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Can an interval be a measure of absolute distance between any given note to another, or is it always used in the context of a scale, where each note in the scale has a corresponding interval value?

If it isn't based on semitones, is it basically giving each note in a scale a certain label?

How does it translate when, for example, we are finding the interval between E and G in the c major scale, or finding the major 3rd of the E note in the c major scale?

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    Intervals are an absolute measure of distance between notes. The distance between E and G is a minor third always, regardless of any scale. Sometimes people talk about "going up a diatonic third from E in C major", or "going up a third from E in C major", or some such thing. This note would be a G, but the note a major third above E is G#, always. – David Bowling Jan 2 '18 at 4:17
  • It might worth noting that an interval name by itself is only ever an approximate measure of the frequency ratio, because of issues relating to temperament. – topo morto Jan 2 '18 at 11:38
  • @David Bowling One could also see intervals from the perspective of a scale, in which case they become relative. en.wikipedia.org/wiki/Interval_(music)#Classification – dfhwze May 8 at 10:49
  • @dfhwze -- Yes; I did mention diatonic intervals in my comment, or are you driving at something else? – David Bowling May 8 at 11:49
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    @David Bowling Thanks for the clarification. I sometimes struggle to find unambigious definitions for certain terms (like 'interval' which could have several meanings). – dfhwze May 8 at 17:16
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As David states, intervals are the measure of the distance between notes. From a Hz perspective, this won't make sense, but from a stave or note name or scale point of view, it does.

Always start with the lower note and count up. So C > E will be a maj3. It is in key C: it's the 3rd note of that major scale. E > G is a min3, as it's a semitone smaller than the maj3 of E > G#, regardless of which key we're looking at. However, the E > G# makes sense as a maj3 as in key Emaj.

As for finding the major 3 of E in the C scale, it isn't there! Because it will always be G#, and G# isn't diatonic in C major.

It is worth looking up a chart of intervals to clear all this stuff up, as it can get confusing, with the distance between two notes (aka interval) having at least two different names. E.g. C > Eb = m3, but C > D# = aug2. What I mean here is the same sounding interval.

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The names of musical intervals are absolute, they aren't affected by context, by their position in the prevailing key. Count up from the lower note. The number of letter names included gives the number. The relationship to the major scale starting on the lower note gives the quality - major (perfect) if it's in the scale etc. - do I need to go over the major/minor/perfect/diminished/augmented definitions here? They're in every theory book.

Now, how you choose to NAME the notes - whether that note 3 semitones above C is to be labelled Eb or D# - that IS affected by context. If we're in C major it's probably more useful to call it Eb, so C - Eb will be a minor 3rd. If we're in E minor, it's almost certainly a D#, so C - D# will be called an augmented 2nd.

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There are different systems for measuring intervals in different contexts.

In tonal or traditional music, an interval is always diatonic. A fifth is always four notes (7 half steps) upward on the staff. An octave is always seven (12 half steps). A third is always two notes up, although it could be either 3 or 4 half steps, depending on whether it is minor or major. The quality (minor, major, diminished, or augmented) of the interval is always stated in terms of the interval itself, not the scale; for example, the distance from C to Eb is a minor third, even if you are in the key of Ab major.

In atonal music, intervals are measured in half steps. For example, a simultaneity containing a C and a G could be written as <0, 7> since there are seven half steps in a fifth. Chords are understood by their intervallic structure, e.g. a typical "triad" will be <0, 3, 7> and a "diminished seventh chord" would be <0,3,6,9>. (Assuming, of course, that the composer is using the common 12-step chromatic scale.) There are other systems as well, such as Forte number.

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