Math and The Chromatic Scale: Loving Music, Loving Math
Harmony occurs in music when two pitches vibrate at frequencies in small integer ratios. Long ago, Greek people realized the concept of harmony occurred when sounds and frequencies are in rational proportion. i.e., One Octave is equal to when the frequency is doubled, and a tripling of frequency brings the key One Octave higher, and is called a perfect fifth. Though not knowing this in relation to “frequency”, ancient Greeks knew this in relation to lengths of vibrating strings; http://www.math.uwaterloo.ca/~mrubinst/tuning/12.html (Why 12 Notes to The Octave?)
1/1 unison C
2/1 octave C
3/2 perfect fifth G
4/3 fourth F
5/4 major third E
8/5 minor 6th Ab
6/5 minor 3rd Eb
5/3 major 6th A
9/8 major 2nd D
16/9 minor 7th Bb
15/8 major 7th B
16/15 minor 2nd C#
The most common conception of the chromatic scale before the 13th century was the Pythagorean chromatic scale. Due to a different tuning technique, the twelve semitones in this scale have two slightly different sizes. (En. Wikipedia. org/wiki/Chromatic_scale) Thus, the scale is not perfectly symmetric. http://strathmaths.wordpress.com/2012/02/22/tipping-the-scales-some-of-the-mathematics-behind-music/. Pythagoras, 13thC Greek mathematician, was famous in geometry for the Pythagorean theorem (en. Wikipedia. org / wiki/Pythagoras). The theorem states that in a right-angled triangle, the area of the square on the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares of the other two sides—that is, a^2 + b^2 = c^2. Pythagoras experimented with a monochord, noticing that subdividing a vibrating string into rational proportions produced resonant sounds. When the frequency of the string is inversely proportional to its length, its other frequencies are simply whole number multiples of the fundamental. (En. Wikipedia. org/wiki/Chromatic_scale)
The term chromatic derives from the Greek word chroma, meaning color, where the total chromatic / aggregate is the set of all twelve pitch classes; an array being a succession of aggregates. Shí-èr-lǜ (Chinese: 十二律 (twelve-pitch scale) is a standardized gamut of twelve notes. The Chinese scale uses the same intervals as the Pythagorean scale, based on 2/3 ratios (2:3, 8:9, 16:27, 64:81, etc.). The gamut or its subsets were used for tuning and are preserved in bells and pipes. In China, the first reference to “the standardization of bells and pitch,” dates back to around 600 BCE. According to ancient scroll/script literature, Pythagoras taught that music was not intended for entertainment, though for calming the mind and bringing about order from chaos of life and the universe using spiritual instruments. Music of the Spheres is one of the phrases used to describe Ancient Greek Pythagorean Music. Here is a sample of what this music sounds like: http://www.youtube.com/watch?v=Bm2Pn_8Oxww This clip is a short educational video on a Pythagorean Tone Generator: Pythagorean Tone Generator: http://www.youtube.com/watch?v=BhqgOH0gDIc
James Hopkins, a student and practitioner of Pythagorean Monochords visually shows us his handmade Monochord Stringed instruments: http://www.youtube.com/watch?v=tbCZO6rPcY8
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