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Is a semitone a half step?

In the same way that there are half steps in the intervals between two adjacent consonants, there is a half step between two adjacent sounds. In the same way that there are half steps in the intervals between two adjacent vowels, there is a half step between two adjacent diphthongs. In the same way that there are half steps in the intervals between two adjacent consonants, there is a half step between two adjacent sounds.

One is tempted to say, Well, yes, but semitones do not have to be identical. For example, if I had made up one long syllable, like the one I just wrote, I could have used different sounds to represent two-thirds of the syllable:

1) diphthong 3 (A)

2) diphthong 3(B)

Musical Link 1 - IB Musical Links Investigation
It is all an illusion. Two-thirds of the syllable can be represented at a different time and in a different way. There is no limit on the sounds that can be used in a semitone, but semitones have to be exactly in tune with respect to the intervals between adjacent sounds.

Semitones are not useful for indicating time or distance. They are not used by humans for this purpose. However, they were, and remain, useful for representing intervals in scales. It was the Greeks who invented the concept of an “octave”. For example, the Greek octahedron has a unit of frequency in its second octave, like this:

1) octahedron

2) diphthong 4 (A)

3) octahedron

4) diphthong 4A

This is not as close as to an interval, but it comes close enough that Pythagoras might have used it as time. However, I believe that his theory would not have been fully developed.

In fact, it is no wonder that we all forget Pythagoras’s time system: all time and place in the world is a series of intervals and ratios. The Greek scientist Eudoxus described the universe as a series of ratios. In fact, the Pythagorean theorem implies that the interval between two quantities equals the ratio of their absolute values.

Now let’s look at a semitone.

The Greek word “semion” was used for the sound in a semitone scale. This meant that we could define a number above the base 3 as a semitone, or