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I'm starting to study music, and I'm reading about semitones and tones.

Looking at a piano, I know there are 8 notes in an octave:

Do – Re – Mi – Fa – Sol – La – Si – Do

Here the Do is counted twice.

Counting the semitones, it looks something like this (homophonic semitones grouped together):

Do – Do# (Re♭) – Re – Re# (Mi♭) – Mi – Fa – Fa# (Sol♭) – Sol – Sol# (La♭) – La – La# (Si♭) – Si

Which gives 12 semitones.

But why is Do not counted again in the end?

7

An octave is an interval composed of 12 semitones. A semitone is an interval, so they are:

C → C#, C# → D, D → D#, D# → E, E → F, F → F#, F# → G, G → G#, G# → A, A → A#, A# → B, B → C.

In Solfège:

Do → Do#, Do# → Re, Re → Re#, Re# → Mi, Mi → Fa, Fa → Fa#, Fa# → Sol, Sol → Sol#, Sol# → La, La → La#, La# → Ti, Ti → Do.

See how you count the intervals, not the notes? The C# is not a semitone, it is a semitone above C.

It's almost correct to say that there are eight notes in an octave. Really, there are eight notes of a major scale within the interval of an octave. This is also true of minor scales, but it's not true of some other types of scale (e.g. pentatonic, blues). The major scale itself only has seven unique notes, but the within an octave, the first gets repeated, so you end up with eight.

So it seems that there's a small error in your count of semitones, and a slight misunderstanding of the relationship between an octave and a scale.

  • 1
    A major scale is considered a heptatonic scale meaning that it is actually a seven note scale and the octave is the eighth note. Just as a pentatonic scale is five notes. – jomki Nov 5 '16 at 4:56
  • @jomki Very true. I didn't phrase that well. I was trying to say that there are eight notes of a major scale within the interval of an octave. I've tried to clarify in the latest edit. – endorph Nov 5 '16 at 5:16
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This is just a weird historic mistake. Apparently some people in the middle ages didn't know zero as a number, and hence labelled the zero-interval with 1 (unison). Continuing this through the diatonic scale ends you up with the label 8 (octave) on the equivalence-class interval. But there are not eight notes in an octave of diatonic scale / white keys, despite the name; there are in fact only seven (the diatonic scale is a heptatonic scale).

On the more recent 12-edo scale which most modern western instruments use to approximate diatonic scales, this mistake was not repeated: the unison in 12-edo consists of zero semitone steps. Hence the octave has the correct label of 12, corresponding to the fact that the 12-edo scale really divides the octave in twelve steps.

  • Yep. This is of course the same mistake that gave us no year zero in our calendar. – Scott Wallace Nov 5 '16 at 18:01
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Count two octaves:

12345678-12345678 = 16 (wrong) Most make the mistake of counting the first number eight twice using it as number one on the second octave only count one number eight note in the middle. That should equal 15 notes.

12345678-2345678 = 15. (correct)

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You have chosen to count do twice in the major scale, only once in the chromatic scale. Count them in the same way, there's no problem.

The major scale spans 8 notes, contains 7 differently-named notes.

The chromatic scale spans 13 notes, contains 12 differently-named ones.

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The historical naming of intervals has mostly to do with the idea that things that are the same are a unity (like the United States of America). A unison (two middle Cs for example) are a unity. The distance between two middle Cs is zero steps but the Cs are called a unison. Calling it a "zero" or "nihil" or "nothing" or something similar doesn't really help. The next note added also doesn't seem to be the first note, it's the second note.

There is an inherent problem between names of objects and names of distances between ordered arrangements of those objects. It's easy to count one, two, three, four for 4 notes or zero, one, two, three for the same 4 notes. Some mathematical representations of music do this. However, I once angered a math teacher (who was saying that "first" and "one" are not related words so why shouldn't zero be called the first number; I just asked how that explained the relationship between four and fourth. and five and fifth, six and sixth, etc.)

The logical problem is that a set of N points in a line form N-1 first differences. If the same set of names is used, something won't match. This problem surfaces in some languages: https://spanish.stackexchange.com/questions/13014/why-does-every-eight-days-mean-once-a-week

  • Right, but what's your point? IMO all this is just reason why we should really always use zero-based indexing. Unfortunately, the words first, second, third (or equivalent in other languages) are so ingrained in culture that this won't happen, but a lot of confusion could be saved if children grew up right from the start with a zero-based system. – leftaroundabout Mar 12 '17 at 19:29
  • Neither zero based nor one based or arbitrary based fixes the fact that N points in a line contain N-1 empty intervals between them. There are numerical systems of music notation (notes named 0-11 for example) which make some things easier. Most of these have octave equivalence which makes it hard to describe the opening note of Also Sprach Zarathustra or Siboney. Music (and the rest of existence for that matter) are hard to describe exactly; things do not scale nor do analogies work very well. We just make do as best we can. (And topologists consider donuts equivalent to a coffee mug.) – ttw Mar 12 '17 at 23:33
  • But zero-based does make it clearer what's the issue: that if you're looking at N elements of fence, the fencepost terminating the last section is really already the starting of yet another section, hence if you count that too you actually have N + 1 fenceposts. Doing the equivalent observation with a “zeroth fencepost that's really already part of the negative-numbered fence sections” is much less natural (pun not intended). – leftaroundabout Mar 12 '17 at 23:46

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