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Given two notes, is the name of the interval based only on the number of semitones between them (absolute) or can the same exact interval have different names related to the key/scale we're talking about at the time (relative)?
(I'm not going to get into whether D♯ and E♭ are the same note, etc, for the purposes of the question, because I don't know enough - hopefully a good answer can make that clear too!)
Intervals are not based just on semitones. They are a combination of distance in semitones and distance in letter name. So as much as you want to avoid the idea that D♯ and E♭ is the same note, you cannot when talking about intervals.
The letter name distance determines the type of interval (unison, second, third, etc.) and the distance in semitones determines the quality. Any A to any E is a 5th and any A to any D is a 4th so while D♯ and E♭ are enharmonic equivalents and may be the same note in some instruments, they indicate different intervals.
Because of this, how things are spelled defines the name of the interval. If a key/scale/chord uses the same spelling of intervals it will have the same name.
Let's compare two chords: Cm (C E♭ G) and C7♯9 (C E G B♭ D♯). Since C to D is a 2nd (or 9th since it's an octave above) and C to E is a 3rd, C to D♯ and C to E♭ are two distinct intervals: an augmented 2nd (augmented 9th) and a minor 3rd.
If the notes are from C to D# it’s an augmented second. If it’s from C to Eb it’s a minor third. Is that what you are asking? Even though the notes are enharmonic (same pitch but different names) the name for the interval is different. They will sound the same though so those two intervals are also said to be enharmonic intervals.
This may at first sound strange, but the number of semitones is determined by the name of the interval but not the other way around. In non-equal-temperament, there may be no enharmonics, D# may (or may not) be identical to Eb (they will be close though.)
In equal temperament, (or by counting piano keys), each semitone is considered to have the same ratio. Often, in some post-Romantic music theory, notes and intervals are identified by semitone count. However, in most Common Practice Period music (and most earlier and lots of modern music), each key is represented by 7 letters (A,B,C,D,E,F,G) and intervals main name are determined by counting this way. So A to C and C to E are both called a third; as the two intervals have different semitone counts, A-C is termed a minor third and C-E is termed a major third.
As music was (and mostly still is) written, naming intervals by note name (rather than semitone difference) carries more information. For example, in the key of C, the interval D-F is a minor third and D-F# is a major third; this naming indicates that for the purposes of the piece being considered, the composer (and performers) should think of F# as a modified F rather than an independent object. (Of course, this is a convention, in the key of D, F# would be the "normal" note and F the modified note.)
One point is that in general (everything is always "in general" or "usually") sharpened notes tend to be followed by a higher pitched note and flattened notes tend to be followed by lower tones. Note that in the same piece, things like dm-G-C and D-G-C chords (ii-V-I and II-V-I or V/V-V-I in Roman Numerals) may both be used to end phrases. Root movements are the same, the only difference is that the D chord appears in two forms. One could even have a d-diminished chord consisting of the notes D,F, and Ab . All these are basically a type of D chord, a G chord, then a C chord even though the intervals may be different.
Another historical relic is that in non-equal-temperament (for example "just tuning"), the ratio of D to C is 9/8 and of E to D is 10/9; this causes the just third C to E to have the ratio 5/4. Equal size whole tones give the C to E ratio as either 81/64 or 100/81. (Thus one has the choice of which intervals should be out-of-tune.)
Music is named because composers write as though intervals may have different sized, but the compromise of equal-temperament means that one is always off (but by the same amount no matter what key.)
It cannot be absolute or unique, due to the fact that any interval name relies on two factors. Like it or not, the D♯/E♭ conundrum must be part of the equation!
The letter name distance gives the number of the interval. C>D is a second. C>D♭ is a second. C>D♯ is a second. C>E is a third. C>E♭ is a third. C>E♯ is a third.
BUT those seconds (C>some D) all have different names - and that's due to the number of semitones between the notes concerned (as you thought).
C>D is called major 2nd. C>D♭ is smaller, so called minor 2nd. C>D♯ is larger, so is called augmented 2nd.
C>E is M3. C>E♭ is smaller, so m3. C>E♯ is larger, so augmented 3rd. If you want more (or less!) C>E♭♭ is a diminished 3rd.
The only intervals with no major/minr name are P4, P5 and P8. When they are smaller by one semitone, they are called diminished; larger by the same, augmented. Same number (4, 5, 8) different word, to signify different number of semitones between them.
To underline all this - when two notes are heard, it is impossible to state what the interval is between them. It may well be a simple 'major 3rd', or 'minor 7th',and context will obviously help, but just two notes in isolation, heard, no. Written down, the interval is obvious every time!
A tone is a frequency distribution with one dominant frequency and many lesser frequencies. A tone is never a pure single "note" in isolation, it is a composition of many frequencies, the strongest one and dominant one being the "name" of the note. This is like calling a city by the name of the tallest skyscraper. It's not calling it Dubai, it's Burj Kalifa. Even though we know the area is actually more detailed and fine-grained, Dubai being huge, when we call a complex frequency distribution a note we are calling this entire province by the name of the tallest building.
Alright, so that's a tone. What's an interval? Interval is the ratio of two tone frequencies. For a perfect fifth the ratio is 3:2 and this is always true no matter what the octave. If we start at A=432 then a perfect fifth up is f*(3/2) or 648Hz.
When we play 432 and 648 together we get a Perfect Fifth two-tone overlap. This is fundamentally what an "interval" represents.
Pythagoras was confronted with a challenge, how to represent this amazing doubling effect in frequency for every octave. If you have 100Hz and double it to 200Hz you get an "octave higher" sound -- the note is the same but it sounds higher. Meaning we have a rainbow of sounds in each Octave that are repeating, slightly higher slightly lower, we may even think of them as brighter or more saturated in the more resonant octaves closer to the human range and reign of our ear.
Pythagoras found an amazing approximation. By creating 12 equally sized pie pieces on the wheel of the Octave, we can have the beautiful 3:2 and the important 4:3 (Perfect Fourth) represented by two distinct keys. The 12-tone system is the most convenient representation that accurately (within a few decimal spots) represents a perfect fourth and perfect fifth with regularity up and down the octaves. The fundamental basis of all modern music theory is the striving to keep 3:2 and 4:3 convenient to play, which is why almost every instrument has 12 buttons per octave.
Today, we can challenge this digitally by returning to the mathematical basis for music, the whole number ratio between tones. Also known as "intervals."
In the same way that you have enharmonic notes (D# / Eb), you can have enharmonic intervals (dimished 5th / augmented 4th or minor 3rd / augmented 2nd). Does this answer your question?
