Mathematical Definitions

Number Sets

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

Number Sequences

1. Natural numbers, or positive integers, an arithmetic progression {WP; OEIS}
   1, 2, 3, 4, 5, ...
2. Harmonic progression
   1, 1/2, 1/3, 1/4, 1/5, ...
3. Powers of 2, a geometric progression {WP; OEIS}
   1, 2, 4, 8, 16, 32, 64, ...
4. Triangular numbers {WP; MW; OEIS}
   0, 1, 3, 6, 10, 15, 21, ...
   
5. Fibonacci numbers {WP; MW; OEIS}
   0, 1, 1, 2, 3, 5, 8, 13, 21 ...
6. 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
• Addition (+) {MW; WP}
• Subtraction (-) {MW; WP}
• Multiplication (x, or *) {MW; WP}, see also dot operator
  
• Division (÷, or /) {MW; WP}
• Exponentiation (see below)
3. Order of Operations: PEMDAS {WP}
4. Fundamental theorem of arithmetic {MW; WP}

Fractions {Calculator: Wolfram Alpha}

1. Fraction, represents a part of a single whole  {WP; MW}
   a/b
• Numerator (a)
• Denominator (b)
2. Ratio, compares the size of two or more quantities {WP; MW}
   a:b, or a/b
3. Reciprocal {WP}
   The reciprocal of x is 1/x
   
4. 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.
5. Reduced fraction, a fraction's simplest form {WP}
   e.g., 4/2 = 2/1; 6/4 = 3/2, 12/6 = 2/1, 80/64 = 5/4, etc.
6. Ratio under octave reduction {XW}
   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.
   
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 = 6/4 = 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}
    
• 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}
12. Divisor, or factor {WP}
    e.g., 2 is a divisor of 6, because 2 x 3 = 6
    
13. Multiple {WP}
    e.g., 6 is a multiple of 2, because 6 ÷ 3 = 2
14. Least common multiple (LCM) {WP; Calculator: Wolfram Alpha}
    e.g., LCM (4, 6) = 12; LCM (3, 5) = 15; LCM (4, 5, 6) = 60, etc.
    
15. 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 (¢) {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:
• Bain, Ratio to Cents {ReginaldBain.com}
• Logarithms Solver {Wolfram Alpha}

Geometry

1. Cartesian coordinate system {WP; MW}
   
2. Circle {WP; MW}
3. Dimension {WP}
• 1D; 2D 3D; 4D; etc.
4. Euclid's Elements (c. 300 BCE) {WP; Texas} (Fitzpatrick 2008)
   
5. Euclidean geometry {WP}
6. Figurate number {MW}
• Triangular number {WP; MW; OEIS}
  1, 3, 6, 10, 15, 21, ...
• Square number {WP; MW; OEIS}
  0, 1, 4, 9, 16, 25, 36, ...
7. Plane {WP}
8. Point {WP}
   
9. Line {WP; MW}
10. Platonic solids {WP; MW}
    
11. Polygon {WP; MW}
    
12. Pythagorean means {MW; WP}
• Arithmetic mean (AM)
• Harmonic mean (HM)
• Geometric mean (GM)
13. Pythagorean theorem {WP; MW}
• Square root of 2 {WP}
14. Spiral {WP}
• Logarithmic spiral
  
15. Tetractys {WP; MW}

Advanced

1. Binary tree {WP; MW}
   
2. Continued fractions {WP; MW}
   
3. Coprime, or relatively prime, integers {WP; MW}
   
4. Farey sequence {WP; MW; OEIS}
5. Fibonacci sequence {WP; MW; OEIS}
   1, 1, 2, 3, 5, 8, 13, ...
6. Golden ratio {WP; MW}
   
7. Graph theory {WP; MW}
   
8. Hasse diagram {WP; MW}
   
9. Linear algebra {WP; MW}
   
10. Mediant {WP; MW}
    
11. Multiplication table {WP; MW}
    
12. Pascal's triangle {WP; MW}
    
13. Prime counting function {WP; MW}
14. Prime factorization {WP; MW}
    
15. Projective plane {WP; MW}
    
16. Rhombus {WP}
17. Rieman Zeta function {XW; MW}
    
18. Set theory {WP; MW}
    
• Set {WP; MW}
  
19. Stern-Brocot tree {WP; WP}
    
20. Strähle construction {WP; HFF}
21. Tessellation, or tiling {WP; MW}
    
• Triangular tiling {WP}
• Hexagonal tiling {WP}
22. Transfinite number {WP; MW}
    
23. Topology {WP; MW}
    
24. Torus {WP; MW}
    

* * *

Image credits: Click on an image to see the Wikipedia credit

Links

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. Urbana: University of Illinois Press. {GB; Full text: TCL; Audio Examples}

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

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

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. Wolfram MathWorld – A Wolfram Web Resource. Available online at: <https://mathworld.wolfram.com>. 

Updated: December 1, 2025