Tuesday, April 10, 2007

Doe a deer, a female deer.

To all readers, even if you are totally estranged from music, which is difficult to believe, you would have heard of the infamous Sound of Music song from which the title comes. So, even armed with as basic an idea as that (there are 7 basic notes that go into making a simple song, and these 7 notes make up what we call a scale), you can probably understand a good part of what's coming.


You can probably tell the difference between a poor singer and a good one, as they are singing to music being played. You don't really know how you can tell, (musically, you can say that the singer is out of tune, or perhaps if you can tell you would say he is singing "flat"i.e. too low, or singing "sharp" i.e. too high) but you can just tell when someone isn't singing well. There's an almost physiological response to poor singing, or to music that is simply out of tune, and likewise there is a feeling of pleasure when you hear an orchestra that is perfectly in tune with each other, or two singers singing in harmony.


Have you ever wondered why we respond to harmony? How do we know when a singer is singing well? How are people able to tell whether an orchestra is playing well, or if a certain instrument is in tune or not? The answer, and I must admit I never fully realised this until quite recently, lies in physics.


If you ask the nearest choir members, or person with an instrument, to play or sing a low "do" and a high "do" together (remember "that will bring us back to doe!", so after "ti" in the music scale, the next note is called "do" again.) you would hear an amazing thing: the two notes sound like one!


As a physicist, if you were able to dissect the voice box of the said choir members, or just simply observe the instruments in action, you would observe something very interesting about the sound produced: The frequencies of the sound would be in the ratio of 1:2.


A quick example for more music familiar people: A concert A has a frequency of 440 Hz. The A above that, A', would have a frequency of 880 Hz. Likewise for any two notes an octave apart. The higher note will be twice the frequency of the lower one.


Now, it turns out that any two sound waves that are produced that have a very nice ratio, for example 2:3 (you can do this by playing "do" and "so", "re" and "la", or "mi" and "ti" together, known as perfect fifths), or even 4:5 ("do" and "mi", "fa" and "la", "so" and "ti" known as major thirds) sound nice. If you don't believe me, you can go try it on any instrument, or simply sing it. On virtually any music that you can hear on the radio, every time you hear two singers singing, most of the harmonies will be in either perfect fifths or major thirds. (Of course this is outrageous to the musically very literate: what of minor thirds, minor and major sixths, and perfect fourths? Undeniably present. But don't forget, in music we can't simply have harmonies, that would make for very boring music. Music is made interesting by disharmonies that are then subsequently resolved into harmonies. So even notes that don't sound good together are useful!)


Okay.


Now for the more musically literate: This is what I'm actually talking about so far: the harmonic series:



Okay. So what is this diagram? Starting from the really low C on the left, if I double its frequency, I will hear the C one octave above it. If I triple the frequency, it would turn out to be the G third from the left, etc. So each multiple of the original frequency will produce a note that gets increasingly less harmonious with the note before it. Of course, 1 and 2 are extremely harmonious: the octave. 2 and 3 form the perfect fifth, 3 and 4 the perfect fourth, 4 and 5 the major third, 5 and 6 the minor third, 6 and 7 the major second, 11 and 12 the semitone. You can see for yourself that the harmonies get less and less pleasant. Dissonances like the tritone occur at 8 and 11 (terrible ratio, 8:11), and augmented fifth (8:13) etc. etc.

We can construct the harmonic series from any starting note of course. Let's try it from A (concert pitch)

A: 440 Hz

A': 880 Hz

E'': 1320 Hz

A'': 1760 Hz

C#''': 2200 Hz

E''': 2640 Hz

G''': 3080 Hz

Okay. Now let us do something here. If G''' is 3080 Hz, then G'' must be 1540 Hz, G' 770 Hz, and G itself 385 Hz. Let us construct a harmonic series from this G:

G: 385 Hz

G': 770 Hz

D'': 1155 Hz

If D'' is 1155 Hz, then D' must be 577.5 Hz, and D must be 288.75 Hz. Now, let us construct a harmonic series from this D:

D: 288.75 Hz

D': 577.5 Hz

A': 866.25 Hz

This means that the A constructed from this harmonic series is 433.125 Hz, a whole 7 Hz different from the original A at 440 Hz!!

What has happened? Long ago, musicians realised this problem: that starting a harmonic series from one note will form notes that are unique to that harmonic series. If you wish to change keys, say from A Major to D Major, if we insist on using the notes formed in the harmonic series of A to play in D Major, the sound would be horribly off key. Modulation was rendered impossible (not to mention to problems this caused for the french horn and other brass instruments: a separate topic, but well worth exploring!), and without modulation, music got boring fast.

Until a certain Johanne Sebastian Bach came into the picture. He had a wonderful solution to this problem: why not make every single note equally out of tune in such a way that ALL keys will sound equally out of tune? Of course, this sounds disagreeable: if it's out of tune, wouldn't we know it's out of tune and hate the music being played?

Turns out that isn't true. He wrote the Well-Tempered Clavier, a set of 24 preludes and fugues in all 24 keys to show that it was possible to do such a thing. Initially, I would think people were rather uncomfortable with the poor tuning, but today we are so used to his system that what we think is in tune today, would probably have been considered badly out of tune in his time!

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