The Physics of Harmony
Fundamental Frequencies
- A musical pitch is the result of a regularly repeating acoustic pressure wave.
- These periodic waves have repeating cycles. The time each cycle lasts is the fundamental period or period of the waveform. The number of cycles per second gives the fundamental frequency of the note in Hertz (Hz).
- The fundamental frequency is ƒ0 ("F zero" or "F nought").
ƒ0 in Hz = 1 ÷ (period in seconds)
(period in seconds) = 1 ÷ ƒ0 in Hz
Harmonics
- Different instruments playing the same fundamental pitch all sound different, and have different acoustic pressure waveforms.
- Acoustic pressure variations cause the vibration of the basilar membrane which is analysed in terms of frequency components. Different patterns of vibration cause the perception of a different timbre even if the sound is at the same pitch.
- Each pitched instrument has a spectrum of harmonics (otherwise known as overtones or partials) which are integer multiples of ƒ0.
| Integer (N) | Overtone Series ((N - 1) × ƒ0) when N > 1 | Harmonic Series (N × ƒ0) | Component Frequency (Hz) |
|---|---|---|---|
| 1 | Fundamental Frequency (ƒ0) | 1st Harmonic | 1 ƒ0 |
| 2 | 1st Overtone | 2nd Harmonic | 2 ƒ0 |
| 3 | 2nd Overtone | 3rd Harmonic | 3 ƒ0 |
| 4 | 3rd Overtone | 4th Harmonic | 4 ƒ0 |
| 5 | 4th Overtone | 5th Harmonic | 5 ƒ0 |
| 6 | 5th Overtone | 6th Harmonic | 6 ƒ0 |
| 7 | 6th Overtone | 7th Harmonic | 7 ƒ0 |
| 8 | 7th Overtone | 8th Harmonic | 8 ƒ0 |
| 9 | 8th Overtone | 9th Harmonic | 9 ƒ0 |
| 10 | 9th Overtone | 10th Harmonic | 10 ƒ0 |
The physical design of many instruments limits the number of harmonics they can generate. Our hearing system provides a limit to how many harmonics we can hear; e.g. a 440 Hz "A" being listened to by someone with an upper limit of 16 kHz would produce 16 000 ÷ 440 = 36 audible harmonics. One octave higher, there would be 16 000 ÷ 880 = 18 harmonics.
Intervals Between Harmonics
- Each harmonic is a sine wave.
- The musical intervals, frequency ratios and staff notation for the first 10 harmonics in the key of C are shown below. All tunings are approximate, as they don't fully match the standard Western tuning system.
The table below shows the musical intervals between adjacent harmonics.
| Octave | 2:1 |
| Perfect Fifth | 3:2 |
| Perfect Fourth | 4:3 |
| Major Third | 5:4 |
| Minor Third | 6:5 |
| Flat Minor Third | 7:6 |
| Sharp Major Second | 8:7 |
| Major Whole Tone | 9:8 |
| Minor Whole Tone | 10:9 |
You can use one ratio to find another. For example, if you wanted to find the ratio for a perfect fourth, you could use the knowledge that a perfect fourth added to a perfect fifth (3:2) makes up an octave (2:1)…
(2 ÷ 1) ÷ (3 ÷ 2) = (2 ÷ 1) × (2 ÷ 3) = 4 ÷ 3
| Harmonic | Ratio to Fundamental |
Scalar Ratio |
Cents from Fundamental (Adjusted) |
Note Name |
|---|---|---|---|---|
| 1 | 1/1 | 1/1 | 0 | C |
| 2 | 2/1 | 1/1 | 0 | C |
| 3 | 3/1 | 3/2 | 702 | G + 2 |
| 4 | 4/1 | 1/1 | 0 | C |
| 5 | 5/1 | 5/4 | 386 | E - 14 |
| 6 | 6/1 | 3/2 | 702 | G + 2 |
| 7 | 7/1 | 7/4 | 969 | Bb - 31 |
| 8 | 8/1 | 1/1 | 0 | C |
| 9 | 9/1 | 9/8 | 204 | D + 4 |
| 10 | 10/1 | 5/4 | 386 | E - 14 |
| 11 | 11/1 | 11/8 | 551 | Gb - 49 |
| 12 | 12/1 | 3/2 | 702 | G + 2 |
| 13 | 13/1 | 13/8 | 841 | G# + 41 |
| 14 | 14/1 | 7/4 | 969 | Bb - 31 |
| 15 | 15/1 | 15/8 | 1088 | B - 12 |
| 16 | 16/1 | 1/1 | 0 | C |
| 17 | 17/1 | 17/16 | 105 | C# + 5 |
| 18 | 18/1 | 9/8 | 204 | D + 4 |
| 19 | 19/1 | 19/16 | 298 | Eb - 2 |
| 20 | 20/1 | 5/4 | 386 | E - 14 |
| 21 | 21/1 | 21/16 | 471 | F - 29 |
| 22 | 22/1 | 11/8 | 551 | Gb - 49 |
| 23 | 23/1 | 23/16 | 628 | F# + 28 |
| 24 | 24/1 | 3/2 | 702 | G + 2 |
| 25 | 25/1 | 25/16 | 773 | Ab - 27 |
| 26 | 26/1 | 13/8 | 841 | G# + 41 |
| 27 | 27/1 | 27/16 | 906 | A + 6 |
| 28 | 28/1 | 7/4 | 969 | Bb - 31 |
| 29 | 29/1 | 29/16 | 1030 | A# + 30 |
| 30 | 30/1 | 15/8 | 1088 | B - 12 |
| 31 | 31/1 | 31/16 | 1145 | B + 45 |
| 32 | 32/1 | 1/1 | 0 | C |
- Harmonics can be plotted on a linear or logarithmic axis.
- The linear scale shows the harmonics to be equidistant, whereas they start to 'bunch up' as the frequency increases on the logarithmic scale.
- The logarithmic scale shows that the pattern of harmonics remains constant when different notes are played. The basilar membrane also responds logarithmically, so the logarithmic scale most closely represents the perceptual weighting given to the harmonics of a pitched note.
- The logarithmic scale shows intervals at a constant distance along the scale. Halving any value has the effect of lowering it by one octave; multiplying a value by 3 ÷ 2 has the effect of raising it by a perfect fifth, and so on.
Tuning Systems
Western tuning systems have 12 notes per octave, with an interval of one semitone between each note. Octaves have a frequency ratio of 2:1.
Pythagorean Tuning
- Built on perfect fifths; each successive note's frequency is derived from the cycle of fifths.
- Twelve perfect fifths take us back to the starting note with a frequency ratio of 3/212 = 129.746.
- Twelve fifths is equal to seven octaves, which has a frequency ratio of 27 = 128.0.
- Twelve fifths is slightly sharp compared to seven octaves by the Pythagorean comma, which has a frequency ratio of 129.746/128.0 = 1.01364.
- If you use the same method but use descending perfect fifths, the resultant note after twelve fifths would be flatter than seven octaves by 1.0136433. All notes in an ascending and descending Pythagorean scale are slightly different.
| Note | C | D | E | F | G | A | B | C |
| Frequency Ratio to C | 1/1 | 9/8 | 81/64 | 4/3 | 3/2 | 27/16 | 243/128 | 2/1 |
| Freq. Ratio Between Notes | 9/8 | 9/8 | 256/243 | 9/8 | 9/8 | 9/8 | 256/243 |
Pythagorean semitone frequency ratio: 256/243
Pythagorean tone frequency ratio: 9/8
Just Diatonic Tuning
All notes of the major scale are taken from the octave (2:1), perfect fifth (3:2) and major third (5:4) intervals.
| Note | C | D | E | F | G | A | B | C |
| Frequency Ratio to C | 1/1 | 9/8 | 5/4 | 4/3 | 3/2 | 5/3 | 15/8 | 2/1 |
| Freq. Ratio Between Notes | 9/8 | 10/9 | 16/15 | 9/8 | 10/9 | 9/8 | 16/15 |
- There are two tone intervals; the major whole tone and minor whole tone. This is correct according to the harmonic series.
- This scale can only be used in one key, as each key requires its own tuning.
- It is possible to compromise by introducing the Pythagorean comma among some of the fifths to make sure that twelve fifths equals seven octaves.
Equal Tempered Tuning
Each semitone is one twelfth of an octave. The ratio is a number which when multiplied by itself twelve times is equal to 2…
r = 12√2 = 1.0595
One cent is one hundredth of a semitone…
c = 100√r = 100√1.0595 = 1.000578
Equal temperament tuning does not satisfy the ratios from the harmonic series, but the tuning discrepancies are within the 5% critical bandwidth criterion, meaning there won't be audible 'beats' within the fundamental notes. However, the harmonics of notes in a chord are often discordant.
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© Matt Bellingham 2003 – 2006
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