Equal temperament and harmonics
Oto-kun
Today in music class, I learned about equal temperament. The other day, my dad taught me about octaves, so the teacher's explanation was very easy to understand. Now, when we talk about equal temperament, it refers to the 12-tone equal temperament, which divides an octave into 12 equal parts, but it seems that many different types of equal temperaments were created before the current 12-tone system, including one with 360 divisions.
father
Wow, that's the first time I've heard that. When you say 360 equal temperament, it means there are 30 notes in between each half step, right? I wonder if music can be made with that. The 12 equal temperament divides the octave into equal parts, establishing one of them as the smallest unit of interval. The reason you can play the same music by shifting it up or down a half step on a keyboard, or using a guitar capo, is that the frequency ratios between the notes are preserved.
Oto-kun
In other words, with the 12-tone equal temperament, it is very advantageous for both instrument making and performance because it allows for transposition and modulation. The teacher also taught me about just intonation. In just intonation, the frequencies of all intervals are simple integer ratios. That's why those chords sound harmonious, right? The teacher said that chords in equal temperament sound somewhat muddy, but I don't really understand that.
father
The reason the frequencies of pure tuning intervals are in simple integer ratios is that they are not something created by humans for convenience in making instruments, like in equal temperament, but rather a law of nature. The combinations of two notes that humans perceive as 'pleasant' have simple integer frequency ratios. These resonant combinations of sounds are closely related to 'harmonics,' which have a significant impact on timbre.
Oto-kun
Harmonics?
father
If you lightly touch the 12th fret, which is exactly in the middle of the guitar strings, and pluck the string, a pure sound will be produced.
Oto-kun
It's the harmonics technique, right? I've started learning guitar, so I know about it. But I wonder how such a sound is produced.
father
When you produce harmonics, observe the guitar strings. If you look closely, you'll see that the strings vibrate in a special way. The point where you touch the string remains still and does not vibrate at all, while the sections on either side create two peaks and vibrate. When you produce harmonics at the 12th fret, the sound that comes out is an octave higher than the sound produced by playing the open string. The 12th fret is exactly half the length of the string, resulting in a sound that is an octave higher due to the second harmonic.
Oto-kun
So, if you produce a harmonic sound at one-third the length of the string, will it produce a sound that is three octaves higher?
father
That's incorrect. Think about it. An octave means that the frequency doubles, right? One octave above 500Hz is 1000Hz, and another octave above that is 2000Hz... If you touch the 7th fret at 1/3, you produce a third harmonic. This relates to the sound of the open string and a perfect fifth. Also, if you touch the 5th fret at 1/4, it produces a fourth harmonic, which corresponds to a sound two octaves above.
Oto-kun
Perfect fifth?
father
The relationship of 1:2 is an octave, and if we consider 1 as the note C, then 2 corresponds to the C one octave higher. In the case of 1:3, the 3 corresponds to the note G, which is a perfect fifth above. This way, because the frequencies of sounds are simple integer multiples, chords resonate beautifully. The open string of the 5th string on a guitar produces the note A. When the open 5th string is struck, it produces a sound at 110Hz. However, in reality, it does not produce just the 110Hz sound, but also sounds at integer multiples of that frequency, such as 2 times, 3 times, 4 times, etc. These overtones are present not only in instruments but also in human voices, car sounds, and all sounds in nature.
Oto-kun
Does the way harmonics are included affect how sound is heard?
father
That's right, it's about the unique timbre that each sound possesses. Generally, sounds that contain many overtones are perceived as having a hard and bright impression, while sounds with fewer overtone components are said to have a round and muffled quality. A sound that contains no overtones at all is called a sine wave, which corresponds to the waveform seen in a sine graph on an oscilloscope. In nature, it seems that it is quite rare for a single overtone component to be emphasized among the complex sounds emitted from a single sound source. Therefore, when there is an emphasized overtone in a certain space, human hearing tends to interpret it as a sound coming from a different source. This is one factor that explains the cocktail party phenomenon.
Oto-kun
Cocktail party phenomenon?
father
I'll talk to you about that theme next time.