Equal temperament and overtones

Today in music class, we learned about equal temperament. Remember how Dad taught us about octaves the other day? That's why the teacher's explanation was so easy to understand. Nowadays, when we say "equal temperament," we mean the 12-tone equal temperament where one octave is divided into 12 equal parts, but apparently, there were many different equal temperaments created before we got to the 12-tone equal temperament we have today. There were apparently up to 360 equal temperaments.

Wow, I've never heard that before. 360-tone equal temperament means there are 30 notes within a semitone, right? Is it even possible to make music with that? 12-tone equal temperament is a system where an octave is divided into integer equal parts, with each part being the smallest unit of interval. The reason you can play the same music even if you shift the pitch of a song up or down a semitone on a keyboard, or if you use a capo on a guitar, is because the frequency ratios between the notes remain the same.

So, with 12-tone equal temperament, you can transpose and modulate, which is a big advantage both in terms of instrument making and performance. My teacher also taught me about just intonation. In just intonation, the frequencies of all notes are in simple integer ratios. That's why those chords sound harmonious, right? My teacher said that chords in equal temperament sound somewhat muddy, but I don't really understand why.

The fact that the frequencies of intervals in just intonation are in simple integer ratios is not something that humans created for convenience when making musical instruments, like in 12-tone equal temperament, but rather a law of nature. The combination of two sounds that humans find "pleasant" to hear has frequencies in simple integer ratios. These combinations of resonating sounds are closely related to "harmonics," which have a significant influence on "timbre."

Harmonics?

If you lightly touch the 12th fret, which is exactly in the middle of the guitar string, and pluck the string, you will get a pure, sharp sound.

That's harmonics, right? I know about it because I've just started learning guitar. But how does it produce that sound?

Observe the guitar strings when you're playing harmonics. If you look closely, you'll see that guitar strings vibrate in a unique way. The middle part where your finger touches remains still and doesn't vibrate at all after you lift your finger. On either side of that, however, the strings vibrate in two distinct peaks. When you play a harmonic at the 12th fret, you get a note that's an octave higher than the note you'd get by playing an open string normally. The 12th fret is exactly half the length of the string, so it's a double harmonic, resulting in a note that's an octave higher.

So, if you play a harmonic at 1/3 of the string length, will it produce a note three octaves higher?

That's wrong. Think about it. An octave means the frequency doubles, right? One octave above 500Hz is 1000Hz, and another octave above that is 2000Hz... If you touch the 7th fret, which is 1/3 of the way up, you get the third harmonic. That's a perfect fifth above the open string. Also, if you touch the 5th fret, which is 1/4 of the way up, you get the fourth harmonic. That's two octaves higher.

A perfect fifth?

A 1:2 relationship is an octave; if 1 is C, then 2 is the C one octave higher. In the case of 1:3, 3 corresponds to a G a perfect fifth higher. This is how chords sound beautiful, because the frequencies of sounds are simple integer multiples. The open 5th string of a guitar produces the note A. When you strum the open 5th string, you get a sound of 110Hz. However, in reality, not only is there a sound of 110Hz, but also sounds with frequencies that are integer multiples of the fundamental frequency, such as 2, 3, 4, and so on. These harmonics are not only found in musical instruments, but also in human voices, car sounds, and all sounds in nature.

Does the way overtones are present make a sound sound different?

That's right, it's about the unique timbre of each sound. Generally, sounds with many harmonics tend to have a hard and bright sound, while sounds with few harmonics tend to have a round and muffled sound. A sound consisting only of the fundamental frequency, without any harmonics, is called a sine wave, and its waveform is like the sine graph seen on an oscilloscope. In nature, it's rare for only one harmonic component to be emphasized within a complex sound emitted from a single sound source. Therefore, if there is an emphasized harmonic in a given space, human hearing seems to separate and interpret it as a sound coming from a different sound source. That's one of the factors that explains the cocktail party phenomenon.

The cocktail party phenomenon?

I'll talk about that topic again next time.