 420.000 Hz was chosen would music sound completely different?

2

There’s no reason why A is 440hz except it has now been defined as such. There is no “natural” frequency for A; it’s just convention. Other times in musical history, it was slightly different.

In other words, A came first. The frequency of A was determined later. And it has gradually risen over the centuries (very slightly).

Once the frequency of A is set, all other note pitches are derived from that. If A is 440, one octave higher is twice A, or 880, etc.

3

Nope. In fact, many people could not tell the difference in isolation. Many movies destined for PAL television had both video and audio stretched from 24 fps (native film rate) to 25 fps (half the native 50 rate). That’s equivalent to the 440 Hz A played at 458.3 Hz. Maybe some people with perfect pitch could tell the difference, but most could not.

Of course, it’s necessary for all the notes to be stretched by the same amount. Just changing a single note would be obvious to most.

4

Wikipedia has some info on pitch standards in Western music.

Basically, for a while, all anyone cared about was that an instrument was in tune with itself, and instruments that were playing together were in tune with each other. It wasn’t such a big deal if you go to one church and their pipe organ is a semitone higher or lower than the church you went to last week across town.

Eventually, instrumentalists started trying to outdo each other by playing a [sub]tiny[/sub] bit higher than the one before, because higher notes often sound brighter than lower ones. Eventually this led to singers complaining when they couldn’t hit the notes they used to. This led, for some reason, to the French codifying the frequency for A (and then a bunch of competing standards).

Nowadays, the reason A is A (mostly) across cultures is probably largely because any instrument that can produce an A will have been made with the current standard as a reference (or, for instruments that don’t have discrete notes like strings or the voice, either training or tuning against a known standard will produce something close to the standard “A”).

As for telling the difference between 440, 420 and 400: If you play one right after the other, the difference is enough that you could tell the pitch changed: the difference between 440 and 420 is almost a half-tone and between 440 and 400 is nearly a whole tone (if A is 440, A-flat is 415 Hz, G is 392 Hz). But if you played a clip where a popular song started less than a half-tone lower than usual, I doubt most people would pick up on the difference. But, singers who are very familiar with particular pieces can often tell when a song starts a half-tone off from where they’re used to.

Here’s a site where you can play various notes on the keyboard and find their frequencies: http://www.seventhstring.com/tuningfork/tuningfork.html You can also set A to be whatever frequency you want and see how different it sounds.

5

As you can see from the Wikipedia article, yes, A=440 is an agreed-upon standard, and it has not always been this way. In Western music, if you just go back a couple hundred years, A could have been set anywhere from around A=380 to A=480. There is a tendency for concert pitch to rise, and many modern orchestras tune their As a little bit higher, from A=441 to A=445, as there is a perception the slightly higher tunings sound “brighter” (partly, I believe, because of the timbre of the strings with higher tension on them, but someone with more orchestra knowledge might be able to chime in. It might just be that a slightly higher A just sounds brighter because it is slightly higher.) Players of Baroque music sometimes tune a good deal lower, A=415.

6

Is there any reason for the number of tones and semitones? Why 7, and not 16? Is it just a compromise between being able to distinguish tones and having enough tones to express music?

7

There is a mathematical relationship between the partitions of the octave that seems to coordinate with what our ears (or at least our Western ears) hear. The most obvious is the 2:1 ratio for the octave, but there are other interesting ones (Wendy Carlos site):

8

This has to do with mathematical relationships between frequencies and their harmonics. An A (440 Hz) has harmonics at 880, 1320, 1760 Hz and so on. Certain frequencies “go together” better than others - obviously the octave (fundamental 880 Hz) is best, followed by the 5th (660 Hz). Most of the seven notes in a normal scale are derived from those harmonics, and the semitones (which then make 12 semitones in an octave) follow. Nowadays you usually tune those notes to be 12th root of 2 apart from each other, so a fifth isn’t actually a fifth (12th root of 2 to the 7th power =/= 1.5). But it’s close enough to be able to play in all twelve keys and have them all sound similarly in tune.

9

Have I got this right? : The absolute frequencies of the notes are not really important, it is the relative frequencies that count. It is simply for convention they have been fixed to a chromatic scale with A=440.000Hz.

So all the different scales of various cultures are not naturally quantised into the chromatic scale, they are forced into the scale by convention?

10

What is so special about the 12th root of 2 that makes the frequencies “go together”? I can understand how harmonics would go together, because the soundwaves superimpose evenly.

11

As others have said, there’s a mathematical relationship between a lot of the notes. Our ears in particular tend to like chords made with the fundamental note and the third and fifth notes of the scale (what are called major chords). This tends to naturally break the notes down a certain way. As for why we have our current 8 note octave, we do it that way because the Italians did it that way. Our scales (do re mi fa so la ti do) are Italian. A lot of words used in music, like piano for quiet and forte for loud, are Italian (the piano, as in the instrument, was originally called a piano-forte, meaning “quiet-loud” because, unlike a harpsichord, it could play notes either quietly or loudly - over the years the “forte” bit got dropped and now we call it a “quiet”). Once the Italians formalized how to write music down in a way that people could understand it, the rest of Europe tended to follow what the Italians had done. We still use that same basic notation today.

That doesn’t mean everyone has always done 8 note scales. Before our current 8 note scales became common, there were pentatonic (5 note) scales. This is a bit more limited, but you can certainly make music with it. Basically, you can do it on a modern piano by restricting yourself to only using the black keys.

12, 16, and 24 note scales have also been used. A lot of eastern music sounds weird to us at least partly because they use different scales than we do

12

Equal temperament.

A perfect fifth at a 1.5 ratio sounds perfect because some of the partials overlap, and because our brain is wired to respond to pitch ratios that occur in harmonic series. 1.5 is the ratio between the second and third partials of a harmonic sound.

There’s a problem, though. Take a violin, and tune the open A string to 440 Hz. This means that at a 1.5 ratio, the open E string is tuned to 660 Hz. Play both string together. Beautiful. Now, tune the open D string to 293.3333 Hz. Play D and A together. Another beautiful fifth.

Now, place your finger on the A string to play a B. Play that B together with the open D string until it sounds perfect. You now have a beautiful major sixth, and your B rings at 488.8888 Hz (293.333 * 5/3). Now play that B with the open E string.

Ouch!

What happened? B and E, that’s a perfect fourth: 4/3. 488.8888 * 4/3 = 651.85. Not 660. If you want to play a B and an E together on a violin, you need to tune your B at 495 Hz. That’s the headache of just intonation. On a fretless instrument like the violin, the player can adjust the pitch of each note (and add vibrato to mask potential problems), but keyboard players are out of luck. Hence equal temperament.

Back to physics and the brain.

The harmonic series describes the relative frequencies present in the motion of a perfect string or column of air. A perfect string is infinitely thin and moves only in one dimension. Real strings are not perfect. If you’re talking about piano strings and vocal cords they’re really not perfect. This means that they behave more like thin plates than perfect strings, and hence their partials are not at integer ratios.

The harmonics of a ringing piano string and a bunch of other sounds are at roughly integer ratios. As far as your brain is concerned, all that matters is being able to tell if two partials were caused by the same event, and so the brain tolerates some amount of deviation from the perfect harmonic series.

So, as far as your brain is concerned, a ratio of 1.498 for a fifth is good enough, and it allows keyboard players to play in any key without risk of playing an interval that isn’t good enough for your brain, as in the violin example above.

13

Correct. I’ll just add that it’s the ratios of the frequencies that need to stay the same, rather than their differences.

[quote]

So it’s like this: Say you pick a note to be “middle C”. This allows us to define a “C” in all octaves, just by multiplying or dividing the frequency by 2 some number of times. But how do we find what the frequencies for other notes should be?As noted above, if you take the frequency of middle C and multiply it by 3/2, you’ll get something that’s a perfect fifth above. So that’s your “G above middle C”; and by the logic noted above, that defines “G” in all octave. Go a perfect fifth above that, and you get a “D”. Go a perfect fifth above that, and you get “A”. And so on.If you keep doing this over and over, after the twelfth iteration you get a frequency that is close to, but not exactly, equal to “C” in some octave. (This is because (3/2)[sup]12[/sup] ≈ 129.7 ≈ 128 = 2[sup]7[/sup]: twelve perfect fifths are approximately equal to seven octaves under this system.)

What I’ve described is the Pythagorean tuning system. Unfortunately, while pieces in C major sound OK under this system, there are some intervals that are out of tune with their “pure” counterparts; for example, the notes in a major third “should” have a ratio of frequencies of 5:4 ≈ 1.25; under Pythagorean tuning, it’s ≈ 1.265 instead. This is enough to be audible (check out the C major chord on the Wikipedia page.) Moreover, if you want to play a piece in a key other than C major, it will sound awful.

To solve these problems, we have to “temper” the scale somehow, by tweaking the frequencies so that they sound better for the type of music we want to play. Historically, there were many systems of “temperament” that were proposed once this problem became clear; the one that won out for the most part was “equal temperament”, in which you make the frequency ratios between successive half-steps in the twelve-tone scale all equal to each other. (BTW, this was not the case in other temperaments that were proposed; the ratio of the frequencies from C to C#, say, would be different from the ratio of G to G#.) If you take this ratio and multiply it by itself twelve times, you should get back to a perfect octave; which means that x[sup]12[/sup] = 2, or x = [sup]12[/sup]√2.

ETA: jovan is obviously a ninja. I’ll make up for my slowness by suggesting further reading: Temperament by Stuart Isaacoff, which discusses the historical development of all this business. It’s a good read.

14

There are even stranger choices like 19 and 31. See also the bottom of that page for other equal temperament scales, going as high as 72.

15

This is a nitpick only if you’re a pedantic nitpicker like me.

Just so there’s no confusion, the 5-note scale mentioned here is not an octave divided into 5 equal parts like our chromatic scale is. It “skips” many of those notes and the intervals are not equal. In the key of C, the notes are C,D,E…G,A…C. It’s like a major scale with tones 4 and 7 missing.

16

Nitpick: it’s a 7-note scale, not an 8-note scale that is used most commonly in Western music. It’s a heptatonic scale. You’re counting the tonic and octave as two notes instead of one. An 8-note scale would be something like the bebop scales, which contain an extra chromatic passing tone (where this passing tone is depends on the type of bebop scale being used.)

17

You could have an eleven-note equal tempered scale using the eleventh root of two, or thirteen notes or whatever. But it just happens that if you use twelve, you get close approximations of many of the important intervals such as the fifth. Not so good on the thirds, but never mind. To get a better approximation, you have to go up to something ridiculous like 31 notes, IIRC. Twelve is a remarkably good trade-off between consonant-sounding intervals and a manageable number of notes.

18

Nitpick of nitpick, I suppose. That would specifically be the C major pentatonic. There’s a whole mess of pentatonics you can make in C. C minor pentatonic, C suspended pentatonic, pentatonics than include semitones (known technically as hemitonic pentatonics; C,E,F,G,B would be a hemitonic pentatonic in C major, for example), and so on. It’s just any five notes within an octave.

19

Thanks, but it’s not really 8 (or 7) notes, is it? There are flat notes as well. I don’t have a keyboard so I can’t count how many notes there really are. So why do some notes have sharps and some don’t? Did our music originate from: what sounds good -> compromises -> our scale today? (sorry to hijack this thread)

Also, one last thing I’ve always wondered: the frets for guitars divide all the strings at the same points. Are the notes played on guitars perfect? What if you use a capo and shift everything?

20

There are twelve tones in the western chromatic scale. The more selective scales that are more commonly used, such as standard seven-note scales or the various pentatonics, are built around certain notes that sound “right”. The fifth, for example (so called because it is the fifth note in seven note scales) - that interval is so consonant, being based on the simple ratio 3/2, that it is used in virtually all common scales. Next up is the fourth (4/3), also very commonly used but it doesn’t appear in the major pentatonic, for example.

Fretted instruments like guitars are inherently equal-tempered, so in fact none of the notes are exactly right, compared to the Pythagorean ideals such as 3/2 for the fifth. The fact that a guitar is equal tempered is what makes it possible to change key using a capo.