For once, let us reduce the question asked by R.K. to:
"Why are there 12 Semitones in an Octave and other musical mathematics"
.
1.1 Preliminary and some Basics
The ancient Greeks recorded the first surviving mathematical investigations in the area surrounding the pitches of sound. In the sixth century BCE, Pythagoras of Samos observed and recorded a connection between mathematics and pleasant musical intervals
Pythagoras
The agreed method by which Pythagoras discovered the consonant intervals was illustrated by a Roman named Anicius Manlius Saverinus Boethius (480-524 CE) in De Institutione Musica. It should be noted that this story is apocryphal. Nearly a millennium had passed in the times between Pythagoras and Boethius, and the Romans had conquered the Greeks, altering their remaining histories. The legend, as Boethius describes it, is paraphrased as follows: One day while walking by a smithy, Pythagoras heard a melodic series of tones coming from the blows of four smiths. He entered, and after much consideration, he decided the differences in pitch between the workers’ sounds was due to the force of the blows. To support his conjecture, he ordered the men to exchange hammers. Much to his surprise, he found the properties of these melodious sounds did not depend on the force exerted by the men. Instead, a specific pitch seemed to be inherently constant within each hammer.
Further investigation led Pythagoras to believe the weights of each hammer determined the tone it would produce. Weighing each hammer, he found the weights of all four hammers formed small integer ratios when compared with the heaviest one: 1/1, 1/2, 1/3, and 1/4. This knowledge of ratios may have led him to later experiment with string lengths.
Duodecimal
The duodecimal (also known as base-12 or dozenal) system is a numeral system using twelve as its base. 60 (sexagesimal) is the product of 3, 4, and 5. 3 is a divisor of 12 (duodecimal), 4 is a common divisor of 12 (duodecimal), 5 is a common divisor of 10 (decimal).
Origin
Languages in the Nigerian Middle Belt such as Janji, Kahugu, the Nimbia dialect of Gwandara, Mahl language of Minicoy and the Chepang language of Nepal are known to use duodecimal numerals.
Natural explanations for the choice of the number twelve include the following:
1) The approximate number of lunar months in an Earth year;
2) The sum of ten fingers on human hands and two feet; or
3) The number of phalanx bones in the four fingers of one hand, with the thumb used as an indicator.
4) In music, the partials of the natural scale.
Dichotomy and Tetrachord
A dichotomy is the splitting of a whole into two parts, in music that is a Tetrachord. The condensed meaning, respectively why dozenal and sexagesimal, I will answer in the following chapter "1.2 The Dozenal Tone System".
Duodecimal fractions are usually simple:
1/2 = 0.30
1/3 = 0.20
1/4 = 0.15
1/5 = 0.12
1/6 = 0.10
1/8 = 0.07:30
1/9 = 0.06:40
1/10 = 0.06
1/12 = 0.05
1/15 = 0.04
1/16 = 0.03:45
1/18 = 0.03:20
1/20 = 0.03
1/30 = 0.02
1/40 = 0.01:30
1/50 = 0.01:12
1/1:00 = 0.01 (1/60 in decimal)
Dozenalism
The case for the duodecimal system was put forth at length in F. Emerson Andrews' 1935 book New Numbers: How Acceptance of a Duodecimal Base Would Simplify Mathematics. Emerson noted that, due to the prevalence of factors of twelve in many traditional units of weight and measure, many of the computational advantages claimed for the metric system could be realized either by the adoption of ten-based weights and measure or by the adoption of the duodecimal number system.
to be cont.
___________________________________________________
Read this in the mean time: Why Twelve Tones?
http://oregonstate.edu/~coolmanr/HS_SeniorProject/Why-Twelve-Tones.html#The%20First%20Scale
.
"Why are there 12 Semitones in an Octave and other musical mathematics"
.
1.1 Preliminary and some Basics
The ancient Greeks recorded the first surviving mathematical investigations in the area surrounding the pitches of sound. In the sixth century BCE, Pythagoras of Samos observed and recorded a connection between mathematics and pleasant musical intervals
Pythagoras
The agreed method by which Pythagoras discovered the consonant intervals was illustrated by a Roman named Anicius Manlius Saverinus Boethius (480-524 CE) in De Institutione Musica. It should be noted that this story is apocryphal. Nearly a millennium had passed in the times between Pythagoras and Boethius, and the Romans had conquered the Greeks, altering their remaining histories. The legend, as Boethius describes it, is paraphrased as follows: One day while walking by a smithy, Pythagoras heard a melodic series of tones coming from the blows of four smiths. He entered, and after much consideration, he decided the differences in pitch between the workers’ sounds was due to the force of the blows. To support his conjecture, he ordered the men to exchange hammers. Much to his surprise, he found the properties of these melodious sounds did not depend on the force exerted by the men. Instead, a specific pitch seemed to be inherently constant within each hammer.
Further investigation led Pythagoras to believe the weights of each hammer determined the tone it would produce. Weighing each hammer, he found the weights of all four hammers formed small integer ratios when compared with the heaviest one: 1/1, 1/2, 1/3, and 1/4. This knowledge of ratios may have led him to later experiment with string lengths.
Duodecimal
The duodecimal (also known as base-12 or dozenal) system is a numeral system using twelve as its base. 60 (sexagesimal) is the product of 3, 4, and 5. 3 is a divisor of 12 (duodecimal), 4 is a common divisor of 12 (duodecimal), 5 is a common divisor of 10 (decimal).
Origin
Languages in the Nigerian Middle Belt such as Janji, Kahugu, the Nimbia dialect of Gwandara, Mahl language of Minicoy and the Chepang language of Nepal are known to use duodecimal numerals.
Natural explanations for the choice of the number twelve include the following:
1) The approximate number of lunar months in an Earth year;
2) The sum of ten fingers on human hands and two feet; or
3) The number of phalanx bones in the four fingers of one hand, with the thumb used as an indicator.
4) In music, the partials of the natural scale.
Dichotomy and Tetrachord
A dichotomy is the splitting of a whole into two parts, in music that is a Tetrachord. The condensed meaning, respectively why dozenal and sexagesimal, I will answer in the following chapter "1.2 The Dozenal Tone System".
Duodecimal fractions are usually simple:
1/2 = 0.30
1/3 = 0.20
1/4 = 0.15
1/5 = 0.12
1/6 = 0.10
1/8 = 0.07:30
1/9 = 0.06:40
1/10 = 0.06
1/12 = 0.05
1/15 = 0.04
1/16 = 0.03:45
1/18 = 0.03:20
1/20 = 0.03
1/30 = 0.02
1/40 = 0.01:30
1/50 = 0.01:12
1/1:00 = 0.01 (1/60 in decimal)
Dozenalism
The case for the duodecimal system was put forth at length in F. Emerson Andrews' 1935 book New Numbers: How Acceptance of a Duodecimal Base Would Simplify Mathematics. Emerson noted that, due to the prevalence of factors of twelve in many traditional units of weight and measure, many of the computational advantages claimed for the metric system could be realized either by the adoption of ten-based weights and measure or by the adoption of the duodecimal number system.
to be cont.
___________________________________________________
Read this in the mean time: Why Twelve Tones?
http://oregonstate.edu/~coolmanr/HS_SeniorProject/Why-Twelve-Tones.html#The%20First%20Scale
.
