Do intervals contain a variable number of steps? Are they absolute, or linked to C-major? [duplicate]

An interval is the distance between two pitches. We name the major intervals based on the number of note names they contain. Since the note letter names have been assigned such that in the C-major scale the 7 notes contain no sharps or flats, it seems that intervals are inherently linked to the C-major scale, not to absolute differences in pitch.

For example, the major third (M3) going from C to E contains 4 semi-tones (C->D=2; D->E=2). In contrast, the M3 going from E to G contains only 3 semi-tones (E->F=1; F->G=2). Thus, it would seem that a M3 is not an absolute unit of distance but tied to the underlying assumption of a "default" to C-Major.

Now my question: my music theory book states an M3 contains 2 steps and a M6 contains 4.5steps. I don't understand. If "step" refers to alphabetical notes, the M3 definition makes sense, but the M6 does not. If a step refers to two semi-tones, then neither definition makes sense. As noted above, an M3 could contain either 3 or 4 semitones (1.5 or 2 steps??) depending on which note you start at. Similarly, an M6 could contain either 9 semi-tones (C-A) or 10 semi-tones (E-C) (4.5 or 5 steps??).

What does my textbook mean by "step"? Why does it imply that intervals are absolute differences in pitch when the number of semi-tones in any given interval varies? Are my example incorrect in some way?

Intervals contain a unique number of diatonic steps. Thus as in the OP, the C-E is a third (intervals also include the endpoints in counting; the number of different notes is counted). However, intervals come in different sizes. Thirds like C-E consisting of 4 half-steps are termed "major" and those like E-G containing 3 half-steps are termed "minor."

Seconds and sixths also come in major and minor varieties. Unisons, octaves, fourths and fifths are called "perfect" as they have no major or minor versions. Perfect fourths C-F contain 5 half-steps and perfect fifths contain 7 half-steps.

There is one more possibility. A widened major or perfect interval is called "augmented" and a narrowed perfect or minor interval is called "diminished." So C-F# is an augmented fourth while C-Gb is a diminished fifth. In equal temperament, these notes are the same (called "enharmonic"); however they have different uses. Augmented intervals tend to expand (upper note goes up and/or lower note goes down) while diminished intervals shrink (upper note goes down and/or lower note goes up.)

Music names exist historically because they describe the way composers wrote. This is why there are "diatonic" and modifications there of of those notes. Naming things from the chromatic scale (12 half-steps) seems simpler but it wouldn't describe how people composed from about 500 to 1900. (Many still write this way.)

• F♯ and G♭ are enharmonically equivalent in every 12-tone tuning system, not only in equal temperament. Also, you forgot to include the seventh, which can also be major or minor. – phoog Jul 3 '20 at 8:27
• I don't understand your term 'diatonic steps'. Intervals contain tones and semitones, which isn't quite the same. Major means larger, minor smaller in this context. It's unfortunate that those terms are used - as they allude to scales, major and minor - which have no real influence on an interval.(There seems to be confusion as though major intervals are from major scales, minor from minor scales, which certainly isn't the real case, although it can appear to be the proving factor, sadly! – Tim Jul 6 '20 at 10:39

I've realized my issue. Tones and steps are synonymous terms (i.e. 1 half-step = 1 semi-tone). We name the major intervals based on how many alphabetical notes the interval contains (e.g. C to E is a M3 as it contains C,D,E). However, we have not yet specified if any notes have sharps or flats (accidentals).

Two of the examples in the original question are incorrect. E-G is not a Major-third, it is a minor-third. E-G# is the M3 (containing 2 half-steps/semi-tones). It contains the note names E, F, and G. It just so happens that one of the notes has an accidental.

The E-C example was doubly wrong. I incorrectly stated E-C had 10 semi-tones. It does not. It contains 8 semi-tones and is not a M6. E-C is a minor-sixth. The major sixth is E-C#.

Saying the interval is named after the number of alphabetical notes it contains holds true, but we keep in mind an interval which starts on a natural does not necessarily end on a natural note.

• As a final thought: at first, the naming convention for intervals seemed at random to me. In fact, if you start at any given letter "X" and follow the majors&perfects. (M2, M3, P4, P5, M6, M7), you will get all of the notes in the X-major scale. M2: 1 step; M3 2 steps; P4 2.5 steps; P5 3.5 steps; M6 4.5 steps; M7 5.5 steps. Because the naming convention is based on the C-major scale. If you start at C all majors and perfects will give you natural notes (no accidentals). – AlexJ Jul 3 '20 at 1:12
• Also keep in mind that an interval doesn't have to start on a natural note. And - intervals are counted from the ower note. And - have a look at inversions of intervals - there's a formula. (C>E compared with E>C, etc.) – Tim Jul 3 '20 at 5:48