BAIN MUSC 726T
Tuning Theory

Mathematical Terms & Concepts

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Number systems N, Z, Q & RNumber Sets

  1. Natural numbers (Natural numbers) {WP}
  2. Integers (Integers) {WP}
  3. Rational numbers (Rational numbers) {WP}
  4. Real numbers (Real numbers) {WP}
  5. Prime numbers {WP; MW; OEIS}


Number Sequences

  1. Natural numbers, also called the positive integers {WP; OEIS}
    1, 2, 3, 4, 5, ...
  2. Powers of 2 {WP; OEIS}
    1, 2, 4, 8, 16, 32, 64, ...
  3. Triangular numbers {WP; OEIS}
    1, 3, 6, 10, 15, 21, ...
  4. Primes {WP; OEIS}
    2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, ...


Arithmetic

  1. Arithmetic {MW; WP}
  2. Operations
  3. Order of Operations: PEMDAS {WP}
  4. Fundamental theorem of arithmetic {MW; WP}


Fractions
{Calculator: Wolfram Alpha}

  1. Ratio {MW}
    a:b
  2. Fraction {WP}
    a/b
  3. Superparticular ratio {WP}
    (n+1)/n, where n is a positive integer; e.g., 3/2, 4/3, 5/4, 9/8, 10/9, etc.
  4. Simplifying (or reducing) fractions {WP}
    e.g., 4/2 = 2/1; 6/4 = 3/2, 12/6 = 2/1, 80/64 = 5/4, etc.
  5. Ratio under octave reduction
    e.g., 3/1 = 3/2, 4/1 = 2/1, 5/1 = 5/4, 6/1 = 6/4 = 3/2, 7/1 = 7/4, etc.
  6. Reciprocal {WP}
    The reciprocal of x is 1/x
  7. Multiplying a fraction by another fraction {WP}
    e.g., 3/2 4/3 = 2/1; 9/8  9/8 = 81/64; 9/8 5/4 = 45/32, etc.
  8. Dividing a fraction by another fraction
    e.g., 2/1 4/3 = 2/1 3/4 = 3/2
  9. Decimal expansion {MW}
    e.g., 3/2 = 1.5; 4/3 = 1.333, 5/3 = 1.667; etc.
  10. Approximation () {WP; MW}
  11. Rounding {WP; MW}
  12. Divisor, or factor {WP}
  13. Least common multiple (LCM) {WP; Calculator: Wolfram Alpha}
    e.g., LCM (4, 6) = 12; LCM (3, 5) = 15; LCM (4, 5, 6) = 60, etc.
  14. Lowest common denominator (LCD) {WP; Calculator: Calculator Soup}
    e.g., LCD (1/2, 2/3) = 6; LCD (5/12, 11/18) = 36; LCD (1/2, 1/3, 1/4) = 12; etc.


Exponentiation

  1. Exponentiation (^) {WP}
    e.g., 2^3 = 2 2 2 = 8;
    e.g., 12-tet: 2^(1/12) ≅ 1.059; 2^(3/12) ≅ 1.189; 2^(7/12) ≅ 1.498, etc.
  2. nth root {WP}

    e.g., 12-tet: 2^(1/12) 1.059; 17-tet: 2^(1/17) 1.042; 31-tet: 2^(1/31) 1.023; etc.
  3. Calculator: Exponents Problem Solver {Wolfram Alpha}


Logarithm

  1. Logarithm (log) {WP}
    e.g., log10(1000) = 3, or 10^3 = 1000; log2(8) =3; i.e., 2^3 = 8; log10(3/2) = log10(1.5) 0.176
  2. Cent (c) {WP}
    c = 1200 log2 (f1/f2)
    e.g., 2/1 = 1200 ; 1/1 = 0 ; 2^(1/12) = 100 ; 3/2 = 702 ; 4/3 ≅ 498 (rounded to the nearest cent)
    e.g., 81/80 = 21.5 (rounded to the nearest 1/10 cent)
  3. Calculators:

Geometry
  1. Euclid's Elements (c300 BCE) {WP}
  2. Euclidean geometry {WP}
  3. Plane {WP}
  4. Dimension {WP}
  5. Line {WP; MW}
  6. Circle {WP; MW}
  7. Pythagorean means {MW; WP}
  8. Pythagorean theorem {WP}
  9. Square root of 2 {WP}


Advanced

  1. Continued fractions {WP}
  2. Prime counting function {WP}
  3. Projective plane {WP}
  4. Rhombus {WP}
  5. Tessellation, or tiling {WP}
  6. Torus {WP}

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Image credits: Click on an image to see the credit



Links

Online Encyclopedia of Integer Sequences (OEIS) https://oeis.org

Wikipedia https://www.wikipedia.org

Wolfram Alpha https://www.wolframalpha.com

Wolfram MathWorld https://mathworld.wolfram.com

References

Benson, David. 2007. Music: A Mathematical Offering. Cambridge: Cambridge University Press. {GB; Website}

Fauvel, John, Raymond Flood, and Robin Wilson, eds. 2003. Music and Mathematics: From Pythagoras to Fractals. New York: Oxford University Press. {GB}

Gann, Kyle. 2019. The Arithmetic of Listening: Tuning Theory and History for the Impractical Musician. Champaign: University of Illinois Press. {GB; Companion Website; Full text: EBSCOhost}

Hardy, G. H. and E. M. Wright. 2008/1936. An Introduction to the Theory of Numbers, 6th ed. London: Oxford University Press.

Loy, Gareth. 2006. Musimathics: The Mathematical Foundations of Music, Vol. 1-2. Cambridge, Mass: MIT Press. {Website; Full text: Vol. 1 Musical Elements: Ebook Central; Vol 2 Musical Signals: Ebook Central}

Marecek, Lynn, MaryAnne Anthony-Smith, and Andrea Honeycutt Mathis. 2020. Prealgebra, 2nd e. Houston, TX: OpenStax. {OpenStax}

Sloane, N. J. A. 1964. The Online Encyclopedia of Integer Sequences (OEIS). Available online at: <https://oeis.org>.

Weisstein, Eric, ed. 2021. Wofram MathWorld A Wolfram Web Resource. Available online at: <https://mathworld.wolfram.com>. 



Updated: January 5, 2024

Reginald Bain | University of South Carolina | School of Music
https://reginaldbain.com/vc/musc726t/