Realizing how little to nothing I know about music theory, I started watching videos on the subject recommended to me by YouTube. Apart from that those videos aren’t really vetted, I don’t know how useful those are, even if they’re accurate. So there’s my caveat to you, the reader.

Today I researched the musical interval, i.e. the ratio in frequency value between musical notes.

In Why Does Music Only Use 12 Different Notes? presenter David Bennett explains how musical notes work in popular and classical music, which most of us listen to on a daily basis.

There are the following 12 notes (semitones) in an octave (using letters, while in some countries other notations are used, like do-re-mi):

  1. C
  2. C#
  3. D
  4. D#
  5. E
  6. F
  7. F#
  8. G
  9. G#
  10. A
  11. A#
  12. B

When a note has a frequency twice as high as another note, the distance (interval) between them is called an octave. Playing two notes an octave apart sounds pleasing (consonant) to our human ears.

Inside an octave there are 12 notes. Playing two of those notes together gives a certain mood (how humans perceive the combination of notes, pleasing or less pleasing). The first of those 12 notes is called the root. Playing the root and the same note an octave higher gives us the most consonant combination of notes.

On a scale from most consonant to most dissonant, there are twelve musical intervals (between brackets is the frequency ratio between both notes in the interval). The intervals numbered 1 - 7 in this list are considered consonant in Western music, 8 and higher dissonant.

  1. Octave, aka 8ve, (2:1)
  2. Perfect 5th, aka P5 (3:2)
  3. Perfect 4th, aka P4 (4:3)
  4. Major 3rd, aka M3 (5:4)
  5. minor 6th, aka m6 (8:5)
  6. minor 3rd, aka m3 (6:5)
  7. Major 6th, aka M6 (5:3)
  8. Major 2nd, aka M2 (9:8)
  9. minor 7th, aka m7 (9:5)
  10. minor 2nd, aka m2 (16:15)
  11. Major 7th, aka M7 (15:8)
  12. Tritone (7:5)

Below that, sounding even less pleasant, there are quarter-tones, and even further divided intervals, that sound even more unpleasant.

Ordered on frequency (how they appear on a piano keyboard, using both white and black keys):

[root] [m2] [M2] [m3] [M3] [P4] [Tritone] [P5] [m6] [M6] [m7] [M7] [8ve]

With 12 notes in an octave we have a limited set of frequencies. It is a practical compromise, really, limited to only the most useful musical intervals, excluding others. However, this compromise is still not enough. With the ratios mentioned above, we can only play in a fixed key (root note). If we want to transpose a piece of music to a different key with the ratios kept intact, the musical intervals sound all weird. This is because this theoretical just intonation doesn’t allow for transposition. To fix it, so we can play in any key we want, the intervals must be changed. This slight alterations in musical intervals is called temperament (tempering the intervals so music can be played in arbitrary keys, not just one single key).

There have been several temperaments over the millenia and ages. The system we use today is called 12 Tone Equal Temperament. Each semitone is a factor of the 12th root of two higher than the previous semitone.

Here are the tempered values of the intervals. They are very close to their just intonation, between brackets:

  1. m2 1.0595 (1.0667)
  2. M2 1.1225 (1.1250)
  3. m3 1.2892 (1.2000)
  4. M3 1.2599 (1.2500)
  5. P4 1.3348 (1.3333)
  6. Tritone 1.4142 (1.4000)
  7. P5 1.4983 (1.5000)
  8. m6 1.5874 (1.6000)
  9. M6 1.6818 (1.6667)
  10. m7 1.7818 (1,8000)
  11. M7 1.8877 (1.8750)
  12. 8ve 2.0000 (2.0000)

Except for the octave, these intervals are slightly out of tune compared to the ideal, but not enough to be noticeable, especially the perfect 4th and 5th are extremely close in value to a just intonation. Using this temperament musicians can play in any key without the music sounding strange. Anyway, the music most of us listen to (Western music) has been already in 12 tone equal temperament for hundreds of years. Most of us (including myself) probably don’t know any better.

It’s a small, yet important part of music theory.