BAIN MUSC 726T

*Tuning Theory*

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726T

- Natural numbers () {WP}
- Integers () {WP}
- Rational numbers () {WP}
- Real numbers () {WP}
- Irrational numbers {WP}
- Prime numbers {WP; MW; OEIS}
- Sieve of Eratosthenes {WP}

Number Sequences

- Natural numbers, also called the positive integers {WP;
OEIS}

1, 2, 3, 4, 5, ... - Powers of 2 {WP;
OEIS}

1, 2, 4, 8, 16, 32, 64, ... - Triangular numbers {WP;
OEIS}

1, 3, 6, 10, 15, 21, ... - 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

- Arithmetic {MW; WP}
- Operations
- Addition (
**+**) {MW; WP} - Subtraction (
**-**) {MW; WP} - Multiplication (
**x**, or*****) {MW; WP} - Division (
**÷**, or**/**) {MW; WP} - Exponentiation (see below)
- Order of Operations: PEMDAS {WP}
- Fundamental theorem of arithmetic {MW; WP}

**
Fractions** {

- Ratio {MW}

*a*:*b* - Fraction {WP}

*a*/*b* - Numerator (
*a*) - Denominator (
*b*) - 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. - Simplifying (or reducing) fractions {WP}

e.g., 4/2 = 2/1; 6/4 = 3/2, 12/6 = 2/1, 80/64 = 5/4, etc. - 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.

- Reciprocal {WP}

The reciprocal of*x*is 1/*x*

- 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.

- Dividing a fraction by another fraction

e.g., 2/1 ÷ 4/3 = 2/1 × 3/4 = 3/2 - Decimal expansion {MW}

e.g., 3/2 = 1.5; 4/3 = 1.333, 5/3 = 1.667; etc. - Approximation (
**≅**) {WP; MW} - Rounding {WP; MW}
- e.g., round 701.995 ¢ to the nearest
cent ≅ 702 ¢ {
**Calculator**: Wolfram Alpha} - e.g.,
round 386.31371 ¢ to 3 decimal places ≅ 386.314 ¢ {
**Calculator**: Wolfram Alpha} - e.g., round 315.64129 ¢ to 1 decimal
place ≅ 315.6
¢ {
**Calculator**: Wolfram Alpha} - Divisor, or factor {WP}
- Least common multiple (LCM) {WP;
**Calculator**: Wolfram Alpha}

e.g., LCM (4, 6) = 12; LCM (3, 5) = 15; LCM (4, 5, 6) = 60, etc.

- 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
**

- 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. - 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. **Calculator:**Exponents Problem Solver {Wolfram Alpha}

Logarithm

- Logarithm (
**log**) {WP}

e.g., log_{10}(1000) = 3, or 10^3 = 1000; log_{2}(8) =3; i.e., 2^3 = 8; log_{10}(3/2) = log_{10}(1.5) ≅ 0.176 - Cent (
*c*) {WP}

*c*= 1200 log_{2}(f_{1}/f_{2})

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)

**Calculators:**- Logarithms Solver {Wolfram Alpha}
- Bain, Ratio to Cents {}

Geometry

- Euclid's
*Elements*(*c*300 BCE) {WP} - Euclidean geometry {WP}
- Plane {WP}
- Dimension {WP}
- Line {WP; MW}
- Circle {WP; MW}
- Pythagorean means {MW; WP}
- Arithmetic mean (AM)
- Harmonic mean (HM)
- Geometric mean (GM)
- Pythagorean theorem {WP}
- Square root of 2 {WP}

Advanced

- Continued fractions {WP}
- Prime counting function {WP}
- Projective plane {WP}
- Rhombus {WP}
- Tessellation, or tiling {WP}
- Triangular tiling {WP}
- Hexagonal tiling {WP}
- Torus {WP}

*** * ***

Image credits: Click on an image to see the credit

**Links**

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