I am trying to understand how much of music theory can be done independently of a specific tuning system.

In equal temperament, all semitones have the same size, but that is not the case in other tuning systems. Nevertheless, as far as I understand, we say that between two adjacent notes in a diatonic scale there are either one or two semitones, even if these semitones are of different size for different note combinations.

This means that semitones can be used as an abstract measure of distance between notes, indepent of a specific tuning system.

I am trying to understand how this extends to chromatic notes. According to Wikipedia, an accidental raises/lowers a note by a semitone. Hence E# is one semitone above E.

But how many semitones are there between E# and F? Since both of them are one semitone above E, I am tempted to say that there are zero semitones between them.

In equal temperament, E# and F are enharmonically equivalent, so saying that they are zero semitones apart is fine. In other tuning systems, however, E# and F may very well correspond to different pitches. In these cases, one would need to think of "zero semitones" as an interval of varying (and hence potentially positive) size.

Is this a valid (and common) approach? If not, is there a better way of measuring the distance between notes indepedently of tuning?

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    "Nevertheless, as far as I understand, we say that between two adjacent notes in a diatonic scale there are either one or two semitones, even if these semitones are of different size for different note combinations." How do you define "diatonic"? Within a pentatonic space, for instance, there are three half steps between some pitches. – Richard Mar 27 '19 at 10:30
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    I'm using the definition from Wikipedia. However, my point is not about the specific number of semitones between two notes but about a semitone being used as an abstract distance measure that may correspond to multiple different frequency ratios for different note combinations in different tuning systems. – Florian Brucker Mar 27 '19 at 11:18
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    Perhaps one might call the difference between E# and F a microtone, but that’s just making up a new word from a related word. I’m not sure it makes sense to think about “the distances between notes regardless of tuning” because the tuning affects the distance! – Todd Wilcox Mar 27 '19 at 11:49
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    To look at this another way: music theory is a broad topic, and the exact "rules" of theory and harmony will differ depending on the underlying temperament. You can't define microtones without having first defined the semitones. – Carl Witthoft Mar 27 '19 at 12:46
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    In anything but 12tet, there's a difference between E# and F, but depending on which temperament, and which key, the difference itself will be different. – Tim Mar 27 '19 at 13:29

The Cent is defined thus:

If one knows the frequencies a and b of two notes, the number of cents measuring the interval from a to b may be calculated by the following formula:
cent formula
Likewise, if one knows a note a and the number n of cents in the interval from a to b, then b may be calculated by:
enter image description here

(from Wikipedia)

In effect, the cent is a 1/100th (on a logarithmic scale) of an equally-tempered semitone - and yet it remains valid, and has the same definition, if talking about pitch differences in the context of other temperaments. It is therefore a suitable measurement for talking about pitch differences without needing to refer to particular frequencies.

It would certainly be perfectly reasonable to talk about the difference between a particularly-intoned E# and a particular F in cents. This is a similar situation to measuring the size of a comma (As mentioned by Dekkadeci on another answer) - as you will see on that page, the comma sizes are indeed measured in cents.

If talking about larger intervals, one could measure in terms of octaves (each of which is 1200 cents), or equal-tempered semitones (100 cents). it wouldn't even be crazy to say that the size of a particularly-tempered semitone was a certain proportion of an equal-tempered semitone, but there's some degree of cognitive dissonance that could be induced by measuring semitones in terms of semitones - so cents is probably more appropriate.

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    Probably best answer. People don't talk about "1.34 semitones". – user45266 Mar 27 '19 at 15:14
  • @user45266 perhaps because, to be fully clear, you'd have to say "1.34 equal-tempered semitones" - and then it is easier to say "134 cents"..? – topo Reinstate Monica Mar 27 '19 at 15:40
  • @user45266 Well... A 1200 key piano is not just difficult to play; it's difficult to think/compose with. And 12 is too little given that even in common practice music people agree that mixing up e# and f is a spelling error. So I'd say we have a "least-count" problem. 1200 too fine 12 too coarse – Rusi Mar 27 '19 at 15:42
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    @FlorianBrucker I don't think "you still need at least one tuning system to talk about cents" is correct. It's true that cents are a way of measuring relative pitch, so you would talk about cents in the context of a comparison. However, you can make many statements involving cents without involving a tuning system at all. Or perhaps I'm misunderstanding what you're saying..? – topo Reinstate Monica Mar 28 '19 at 9:00
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    @FlorianBrucker I do agree that "without a tuning system there is no way to get from an abstract note name like E#4 to a frequency". It's only once you know the tuning system/temperament that a note name maps to a specific frequency. However, cents aren't directly defined in terms of note names at all, so you don't need a specific tuning system to talk about cents. – topo Reinstate Monica Mar 28 '19 at 9:03

...semitones can be used as an abstract measure...

If you give that as the basis for your question, the answer is zero semitones between E# and F.

The other answer give details based on specific tuning systems which you specifically want abstracted in your answer.

Given the way you asked the question, I think you already knew the answer. I'm not sure why you asked.

  • Midi note numbers are equivalent to semitones and have the advantage of directly supporting addition and subtraction. – guidot Mar 27 '19 at 14:52
  • Perhaps I should have mentioned my motiviation: as a beginner in music theory I often struggle with clearly separating different concepts. Therefore I'm taking extra care to understand how, for example, the abstract idea of an interval relates to its practical implementation via a specific tuning and a particular instrument. My (probably naive) assumption was that one could do music theory in a purely abstract fashion -- like the symbolic computations in algebra where you don't really care about the actual numbers but about relationships and operations... – Florian Brucker Mar 28 '19 at 8:18
  • ... Judging from the other answers, it seems that this is not as easily done as I've thought. – Florian Brucker Mar 28 '19 at 8:19

In a system where E♯ and F are slightly different there is certainly not a half step between them, not even close. The difference is very small. On a violin E♯ would be intonated in relation to F♯ while F would be intonated in relation to E. Thus the E♯ can be a bit sharp and the F can be a bit flat. But the violin player also needs to listen to what else is going on and intonate accordingly.

Anyway, the difference between E♯ and F has absolutely nothing to do with a half step.

EDIT: Here is a quote from my own comment below just to clarify which interval it actually is:

The interval between E♯ and F is a diminished second.

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    @FlorianBrucker - I think your statement "one can talk about notes and the intervals between them without any reference to frequencies" is a fallacy. As Lars points out, your theory just won't work in reality. – Doktor Mayhem Mar 27 '19 at 12:12
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    @FlorianBrucker - Regarding "what do you call the interval between E# and F, without referring to a specific tuning system?", I'd call it a comma. – Dekkadeci Mar 27 '19 at 14:57
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    This old answer on comma is useful (I think!) music.stackexchange.com/q/33363/13649 Though it needs supplementing with the fact that the syntonic comma is more central to western common practice than the Pythagorean.... A technical difficulties given how close both are – Rusi Mar 27 '19 at 17:03
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    @FlorianBrucker the interval between E♯ and F is a diminished second. – Lars Peter Schultz Mar 27 '19 at 18:56
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    @FlorianBrucker Talking about the size of intervals in an abstract way without referencing a tuning system isn't necessarily meaningless - it depends what you mean by "size of interval". "Interval" in music could refer to a pitch difference, a frequency difference, or a difference between named notes (such as B and C). Named notes are a very abstract way to talk about music theory, from the perspective of the diatonic scale - and like most abstractions, it is a simplification, and breaks down when you try to use it from a perspective that the abstraction isn't designed for. – topo Reinstate Monica Mar 28 '19 at 10:57

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