: Why does standard notation not preserve intervals (visually) (Disclaimer: I don't know much about music theory but ...) The notes A, B are a whole step apart, and B, C are a semitone apart, yet their distance in standard

I like to think of it as all being relative to C major and really the the fundamental thing is the major scale. So in some ways its like a mapping to the white keys of the piano and you get pretty routine at associating lines and spaces of treble clef to section of piano, and then you have little notation for sharps and flats pretty nicely, and even little "valves" to hold the note sharp and flat per say.

What youd like is a bit closer to the metal, just strait diatonic, 12 notes. This is nice but in music (western, tonal) were after something a abstracted from that, the major scale and really circle of fifths. So when you abstract you usually remove unnecessary things. So instead of dealing with things in terms of notes, now we use 12 entire major keys and their notes. But what's better is you still get all twelve diatonic notes, but it's hard on the notation with sharps and flats everywhere. But it's okay because too much chromaticism and you begin to lose a tonal center.

How tonal centers play a role in music is complicated and matters for separate discussion. Point is most music can be viewed as being in some key, with modulation to different keys for desired effect relating to modes. I like to think of keys as absolute things in relation to other keys, with their own moods and feeling. Thus F major is different then E major even though they share the same pattern. C major is the most familiar key for many reason and so it acts like the foundation. It makes sense to annotate and analyze music relative to C major which is the choice taken by standard music notation.

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If you look at a "piano roll" display in a sequencer or other MIDI program, I believe looks like the system you're describing, ie, the visual distance is always analagous to a particular interval

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Why does standard notation not preserve intervals (visually)

It does, but I think you are probably not accustomed to reading it, or how it was developed.

Let's first make an analogy with something familiar: reading English.

What is the meaning of "right" versus "right?" I can read the words, but only reading the single word isn't going to tell me the meaning. Of course we all know we need to read the context around the word. "His answer is right." "I write with my right hand." (Couldn't resist mixing it up more with a homophone.)

Following the analogy I can ask "why doesn't written English preserve the meaning of words?"

The answer is: "it does." But it doesn't preserve the meaning with single words. Complete meaning is only communicated in context.

Back to music notation...

Visually, you are mistaking this...

...to mean lines to spaces are fixed sizes of whole-steps or half-steps.

That is not how it works. The line and space represent steps on a diatonic series. The diatonic series is an asymmetrical series of whole and half steps represented by letters A-G and the staff lines and spaces do not have fixed letter identities.

You must use clefs to know the letters of the staff lines and spaces and consequently the size of intervals between staff lines and spaces...

...where the middle line to the space above is B to C a half-step or...

...where the middle line to the space above is C to D a whole step.

I suppose the use of clefs is already understood by the OP. The question instead may have been "why use a system of clefs to know what the intervals and letter names of the staff?" It does seem to be a confusing system.

I think the reason for all the clefs is historical in two ways: 1) it evolved from notation for a single melodic vocal line (plainchant), and 2) it is harder to read ledger lines than a 4 or 5 line staff.

It's interesting to add that back during plainchant (Dark ages, Medieval era) there wasn't even a notion of absolute pitch for the tones of the staff. The tonic - or the selected range - was sung at whatever pitch was comfortable for the singer. A lot has changed with notation!

Depending on the range of the instrument a clef is selected that will put most of the notation on a 5 line staff, the clef gives a reference pitch like G or F and that let's you know where the diatonic whole and half steps lay. The system isn't immediately intuitive and requires study. That's the reality. You can look for some alternate notation system - they do exist. But you will need to learn that too, and there may not be much available music depending on the system. Going back to the language analogy you can compare to the idea of Esperanto versus English. Alternate systems might be more logical (debatable?) but not practical.

EDIT

...It's unnecessarily difficult to identify intervals like fifths or thirds.

One little addition about that.

The interval number will actually stay the same when counting lines and space regardless of clef, key signature, or accidentals. Ex. if you are on a line and consider that starting point 1 and then go up space 2, line 3, space 4, line 5. We have ascended to the 5th step and the interval number is a fifth, 5. Sames goes for spaces, but let's just stick to an example with lines. All of these are fifths...

...even with the various key signatures, placement of notes within the key, and the crazy accidentals of the third example they are all fifths.

Why? Because the lines and spaces represent the diatonic letters and we count those steps to first get the interval number: E to G, G to D, B to F. Count them all on the letter series and you will see they are all 5 steps.

After the interval number we determine the interval quality. Those qualities are major, minor, augmented, and diminished. You can also have double diminished, and other complex intervals. At this point you do need to read the key signature and accidentals to know the specific quality.

But, rest assured that a fifth is always a fifth - of some quality - by the staff/letter counting. The three examples above are...

E to Bb a diminished fifth d5
G to D a perfect fifth P5
B? to Fb a double-diminished fifth dd5 (sorry, that's really ugly, enharmonically it's a perfect fourth! I did it on purpose to illustrate the point, it's not very normal)

The same idea will apply with all the other intervals.

Key signatures with many sharps or flats and highly chromatic chords are difficult to read. The resulting enharmonic spellings like Gbb which is enharmonically F? are difficult, no doubt. But that is the extreme end of difficult notation.

If you really want to work on reading skills, work up gradually from simpler music. Mozart minuets or Czerny's Recreations are a good starting place. Simple key signature, mostly diatonic harmony. Schubert's various dances for piano would be a nice step up in complexity.

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A shallow answer as to why standard notation does not preserve intervals visually in terms of numbers of semitones is that modern standard notation has evolved to encompass the ideas that:

the octave is split into 12 identifiable notes, each a semitone apart
every particular piece of music would be based around a diatonic scale, which represents a 7-note subset of those 12 identifiable notes in the octave in a particular pattern which itself has uneven sizes of intervals.

The stave in standard notation without a key signature depicts the C major scale and its modes (all of which are Diatonic scales). Adding a key signature allows it to depict other diatonic scales, and the use of accidentals allows all possibilities in the 12-note (chromatic) scale.

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It's actually easier to find 3rds and 5ths. 3rds are on the next line or space, and 5ths are on the next but one. As in F is on bottom space in treble clef, A (3rd higher) is on the next space. In key E, with 4#, E is on the bottom line, G# (3rd) is on the nexy line up, and B (5th) is on the next line up from that. (Next but one from E). Once the key signature is established - it's at the beginning of each line - there's no problem. I think if there was always a problem for the last few hundred years, something would have been changed.

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It DOES preserve intervals (visually). What it does NOT tell you is whether those intervals are major or minor (or augmented or diminished). The distance of a space to its adjacent line will always be a second of some sort. This is because in part of history (which requires a discussion of Church modes and the history of notation), and partly because the notes on the staff change depending on what clef I give it as well as what key signature I give it. I will always know that a line to the next line, or a space to the next space, will be a third of some sort, but I only know what kind of third (and also what the notes are) if I am given more information, namely clef and key signature.

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