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

Circle progression

A progression along the circle of fifths[1]

	Vi–ii–V–I chord progression
File:Circle progression just intonation I IV V I.mid	Just intonation
File:Circle progression Pythagorean tuning I IV V I.mid	Pythagorean tuning
File:Circle progression quarter-comma meantone I IV V I.mid	Quarter-comma meantone
File:Circle progression Werckmeister temperament I IV V I.mid	Werckmeister temperament
File:Progression en cercle I IV V I.ogg	Equal temperament

Math

Images

xx

Natural scale ratios -Large Print

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

• w: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

1. w:Countable_set
2. w:Pairing_function#Cantor_pairing_function
3. w:Uncountable_set
4. w:Infinity
5. w:Infinite set
6. w:Infinity
7. w:Cardinality
8. w:Cantor's diagonal argument
9. w:Aleph number
10. w:Cardinality_of_the_continuum
11. w:Axiom of countable choice
12. w:First uncountable ordinal
13. w:Concert pitch