OK, let me preface this by saying I am an engineer and tend to approach things this way. I have started playing the guitar and have become interested in the underlying science of music and this thread (http://boards.straightdope.com/sdmb/showthread.php?threadid=173452) broght the question to mind.
I understand that a note one octave higher is twice the frequency, something that the human ear (other than mine apparently) can identify relatively easily. So where do the frequencies of notes come from. Who decided in the first place? Why are they what they are? The frequency range between notes seems very odd. It would seem that for the human ear to “like” them they should follow a more ordered progression.
Also, could you please explain thirds, fifths, etc. I have looked at several web sites but have either found things that require a vocabularly I don’y have or are just designed to tell you how to do things, not why.
well, iirc (from high school music class ages ago) tones were decided from the use of “modes” (which are a form of scale). The notes were picked because they sounded right.
A third can be major or minor- if it’s major, it’s the distance from the tonic (first note) to the third note of a major scale (eg Ionian mode, C major scale) If it’s minor it’s the distance from the tonic to the third note of the minor scale (which is a semitone less than in the major scale because the minor scale has a flattened third note). A fifth is always perfect because it’s the same whether the scale is major or minor.
As for notes to follow an ordered progression- I actually quite like a whole tone scale (where all the notes are a tone apart), although other people seem to find this jarring and dissonant. An example of a whole tone scale can be found throughout the musical “The Phantom of the Opera” by Andrew Lloyd Webber.
The progression of notes in a scale is ordered, it’s measured, it’s just not as regular as you might expect.
Joining a music theory class might help explain a lot of this stuff for you.
The reason the notes are where they are is that it’s based on what the ear does like. The ear likes thirds and fifths. Since you are starting to play the guitar, try to hit harmonics on the strings and see which ones tend to be the easiest to produce. That is what our music is based around. Then you have to fill in the blanks with some sort of evenly spaced (as far as your ear is concerned) notes. Western music settled on one scale. Eastern music tends to use a different one, which is why music from india sounds so strange to us, aside from the fact that they are often using different instruments. IIRC the more common scale used in eastern music has twice as many notes as ours, which means they divided it pretty much the same way but went twice as far with dividing it up into notes as we did.
The designation that A = 440 Hz is purely arbitrary. You’ll find many examples of music that were recorded with all instruments tuned to each other, but not to any fixed standard, so their “A” won’t be 440 Hz at all.
As for who decided what, the largest influence on western music was originally Italy. Music today is taught with a lot of Italian terms in it. Then over the centuries influences from other cultures was added into the mix, producing what we have today.
In these days of instant gratification on the Internet, people tend to forget about books. There are lots of good books on this topic.
When a sine wave tone is played, things vibrating in sympathy with it (like your eardrum) produce an overtone series, which is whole-number multiples of the original tone. (This is also true of an ordinary object vibrating at some fundamental, since nothing natural really vibrates as a sine wave AFAIK.) The octave is the 2:1 ratio, as you mention. The overtone series for an octave overlaps the overtones of the note an octave lower (although the intensity of each overtone is different). So you don’t have any conflicts, it sounds “good”. Other musical intervals also have various ratios (I don’t remember them without looking them up) so as to also have overlaps with the overtone series.
When two frequencies close together are played, their frequencies add together to produce an apparent tone that is the average of the two frequencies, which pulsates at a rate equal to the difference between the two frequencies. This pulsating can be very irritating to the human ear as it reaches higher rates. For an example, play a minor 2nd interval on your guitar. You can do this by playing Eb at the 4th fret on the 2nd string and playing the E on the open 1st string at the same time. Ouch. This interval is rarely used in music and if it is, it is generally resolved quickly. You might hear it most often in jazz. Any other pair of notes will generate some of the same overtones, or some overtones that create beats, or some combination. In general, the faster the beats, the more irritating the result.
The overtone series is different depending on where you start, so to play perfectly in key an instrument with discrete notes should have a set of notes for each key (not important for the violin, for example, which does not have frets). This was found to be slightly impractical, so methods of tempering were developed. Tempering is fudging the notes so that, for example, you have a single A that is close enough to all the A’s in every key for practical purposes. Several methods were developed but the one that survives today is the equal temperement system. In equal temperement, the scale is divided into half steps that all have exactly the same ratio (a half step is the ratio of the 12th root of 2 to 1).
After you become accomplished on the guitar you will find that the guitar has some inherent tuning problems beyond tempering. Too much of a tangent to get into here, although one guy named Buzz Feiten has a patented system that claims to solve this problem for the guitar.
After the octave, the fifth is considered the most, uh, harmonious, interval. The frequency ratio of a fifth is 3/2 - another whole number ratio. Thirds are a ratio of 5/4. I’m not sure where the other notes line up.
If you take a fifth and go up another fifth, you’ll end up on the second. So that second is a ratio of 9/4 above the initial starting note. But it’s one octave higher. A second in the same octave would be a 9/8 interval. Beyond that, I think the scale was divided into roughly equal steps.
All of those neat ratios only apply to just intonation. As others have said, Equal Tempermant tweaks the frequencies so that you can play melodies in different keys without retuning the instrument. Our ears have gotten used to the tweaked scale, and accept it as normal. But some people still use Just Intonation scales and tunes played in that scale sound just a little more “in tune” than normal.
J. S. Bach wrote “The Well-Tempered Clavier” back when the equal tempered scale was a new invention, partly to demonstrate it’s benefits. (Self-nitpick: I think there are small differences between the well-tempered system at Bach’s time and the equal temperment that we use today. But the goal of an instrument that could play all key signatures was the same.)
Just a quick addition. In Western music, the person who first “decided” on the intervals we now use was the Greek philosopher Pythagoras (of Pythagorean Theorem fame), who based them on the mathematical ratios of vibrating strings.
In fact, it’s this discomfort, and the nature of the “resolution” which underlies the harmonic structure and rhythm of western music. In fact, it’s second nature to most folks after only a few years of hearing music, to have a sense of that. Play the next to the last chord of many pieces of music and notice how you have the inner sense that something should come next. This is because the next to the last chord often contains an interval, (it’s called the augmented 4th) which creates (in western ears) a dissonance and when the last chord sounds, music theorists will say that the chord has resolved, and the dissonant interval has changed to a more consonant one - a sixth.
Don’t know if this was your question also,but I remember watching a program that explained how the notes first started.If I remember they took a certain length of some metal and tapped it and it gave off a ring,that was called eventually middle c I guess. then they cut a third of the length and that was the next note and over each successive piece until they came full circle,you can do the same with glasses of water .But somehow had a prob in each 4 notes or something that is why there appears to be no flat between e and f…b and c
I guess someone else here can find an informative link.
virtually yours
flight, you don’t need a musician to explain this to you, you need another engineer.
Imagine that you strike an A on the piano. It’s tuned to 440Hz, but the vibration has overtones, all multiples of 440Hz. The second multiple 880Hz is just the next A’, an octave higher, and the fourth multiple is double that, the next A". The sixth multiple is just twice the third, so they are also the same note, one octave apart. So, in the first six overtones, you really only have three notes, the 4th, 5th, and 6th. The ratio between their frequencies is 4:5:6. They sound good to us because every damn string we’ve ever struck has sounded them together. That, and our ear is probably constructed the same way.
We call it a major chord, and designate the notes as A, C#, E, the A major chord. All major chords have the ratios 4:5:6. If you start with three major chords, FAC, CEG, GBD, you can define all seven notes, ABCDEFG. That’s the just intonation that heresiarch mentioned. However, if you follow this scheme up and then try to drop down by octaves, you find that it doesn’t quite match up. That raises havoc when you try to shift keys.
The reason for that is the same reason that 2[sup]10[/sup] doesn’t quite equal 10[sup]3[/sup]. But, as all engineers know, it’s close enough. And musicians have created a similar fudge factor, called tempered intonation. They divided the octave into twelve steps (it could have just as well been 31), so that the ratio between each step is exactly 2[sup]1/12[/sup]. That makes it easy to shift keys. Pianos and guitars are tuned this way.
The fourth step is 2[sup]4/12[/sup], or 1.2599, which is apparently close enough to 5/4 that it’s agreeable. And the seventh step is 2[sup]7/12[/sup], or 1.4983, which is almost as close to 6/4, so that works too. Thus, in a major chord, you just start with a note, go up 4 steps, then 3 more. The ratio of their frequencies will be close to the harmonious 4:5:6. When you form an G major chord on the guitar, notice that all six notes are either G, B, or D.
I invite all musical criticism, so we can get the engineering right.