But then, in which case can it be musically correct to say "F-A# fourth"? It goes against rules.
The short version of this super long post:
* We should start a new thread, or continue this over PM's/email
This is WAY offtopic, this should even be in a different forum!
* We are now discussing "spelling and grammar" in music
* Musical notation attempts to communicate what the composer wants.
* There are and have been different conventions, acceptable standards, customs... Those change with time and what is common at one point was/will be rubbish at some other. An example of this could be dots BEFORE notes to add one half of the value. We are used to see them AFTER the note, but the other version was used in the 18th century (Couperin, Rameau [you can even find them in his famous treatise]). Such a weird usage of dots indicated that the time should be taken from a note that would be found before the marked one (and not after, as we are used to see these days).
* There are no rules in music
* Notation is meant to make things easy, not hard. There are conventions in musical notation, not rules. And this conventions change with time.
* Conventions in notation are there for a reason. When conventions make no more sense, they are discarded.
I present a small silly example of non tonal music (ok, it is not music, but it still is an example), and also present enharmonic modulation in the 19th century as examples of why there is no need to stick to previous conventions when things can be made easier focusing on what is actually there (and not in what was usually there in previous styles and ideologies).
I'll start by saying there are no rules in music. "Rules" have been a strategy used by teachers since antiquity: simple instructions for students to follow. Axioms are some times called rules, and axioms are required to create models of systems.
In music, we have organization. New ways to organize sound have come and gone. Some evolved and became new systems, some were not kept (for better or worse).
Music theory is created AFTER these systems have existed for some time. It is also possible to create a theoretical framework and then compose based on that (I guess this has been less common, and is usually nothing but a fad).
Music theory tries to identify patterns and structures in composers' techniques. Music theory has come with descriptions and models to understand and explain how music works.
There has never been anything stopping composers or musicians from doing whatever they want. Social constraints, conventions among peers, prejudice, the common denominator in taste, physical limitations, economic problems... those have an impact in the way musicians do their thing, but there's nothing stopping anybody from doing anything.
The famous treatises do not contain hard rules written in stone, they offer suggestions, examples and do their best to explain how things work. They come with some "rules of thumb" and clearly mention those are suggestions that will usually give good results. And that once the composer knows what he is doing, he is free to do as he pleases (as long as it yields the desired results). The actually want people to start from what is presented and go further ahead. They don't threat people with eternal suffering in a lake of fire. They have no reason to do that, and there's no way to punish the heretic.
Teachers create and "enforce" rules as a didactic strategy (there's evidence of that since antiquity, what we know as grammar originated from the way the latin teachers taught their students how to speak and write "proper" latin). Once people know what they are doing, they don't require rules. It becomes second nature, and we are able to see what is happening and what we want to do (obviously, if we are trying to write something in the style of Palestrina, parallel fifths are rubbish, but if we want something impressionistic, then it can work pretty well).
Now, I have a small example.
Our theoretical framework: assume we are going to write some music, using 12TET. This music is not tonal, and the main way to organize sound in this work is not "vertical". That is, we are not concerned with "chords" or vertical intervals, we want several independent parts to be played together and we are more concerned with the intervals between two adjacent notes in each independent part.
Why are we going to do that? Why not. Just because that's the idea we got, and it seems to be a good idea (just like Rameau thought it made sense to work with "chords" and not just intervals, or just like Schoenberg thought it could work to use series of 12 notes instead of major scales). This idea is not exactly new, either.
In that image, the same thing is written twice, but with different "spellings".
If we are not concerned with the vertical combinations (that is not the intent of this work), why should I write the last two notes of the first bar as Bb and F? Read one part at the time, they are easier to read without the extra natural signs (zero natural signs vs. 3 in each part).
In this example, it is trivial to read both versions, but what if things get more complicated? (complicated rythmic patterns, big jumps in each part and lots of notes everywhere).
My point is: What is more important? Writing things in an easy to read way, or trying to make this fit some model that has nothing to do with the intent of the music? This music is built of "horizontal parts", independent parts that are played together.
What if this music is composed using random numbers? If there happens to be a fifth at the end of that bar, it just happened, nobody wanted or was planning to have that fifth there. It is a byproduct. That's what the random numbers indicated, so that is what we write. How should we notate it? In the most complicated possible way? In the most readable manner?
I am writing in a horizontal way, writing rows of notes, and am not concerned with vertical matters.
You can say there is a fifth. Yes, both A#-F and Bb-F will be a fifth when played in 12TET (there can't be any difference). But I am not concerned with intervals, we can use the term fifth at some rehearsal just because it is a term that pretty much all musicians know (it is good to have names for things). But when I was writing this, I was not concerned with such a thing.
Is it correct to say that the distance between the last notes in both parts is a fifth (ignoring the octaves, of course)? Well, yes. When both notes are played, what we will hear is exactly what we call a fifth in 12TET. A#-F = Bb-F
Is there any reason to say those are not the same fifth in 12TET?
What's the point in saying it is a dimished sixth? What do we accomplish by doing that? Why should I say that's the proper name of this interval? In this context, intervals are not relevant to the structure of the music. The semantic content that was important for music in the common practice period is not relevant to this music.
So, why should I say this or that interval is not a fifth, or a fourth, or an augmented third? Same frequencies, and no semantic requirements to force us to use one name or the other.
If I were talking about maths, why should I say "it's the squared root of four" instead of saying "it's two"? If it is the same thing, I'd have to have a good reason to use the most complicated version. Well, what reason is there for me to do that in this musical example?
If we were concerned with chords and intervals, I'd go for Bb-F, if only because we are used to see a "fifth" spelled that way (it is a very very common thing to see). That would be a good reason. I agree It'd be easier to read, and I'd agree that writing Cbb-E# won't help anybody. That would make this thing harder to be read.
Of course, after doing that (writing Bb-F), I'd like to check the rest of each individual part to see if it is readable (or if I have a valid reason to write things in a specific different way).
Why don't I do both things? Why don't I try to keep Bb-F and get as few natural signs as possible. Well, why should I? If I have a very readable part, why should I do things differently? Isn't the intention of musical notation to communicate? Well, I am communicating what I want (I have no intentions in communicating something that was important for earlier music, but completely irrelevant for this).
In the theoretical framework for this example, just intonation doesn't exist (we are in 12TET). Chords and vertical intervals are not important (just because we don't want them to be), and the actual vertical combinations are pretty much byproducts of the main organization of the music (which is not tonal).
So I ask, what rules say that what I am doing here is wrong? Where can we find a list of those rules? Who created them, and why? Why are those rules relevant to this example?
The rules of tonal music? This is not tonal music.
Should we keep writing in a "traditional" way just to keep things being always the same? That is pretty much the opposite of what A LOT of composers from the late 19th century to our days did. Some were intentionally going against tradition, trying to make things different no matter what. Some were just finding new things because that's where they work took them, some were just lucky...
In any case, music changed, and the conventions changed. Just like language has evolved.
We could have used pianola rolls to write down this music, but this was written in western musical notation because it is a useful way to communicate music. This kind of notation is very common and we don't need to reinvent the wheel. Producing an easy to read score is a very good idea, specially knowing that things will get ugly.
We produced a readable score, that contains all the information we wanted it to have. We want musicians to be able to decipher this and create music.
If we are writing chords, we want them to be readable. If we are writing chromatic scales, we want them to look tidy and easy to follow... In this case, we are not concerned with enharmonics, because we are in ET.
What rules am I breaking?
Musical notation represents sound. It attempts to communicate something. The very same notation has been used in extremely different music (Schoenberg vs Monteverdi, early chants vs fake books), each with it's own semantics and organization.
In the 19th century, we can find enharmonic modulations (think of Chopin's common Db major <---> C# minor). What once was a problem, became some kind of shortcut. (Db is NOT C#, or is it? It is the same note in a keyboard instrument with 12 notes per octave... that applies for many different tuning systems)
There are very few examples, if any, of music written in Db minor, the minor relative of Fb major (a scale that should have a key signature that includes one double flat).
Was Chopin (and all the other composers who did that) wrong? Was he breaking the rules of what is musically correct? Db is not C#, is it? Was he musically incorrect?
Can we find such a thing (Db as C#) in any music from the 16th century? Or from the 1600's?
Why did he used such an incorrect equivalence in his music?
Db minor would not be of much help communicating his intentions, C# minor works fine. Is it musically correct, or not?
In later composers:
If you see some of Schoenberg's dodecaphonic music, you can find C# next to Db. You can find the same chord "spelled" in different ways (some look weird, some look like normal triads). He was not after tonal chords.
If you see Ligeti's etudes (the devil staircase, for example), you can see chords and scales are written to be readable. Tritones are written in the most convenient way (augmented fourth and diminished fifths are irrelevant), chromatic scales are very easy to spot, all intervals are written to be easy to read.
If a composer is using quartal harmony, it helps to be able to see the fourths easily (yes, I accept tradition plays an important role).
If a composer wants independent musical lines, he concentrates on that. Augmented thirds or fourths... Sixths or fifths... If that is not relevant, the important stuff takes priority. This one might be uncommon, but what reasons are there to see this goes against some rules (what rules)?
If the composer wants to add some extra meaning to a symbol, that is just fine. An example of this: explicit instructions saying that when a chromatic scale is written with sharps, it is meant to be cresc. and accel. and when it is written with flats, it is meant to be dim. and rall.
There could be some extra musical meaning: if an idea is written with sharps, then it is associated with X character (What do I mean by character? Think of Berlioz's Fantastic Symphony). If it is written with flats, it is associated with the other character, Y.
If any music is written with this in mind, and find that there is some F-A# interval, can we say it is not a fourth just because it's A# and not Bb? What about the other semantic content?
Can we say that is musically incorrect? Can we say that only the notation used by Mozart is correct? Mozart didn't have this in mind. And somebody composing in this manner will probably not have tonal conventions of the 18th century in mind.
The difference between intervals might not be part of the musical language, but the extra musical content represented by the A# is important.
Does it sound like I am just making up crazy reasons to say F-A# is a fourth? Even crazier things were done in the 20th century.