In this way, a single tuning can be used regardless of the key being chosen, or the music being played.Įqual tempering is a system for breaking up each octave into twelve equal semi-tones. This system - called equal tempering, a version of well tempering - spaces all twelve notes of the octave equally. Over the last couple of hundred years, a more universal system has been used instead. However, this system required different tunings depending on which notes you were planning to play, or in which key your music was written. This would make the X's line up perfectly, so the notes would fit together exactly right. For example, when playing in the key of C, the note Middle G would be tuned so that its frequency was exactly 3/2 times that of Middle C. tuned musical instruments specially for each key. What are their frequencies? How do they fit in?Įarly musicians - as far back as the Greek mathematician and musician Pythagoras of the sixth century B.C.E. There are eleven other notes - C-Sharp, D, D-Sharp, E, and so on - squeezed in between Middle C and High C. On the other hand, High C is twelve semi-tones above Middle C. Remember that High C has a frequency which is twice as large as Middle C. So why is it that the notes C and G fit together well, but the notes C and F-Sharp do not? Who decides which notes will have their X's line up well, and which notes will not? The answer is, mathematics decides! Why do these three notes - C, E, and G - sound so sweet together? Let's have a look. It is the basis for music as diverse as Row, row, row your boat, and the symphonies in C Major of Mozart and Beethoven and Schubert. When all three notes are played together, they form the "C major chord", which is a sweetly harmonious, happy sound, like a barbershop Middle E is four semi-tones (a "major third") above Middle C, and three semi-tones (a "minor third") below Middle G. To get a really pleasing sound, let's add a third note - E. Striking a chordĬombining the notes C and G produces a sound which is fine, but not very exciting. These fundamental truths apply regardless of your favourite musical style. But it isn't so close either, and anyway, seven is too large a number of time periods to have to wait.) So, combining C with F-Sharp isn't so great, but combining C with G works (Well, it is true that five time periods for C isn't so far off from seven periods of F-Sharp. They have no simple relation to each other. This time, the X's just don't line up well. But why is this? Well, let's examine their wave graphs together: In fact, these notes form a "fifth" interval and fit together well. But which notes sound good together, and which ones don't?Ĭonsider the notes Middle C and Middle G. More interesting combinations like chords result if we bring in other notes, too. However, the result sounds sort of hollow, or even boring. If you play Middle C and High C together, then there is no discordance at all. Notes an octave apart do indeed fit together well. In principle, we could keep increasing the octaves, and doubling the frequencies, forever - but after a certain point, the notes would be so And the note an octave above that one, has a frequency eight times that of Middle C. The note two octaves above Middle C (sometimes called High High C) has a frequency four times that of Middle C. The same pattern continues as we increase the octaves. Playing that same string with the twelfthįret pressed produces a High E - a note one octave higher, with frequency twice as high. For example, the first string of a guitar is normally tuned to Middle E. This leads to a frequency which is twice as high, and thus corresponds to a note one octave higher. This makes it vibrate precisely twice as quickly. There, pressing a guitar string at the twelfth fret cuts the string precisely in half. We can draw a graph by puttingĪn X at every time when a pocket of air arrives: Equivalently, the pockets of air arrive so quickly that one pocket strikes your ear every 0.00382 seconds. That means that when Middle C is played, 262 pockets of higher air pressure pound against your ear each second. This note has a frequency of about 262 Hertz. With music, the frequency at which these pockets strike your ear controls the pitch that you hear.įor example, consider the note called "Middle C" (usually the first note learned in piano lessons). (Or, as the horror movies would say: in space no one can hear you scream.)Ī sound wave creates minute pockets of higher and lower air pressure, and all the sounds we hear are caused by these pressure changes. In fact, sound progresses as a wave through the air, and sound cannot be produced without an atmosphere. Music appears to be transmitted by magic, escaping from your expensive stereo - or a loudly passing car radio, or a guitar-strumming maestro - and accosting your eardrums in one fell swoop.
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