Draft:Sing free/Prelude and Fugue in C major (ear training)

Simple numbers
... For example, a note played at 200 cycles per second will make a perfect fifth with a note played at 300 cycles per second. The whole numbers involved in the ratio are 2 and 3:


 * $$\frac{300}{200}=\frac 3 2 = 1.5$$

While this ratio of 3:2 makes "perfect harmony", the tempered ("scientific") scale fails to achieve this fraction. Instead, a compromise must me made:
 * $$2^{7/12}=1.49830707687668149879928...$$

List fractions involving small integers
The harmonious nature of intervals with fractional ratios also involves how large the numbers are. For example, the ratio 2/1 involves the two smallest whole numbers. It is also the most fundament interval, namely the octave. If we restrict ourselves to fractions less than 2, the next simplest fraction is 3/2, which is the fifth. Things start to go wrong as the numbers get larger. For example, 7/4 is the tritone, which has been called the "devils triad". Since there are an infinite number of fractions between 1 and 2, we need systematic procedure to label them from "small" to "large".

Links

 * Syntonic comma 81/08 or (around 21.51 cents)


 * Further information on Wikipedia: Tritone, and Harmonic seventh chord

CONVERT TO SEE ALSO

 * https://www.youtube.com/watch?v=7GhAuZH6phs
 * http://www.n-ism.org/Projects/microtonalism.php
 * http://www.christopherstembridge.org/cromatico.htm

Changed mind. Will try to get all this a a sing free subpage

 * Countable_set
 * Pairing_function
 * Uncountable_set
 * Infinity
 * Infinite set
 * Infinity
 * Cardinality
 * Cantor's diagonal argument
 * Aleph number
 * Cardinality_of_the_continuum
 * Axiom of countable choice
 * First uncountable ordinal
 * Concert pitch