HOME > Ono Sokki's Patio > Equitempered scale and harmonic overtone

Ken: I learned “equitempered scale” in a music class today. Dad, you told me about an octave the other day so what the teacher explained in the class was very familiar to me. In these days, it represents “twelve equitempered scale” which divides one octave into twelve notes. There was a variety of equitempered scales before today’s style. It is said that there was even a “360 equitempered scale”.

Dad: I have never heard it before. What is specific with it is there are no less than 30 notes between each semitone. I wonder if it makes music up. “Twelve equitempered scale” is an equally divided equitempered scale dividing an octave with an integral number and make the resultant note to be the smallest unit. You can play music in the same way even if you put off music by a semitone on a keyboard, or put capotasto on a guitar because the ratio of frequencies is kept as it is.

Ken: You can do transposition or modulation in the “twelve equitempered scale” and it is very convenient to make a musical instrument and play music. The teacher also told us about just temperament. Any pitch of sounds in just temperament has frequencies of notes which are related by whole number ratios. So the summational augmented chords sound harmonious. Chords of equitempered scale sounds cloudy somehow, the teacher said, though I don’t know well.

Dad: Why the frequencies of just temperament have simple ratio of integral numbers is that it is the rule of nature, not the one which was made artificially for making instruments as twelve equitempered scale. The frequencies of two sounds whose combination a man feels easy are related by whole number ratios. The combination of sounds has close relationship with “harmonic overtone” which is very effective to “tone”.

Ken: What do you mean by “harmonic overtone”?

Dad: If you flip the guitar with your finger on a 12 fret, the center of the string, it makes a very pure sound.

Ken: Harmonics play, isn’t it? I know what it is as I started to learn guitar. But how does it sound that way?

Dad: Look at the string of a guitar closely when you play the guitar with harmonics. Looking at it, the string vibrates in a peculiar manner. The center you put a finger on does not vibrate at all even after releasing your finger off it. The strings on both sides vibrate making two waves. In playing harmonics with 12 fret, it sounds one octave higher than playing with open string. 12 fret is just a half of a string or double sound and is one octave higher.

Ken: If you play harmonic sound with shorting a string by one third, can you sound three octaves higher?

Dad: It is not right. In an octave, each frequency becomes double. What is one octave higher than 500Hz is 1000Hz and the one another octave higher is 2000Hz… If you play with 7 fret of one third, it sounds three times sound. This has a relation of perfect fifth with open string sound. If it is 5 fret of one forth, it would be four times sound. This is two octaves higher.

Ken: Perfect fifth?

Dad: In a ratio of pitch whose proportion of one part to two (1:2) for example, if you assume “1” for “Do”, “2” represents “Do” one octave higher. In a proportion of one part to three (1:3), “3” corresponds to “So” one perfect fifth higher. The frequencies of notes related by whole number ratios make harmony. The open sound of 5th string of a guitar is “A” or “La”. When you flip the string, it sounds 110Hz. But actually there are not only 110 Hz but also frequencies of notes related by whole number ratios against tonics such as twice, three times or four times of 110Hz. This harmonic overtone is included not only in instrument sound but also voice of human, sound of vehicles and all natural sounds.

Ken: Is sound different depending on how harmonic overtone is included?

Dad: Yes. That is to say, distinctive tone each sound has It says sound with lot of harmonic overtone has a stiff and bright impression and sound with less harmonic overtone sounds rounded and hollow. Sound with only tonic which has no harmonic overtone included is called sign curve and it shapes wave pattern of sign chart displayed on an oscilloscope. In nature, it is rare that only one harmonic overtone factor out of complex sound given out from one source of sound is emphasized. So if there is an emphasized overtone in a room for example, a man recognizes it as a sound from another source. This is a key to explain cocktail party phenomenon.

Ken: Cocktail party phenomenon?

Dad: I leave it for next time.

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Equitempered scale is a prosody which equally divides one octave into twelve tones. So X, in the formula of X12=2 is to be that rate. The X equals 1.059463094…. It is easy to understand if explained financially that it will be twice as large as a capital at a 6% annual rate in 12 years. See comparison to the rate of just temperament below. For example, sound “Mi” which is 1.25 times than “Do” of tonic in just temperament is to be 1.26 times in equitempered scale and there is approx. 0.8% difference. Cloud of harmony in equitempered scale results from the difference.

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