Consonance Theory Explained |
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A musical tone makes is a sound containing a dominant frequency of vibration. For this discussion you might want to imagine a sine wave. This frequency is measured in cycles per second or Hertz, abbreviated, Hz. When two or more tones are sounded and their frequencies share a common factor, our ears perceive them as consonant. For instance: If an A is sounded at 440 Hz along with an E at 660 Hz, the frequencies are in a ratio of 660 to 440 or 3 to 2 (3/2) and the human ear will perceive this as consonant.
The ear will hear any small whole number frequency ratio as consonant. Tuning systems based on these simple intervals are called Just Intonation systems. Here's a list of small whole number ratios commonly considered consonant:
| Ratio | Cents | Note | Name |
|---|---|---|---|
| 1/1 | 0.00 | C +0 | unison |
| 6/5 | 315.64 | Eb +16 | minor third |
| 5/4 | 386.31 | E -14 | major third |
| 4/3 | 498.04 | F -2 | perfect fourth |
| 3/2 | 701.96 | G +2 | perfect fifth |
| 8/5 | 813.69 | Ab +14 | minor sixth |
| 5/3 | 884.36 | A -16 | major sixth |
| 2/1 | 1200.00 | C +0 | octave |
An octave is created when one tone vibrates at twice the frequency of another one. The sound is so similar we use the same note name but say that it's up or down an octave. Multiplying a ratio times two moves it up an octave. Dividing by two brings it down an octave.
By the same token, 3/2 is a perfect fifth up from 1/1 and 2/3 is a fifth down. If you want to talk about all these notes within the higher octave (the one containing the 3/2), you must bring the 2/3 up to that octave by multiplying it times 2 to yield 4/3, or a perfect fourth.
It is customary to analyze scales in the octave from 1/1 to 2/1. By definition, all ratios in this octave are between 1 and 2 (convert the ratios in the above chart to decimals to prove it).
| Sample Scale | |
|---|---|
| Ratio | Note |
| 1/1 | C +0 |
| 9/8 | D -18 |
| 6/5 | Eb +16 |
| 5/4 | E -14 |
| 3/2 | G +2 |
| 16/9 | A# -4 |
As I said before, small whole number frequency ratios represent consonant intervals. How small is small? Generally, odd numbers less than 15 are considered small. The highest odd number which is part of a JI tuning system is called the odd-limit of that system. The sample scale shown at right is a 9 odd-limit system because the highest odd number is 9. Odd-limits are useful for analyzing consonance in a Just Intonation system. A list of 13 odd-limit Just Intonation ratios can be found here.
Consider again the sample scale shown at right. There is a huge gap between the 3/2 (G) and the 16/9 (A#). We could arbitrarily add any note to fill the gap, but most people use an interval already in their scale. We could stack two 5/4 to give us 25/16 or G# - 27 (5/4×5/4 = 25/16). This would be 25 odd-limit and be dissonant against the 1/1, but it's a consonant 5/4 away from E. Extending the scale this way involves another type of limit called prime-limit.
Prime numbers are divisible only by 1 and themselves. The sample scale at right is 5 prime-limit because the highest prime factor of any ratio in the scale is 5. 9 is a larger odd number than 5, but it's highest prime factor is 3 (9 = 3×3). Even when we add 25/16 to this scale, it's still a 5 prime-limit scale. Technically, there are an infinite number of prime-limit ratios in the 3 prime-limit (or higher prime-limit) but only the smaller ones are consonant. To prove this to yourself, consider the three prime-limit ratio: 243/128 (243/128 = 3×3×3×3×3 / 2×2×2×2×2×2×2). Since this ratio is 243 odd-limit, it is very dissonant.
When talking about limits in the context of scale design, prime-limit is usually assumed. When talking about limits in the context of consonance, odd-limit should be assumed.