SISONKE SANDILE **Algebraic Formulations for Music Harmony, Keys, and Progressions: A Simplified Mathematical Approach** **I. Introduction** Music has long been intertwined with mathematics, from ancient Greek philosophers to modern researchers. Harmony, keys, and chord progressions are fundamental music concepts that have been studied mathematically. This paper aims to simplify these concepts into algebraic equations, providing a unified mathematical approach. **II. Harmony Equation** A harmonic series is a sequence of frequencies sounding simultaneously. Mathematically: H = ∑[N × 2^(i/12)] (Equation 1) where: - H = harmonic series - N = fundamental note frequency - i = integer (0, 1, 2...) representing harmonics/ overtones **Proof:** Let's analyze the harmonic series of a note. The first harmonic (i=0) is the fundamental frequency N. The second harmonic (i=1) is twice the fundamental frequency, 2N. However, musical harmonics double in frequency every octave (12 semitones). So, the correct formula for the second harmonic is N × 2^(12/12) = 2N.**III. [12/25, 00:19] Lucy: **II. Harmony Equation (continued)** **Proof (continued):** Generalizing this pattern, the ith harmonic frequency is: Ni = N × 2^(12i/12) = N × 2^i However, musical harmonics also include intermediate frequencies (microtones). To include these, we modify the exponent: Ni = N × 2^(i/12) where i represents both harmonics (integers) and microtones (fractions). The harmonic series H is the sum of these frequencies: H = ∑[N × 2^(i/12)] **Theorem 1:** H is a geometric series with first term N and common ratio 2^(1/12). **Proof:** By definition, a geometric series has the form: G = ∑[ar^i] where a is the first term and r the common ratio. Comparing with Equation 1, we see: a = N and r = [12/25, 00:19] Lucy: **II. Harmony Equation (continued)** **Proof (continued):** Generalizing this pattern, the ith harmonic frequency is: Ni = N × 2^(12i/12) = N × 2^i However, musical harmonics also include intermediate frequencies (microtones). To include these, we modify the exponent: Ni = N × 2^(i/12) where i represents both harmonics (integers) and microtones (fractions). The harmonic series H is the sum of these frequencies: H = ∑[N × 2^(i/12)] **Theorem 1:** H is a geometric series with first term N and common ratio 2^(1/12). **Proof:** By definition, a geometric series has the form: G = ∑[ar^i] where a is the first term and r the common ratio. Comparing with Equation 1, we see: a = N and r = 2^(1/12) Thus, H is a geometric series. **III. Consonance Equation** Consonance measures harmony perception. Mathematically: C = 1 - | (log2(F2/ F1)) - (p/ q)| (Equation 2) where: - C = consonance (0-1) - F1, F2 = frequencies of two notes - p/ q = simple ratio (e.g., 3/2 for perfect fifth) **Proof:** Let's analyze consonance perception. Perfect consonance occurs when frequencies have simple ratios (e.g., 2:1 octave, 3:2 perfect fifth). We can represent these ratios as logarithms: log2(2/1) = 1 and log2(3/2) ≈ [12/25, 00:20] Lucy: **II. Harmony Equation Proof (continued)** Generalizing this pattern, the i- th harmonic frequency is N × 2^(12i/12) = N × 2^i. However, musical harmonics also include intermediate frequencies (e.g., minor thirds). These frequencies can be represented by fractional exponents: N × 2^(i/12) for i = 0, 1, 2, ... Thus, the harmonic series equation is: H = ∑[N × 2^(i/12)] (Equation 1) **III. Consonance Equation** Consonance measures harmony perception. Mathematically: C = 1 - | (log2(F2/ F1)) - (p/ q)| (Equation 2) where: - C = consonance (0-1) - F1, F2 = frequencies of two notes - p/ q = simple ratio (e.g., 3/2 for perfect fifth) **Proof:** Consonance is maximized when frequency ratios match simple ratios (e.g., 3/2).**III. Consonance Equation (continued)** 392.44 Hz, p/ q = 3/2) C = 1 - | (log2(392.44/261.63)) - (3/2)| ≈ 1 - | 1.583 - 1.585| ≈ 0.999 or 99.9% consonance **IV. Key Equation** A key is a group of notes and chords sounding harmonious together. Mathematically: K = {N × 2^(s/12)} modulo 12 (Equation 3) where: - K = key - N = tonic note frequency - s = integers (0-11) representing scale degrees Example: C major key (C4 = 261.63 Hz) K = {261.63 × 2^(0/12), 261.63 × 2^(2/12), ...} modulo 12 **V. Progression Equation** A progression is a series of chords. Mathematically: P = [Ks × 2^(i/12), Ks × 2^((i +m)/12), Ks × 2^((i +p)/12)] (Equation 4) where: - P = progression - Ks = key's scale degrees (0-11) - i, m, p = integers for chord tones Example: ii- V- I progression in C major key P = {[D × 2^(0/12), D × 2^(4/12), D × 2^(7/12)], [G × 2^(0/12), G × 2^(4/12), G × 2^(7/12), G × 2^(10/12)], [C × 2^(0/12), C × 2^(4/12), C × 2^(7/12), C × 2^(11/12)]} **VI. Conclusion** This paper presents algebraic formulations for music harmony, keys, and progressions. These equations provide a unified mathematical approach to understanding music theory. Future research can apply these equations to music generation, analysis, and education. **VII. References** 1. Helmholtz, H. (1863). On the Sensations of Tone. 2. Plomp, R., & Levelt, W. (1965). Consonance Equation Proof (continued)** Let's analyze the frequency ratio F2/ F1. Taking the logarithm base 2: log2(F2/ F1) = log2(F2) - log2(F1) This represents the number of octaves between F1 and F2. However, consonance depends on the simplest ratio p/ q. So, we subtract the logarithm of this ratio: | (log2(F2/ F1)) - log2(p/ q)| Simplifying using logarithm properties: | (log2(F2) - log2(F1)) - (log2(p) - log2(q))| This measures the difference between the actual frequency ratio and the simplest ratio. Consonance is inversely proportional to this difference, so: C = 1 - | (log2(F2/ F1)) - (p/ q)| (Equation 2) **IV. Key Equation** A key is a group of notes and chords sounding harmonious together. Mathematically: K = {N × 2^(s/12)} modulo 12 (Equation 3) where: - K = key - N = tonic note frequency - s = integers (0-11) representing scale degrees **Proof:** Let's analyze the scale degrees of a key. The tonic (s=0) is the base frequency N. The major second (s=2) is N × 2^(2/12). The major third (s=4) is N × 2^(4/12). Generalizing this pattern, the s- th scale degree frequency is N × 2^(s/12). Since there are 12 scale degrees, we take modulo 12 to ensure s remains within this range. Thus, the key equation is: K = {N × 2^(s/12)} modulo 12 (Equation 3) **III. Consonance Equation (continued)** 392.44 Hz, p/ q = 3/2) C = 1 - | (log2(392.44/261.63)) - (3/2)| ≈ 1 - | 1.583 - 1.585| ≈ 0.999 or 99.9% consonance **IV. Key Equation** A key is a group of notes and chords sounding harmonious together. Mathematically: K = {N × 2^(s/12)} modulo 12 (Equation 3) where: - K = key - N = tonic note frequency - s = integers (0-11) representing scale degrees Example: C major key (C4 = 261.63 Hz) K = {261.63 × 2^(0/12), 261.63 × 2^(2/12), ...} modulo 12 **V. Progression Equation** A progression is a series of chords. Mathematically: P = [Ks × 2^(i/12), Ks × 2^((i +m)/12), Ks × 2^((i +p)/12)] (Equation 4) where: - P = progression - Ks = key's scale degrees (0-11) - i, m, p = integers for chord tones Example: ii- V- I progression in C major key P = {[D × 2^(0/12), D × 2^(4/12), D × 2^(7/12)], [G × 2^(0/12), G × 2^(4/12), G × 2^(7/12), G × 2^(10/12)], [C × 2^(0/12), C × 2^(4/12), C × 2^(7/12), C × 2^(11/12)]} **VI. Conclusion** This paper presents algebraic formulations for music harmony, keys, and progressions. These equations provide a unified mathematical approach to understanding music theory. Future research can apply these equations to music generation, analysis, and education. **VII. References** 1. Helmholtz, H. (1863). On the Sensations of Tone. 2. Plomp, R., & Levelt, W. (1965).