Mathematical relationship between chords in music (for BAA GNU Radio Meeting homework)
To change a C major chord into a D major chord, you need to multiply the frequency of each note by (2^{2/12}) (the twelfth root of two raised to the power of two), which is approximately 1.122.
To change a D major chord (D-F#-A) to an E major chord (E-G#-B), you need to raise each note by a
whole step (2 semitones); mathematically, you multiply each note’s frequency by the twelfth root of 2 (approx. 1.05946) raised to the power of 2 (or about 1.122), effectively shifting the entire chord up by a factor that makes the new notes sound correct in the E major key.
In music theory and physics, moving from a
C Major chord to a D Major chord involves shifting the fundamental frequencies of each constituent note up by a major second (two semitones).
1. Frequency Change per Note
In the standard 12-Tone Equal Temperament (12TET) tuning system, every semitone represents a frequency ratio of
21/122 raised to the 1 / 12 power
21/12
(approx. 1.05946). Because a major second consists of two semitones, the frequency of each note in a D Major chord is approximately 1.12246 times higher (
22/122 raised to the 2 / 12 power
22/12
) than its corresponding note in the C Major chord.
| Chord Component | C Major Notes | Frequency (Octave 4) | D Major Notes | Frequency (Octave 4) |
| Root | C | 261.63 Hz | D | 293.66 Hz |
| Major Third | E | 329.63 Hz | F# | 369.99 Hz |
| Perfect Fifth | G | 392.00 Hz | A | 440.00 Hz |
2. Comparative Ratios
- Just Intonation: In older tuning systems or choral settings, a major chord follows a specific frequency ratio of 4:5:6.
- The Shift: When shifting from C to D, the entire 4:5:6 “shape” moves up the frequency spectrum.
- Logarithmic Perception: Human ears perceive pitch logarithmically, meaning we hear the ratio of change (multiplication) rather than an additive difference in Hertz. For example, the gap between C4 and D4 is ~32 Hz, while the gap between C5 and D5 is ~64 Hz, yet both are heard as the same musical interval.
3. Summary of the Transition
- Root: Increases by a factor of ~1.122.
- Third: E (329.63 Hz) becomes F# (369.99 Hz).
- Fifth: G (392.00 Hz) becomes A (440.00 Hz).