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What is the procedure for measuring the interval between two pitches?

How do I know if something is a fifth, sixth, or something else? Why, for instance, is C to D a second even though we only go up one note?

And how do I know if it the interval major, minor, or something else?

Lastly, I've heard people say that I shouldn't measure intervals by just counting half steps. Why is this a bad idea?

  • Your previous activity on this stack makes me feel like you know this stuff already. Is that somehow not the case? Are you asking for someone else? Sorry to ask, I just find it odd. – Todd Wilcox Aug 27 '18 at 1:07
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    @ToddWilcox I was looking for a canonical answer to this question to refer to another user, but I couldn't find one. In chat I asked Dom if he knew of one, and he couldn't find a suitable one either. We thought that it would be helpful to have a thread dedicated to it that we could use at in the future as a reference. – Richard Aug 27 '18 at 1:09
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First, I suggest looking this up in the Wikipedia but I'll go into it a bit.

Intervals are described using both endpoints, that is the notes spanned, like using "Ocho días" for "week" in Spanish. Thus the interval C-C is a unison; C-D is a second, C-E is a third, ... C-B is a seventh, and C-C (the next higher C) is an octave. It's a funny linguistic thing as the word unison (in most languages) refers to a single object. C-C contains 1 note. C-E can be considered as the C,D,E and thus is a third.

Second point, there are only 7 different letters used even though a chromatic scale has twelve notes. C-D is a second though the note C# (or Db) is in between; E-F is also a second. The term "major" is used for a "nominal" second containing a two half steps; "minor" is used for a second containing a single half step. It's complicated because it grew over time and the various terms added gradually.

The rationale for having such a complicated nomenclature is that originally, the half and whole step intervals were not all equal. See the Wiki article on Temperament. The 8 nominal intervals spanning an octave have assigned ratios. In the table below, the ratio is of the second note to the first.

C-C is 1/1 C-D is 9/8 C-E is 5/4 C-F is 4/3 C-G is 3/2 C-A is 5/3 C-B is 15/8 C-C is 2/1

Unfortunately, these ratios cannot be extended exactly. There is a problem in that the ration of D to C is 9/8 but that of E to D is 10/9; these are close but not identical. Thus no consistent naming of intervals by frequency is possible with the above ratios. If one uses an equally tempered scale, then each half step has ratio of the twelfth root of 2. Many instruments (winds, strings, voices,...) do not naturally produce this equal temperament so the historical names of intervals and notes is still in use.

  • Yep. Calling C to D a "second" rather than a "first", counting both endpoints as you say, is called "the fencepost problem", which comes up all the time in various guises. If you have twelve lines spaced an inch apart, how far from the first to the last line? Eleven inches. We have the opposite problem in numbering harmonics- typically, the octave is called the "first harmonic", when it should really be the second- the fundamental is the first, because it's one times the starting frequency, the octave is two times, and so forth. Conventions are not always logical. – Scott Wallace Aug 27 '18 at 7:55

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