Perception of sound
13 Musical intervals and temperament
Musical intervals in equal temperament
All musical intervals including the octave are built on frequency ratios. Modern Western music uses a system called equal temperament (ET for short). The table below shows the frequency ratios for all intervals from unison up to an octave. (Unison is the musical name for the “interval” between two identical notes).
Interval name | # semitones | f_{2}/f_{1} (exact) | f_{2}/f_{1} (approx.) | f_{2}/f_{1} (approx.) |
Unison | 0 | 1 | 1 (exactly) | 1:1 |
Semitone | 1 | 2^{(1/12)} | 1.06 | – |
Whole tone | 2 | 2^{(2/12)} | 1.12 | – |
Minor third | 3 | 2^{(3/12)} | 1.19 | – |
Major third | 4 | 2^{(4/12)} | 1.26 | 5:4 |
(Perfect) fourth | 5 | 2^{(5/12)} | 1.33 | 4:3 |
Tritone | 6 | 2^{(6/12)} | 1.41 | – |
(Perfect) fifth | 7 | 2^{(7/12)} | 1.50 | 3:2 |
Minor sixth | 8 | 2^{(8/12)} | 1.59 | – |
Major sixth | 9 | 2^{(9/12)} | 1.68 | – |
Minor seventh | 10 | 2^{(10/12)} | 1.78 | – |
Major seventh | 11 | 2^{(11/12)} | 1.89 | – |
Octave | 12 | 2 | 2 (exactly) | 2:1 |
Equal temperament divides the octave into twelve identical intervals (called half-steps or semitones). As a result, each semitone corresponds to a frequency ratio of [latex]2^{(1/12)}[/latex]. What this means is a note a semitone above [latex]100 Hz[/latex] has a frequency of [latex]2^{(1/12)} *100 Hz[/latex], or roughly [latex]106 Hz[/latex]. Note: [latex]2^{(1/12)}[/latex] can also be written as [latex]\sqrt[12]{2}[/latex] and is roughly equal to 1.06.
Why [latex]\sqrt[12]{2}[/latex]? Before diving into the explanation, make sure you understand the following Stop to Think.
Stop to think
What is the frequency ratio that corresponds to a musical interval of three octaves?
When you “stack” musical intervals, the frequency ratios multiply together– not add. A semitone is a factor of [latex]\sqrt[12]{2}[/latex] because going up twelve semitones in a row must be the same thing as going up one octave. If you multiply [latex]\sqrt[12]{2}[/latex] by itself twelve times in a row, you get two- exactly the right frequency ratio for the octave.
Other intervals in ET are built by “stacking” semitones. Each semitone you go up (or down) increases (or decreases) the frequency by a factor of [latex](\sqrt[12]{2})[/latex]. Go up (or down) certain number of semitones, the frequency increases (decreases) by a factor of [latex](\sqrt[12]{2})^{n}[/latex] where [latex]n[/latex] represents the number of semitones you go up. You can combine these ideas into a single equation:
[latex]f_1= f_2 *(\sqrt[12]{2})^{n}[/latex]
Here’s the same equation written with fractional exponents (rather than root signs);
[latex]f_1= f_2 *2^{(n/12)}[/latex]
In the equation, [latex]f_2[/latex] represents the frequency you want to find and [latex]f_1[/latex] is the frequency of the note you start with. Use positive numbers for [latex]n[/latex] if the frequency you want to find is higher than the frequency you are given. Use negative numbers if you are going down in pitch.
Charts with note names and corresponding frequencies are freely available on the web- search on “note name frequency chart” to see a selection. Almost all of these charts are generated using the equation above, using the frequency 440 Hz for the note A4 as the starting point. (This standard note is often called A440).
Musical interval example
Example: Musical intervals
QUESTION:
Find the frequency of the note a minor third above 440 Hz.
SOLUTION:
Identify important physics concept: Musical intervals in ET are based on the equation:
[latex]f_1= f_2 *2^{(n/12)}[/latex]
List known and unknown quantities (with letter names and units):
[latex]f_1=440 Hz[/latex]
[latex]n=3[/latex]
(because there are 3 semitones in a minor third)
Do the algebra: There’s no algebra to do- the equation is already solved for the frequency ([latex]f_2[/latex]).
Do unit conversions (if needed) then plug in numbers: No unit conversions needed- everything is in Hz.
Then plug the number into the equation:
[latex]f_1= 440 Hz *2^{(3/12)} \approx 440 Hz*1.19=523 Hz[/latex]
(The 1.19 value was taken from the chart on this page. You can use your calculator to get a more exact result).
Reflect on the answer:
- Answer is above 440 Hz, as expected. Answer is less than 880 Hz, too. (Since a minor third is less than an octave, I expect the answer to be less than double 440 Hz).
Musical intervals in non-ET systems
Equal temperament is the most common system for musical intervals but it is not the only one. Early temperament systems based musical intervals frequency ratios based on whole numbers. Pythagorean temperament defines a perfect fifth as a frequency ratio of 3:2 and defines all other notes in terms of the perfect fifth. Just temperament defines all intervals in terms of fractions of whole numbers. The advantage of these older temperaments is that musically important intervals (like perfect fifths and major thirds) are perfectly in tune with the overtones wind and string instruments produce. The drawback is that non-ET systems do not handle key signatures changes well. In non-ET systems, musical notes produced in one key signature (say D major) do not produce the same musical intervals when music of a different key signature (say C major) is played.
Equal Temperament is a compromise. ET sacrifices the proper intonation of some musical intervals in favor of having musical intervals that are the same in all key signatures. Even though none of the ET intervals are exactly “in tune,” many important intervals (like the perfect fifth and the major third) have ratios that are close to whole number ratios. The result is a system that sounds reasonably good in all keys. (Purists complain that ET sounds equally bad in all keys).
In practice, musicians subtly adjust tuning as they play or sing (whether they intend to or not) so that the sound is pleasing, especially at important moments in the music. Often the result is a hybrid of temperaments. To learn more and hear differences that temperament systems can make, visit B.H Suit’s web page on Just and Equal Temperament. ^{[1]}
Stop to think answer
A musical interval of three octaves is a frequency ratio of eight. Remember that each octave doubles the frequency. If you start at 100 Hz, one octave up is 200 Hz; another octave up doubles the frequency again (to 400 Hz) and the third octave doubles the frequency yet again (to 800 Hz). Mathematically, three doublings is a ratio of two to the third power: [latex]2*2*2=2^{3}= 8[/latex].
Overtones and musical intervals
Most musical instruments produce complex tone- a mix of the fundamental and overtones. Without exception, the overtones on any given instrument are at multiples of the fundamental frequency. This also means that the overtones of any given instrument are at specific musical intervals from the fundamental.
For instance, the spectral content of an ideal string is f, 2f, 3f, 4f… Since the first overtone of a string is always double the fundamental, the first overtone is always one octave above the fundamental. The second overtone is three times the fundamental- or an octave and a fifth above the fundamental. The table below shows the first seven overtones for strings:
Frequency | Musical interval relative to fundamental | |
Fundamental | 1f | unison |
1st overtone | 2f | octave above |
2nd overtone | 3f | octave and a perfect fifth above |
3rd overtone | 4f | two octaves above |
4th overtone | 5f | two octaves and a third above |
5th overtone | 6f | two octaves and a fifth above |
6th overtone | 7f | two octaves and about a minor seventh above |
7th overtone | 8f | three octaves above |
The basic principle holds for all musical instruments, because overtones are at fixed ratios above the fundamental. (Brass players might notice that the overtone intervals for strings above apply to brass instruments, too).
- Suits, B. H. (1998). Scales: Just vs. Even Temperament. Retrieved from https://pages.mtu.edu/~suits/scales.html ↵