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I'm a beginner at music, so I apologize if this is a dumb question. I've been trying to figure out why (in a 21/12 equal temperament tuning) music theory is based on 7 distinct notes (A,B,C,D,E,F,G) instead of the 12 semitones.

Here are a few things that bug me and make music theory very confusing for me:

  • It seems very redundant to have both sharps and flats (not to mention double flats and double sharps)
  • All pitch classes seem as fundamentally important, why are 5 of them second class citizens and not assigned a proper letter?
  • Why not name intervals by their actual distance (let's say 4 semitones, for example), instead of having to see what the base note is to figure out if you should call it a doubly augmented second, a major third, a diminished fourth, etc?
  • Is there an advantage to memorizing unpleasant things like the circle of 5ths instead of just doing mod 12 arithmetic?
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    An entirely different approach to music where these questions don't come up, but you intuitively learn answers to them: start playing music on the white keys of the piano. Gradually start incorporating the black keys into your playing. You learn to create music, and you don't need to ask these questions at any point. But good luck trying to make sensible music with your modulo 12 arithmetic. ;) With that "better" system you'd probably spend a lot of time and eventually "invent" the white keys of the piano. – piiperi Reinstate Monica Jun 27 at 6:36
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    Spoiler alert : the "unpleasant circle of 5ths" is really just mod 12 arithmetic, and you don't need to memorize anything in order to use it. ;) That being said, your question is insightful and it really resonates with me. – Eric Duminil Jun 28 at 19:47
  • @piiperiReinstateMonica Isn't set theory essentially a description of music in terms of modulo 12 arithmetic? I don't think Alexbib's idea is as novel as they think it is, nor as unconventional as you think it is. – Eliza Wilson Jun 29 at 18:33
  • @ElizaWilson the idea is not novel or unconventional at all, it's more like, if you know how music works and you try to model it as maths and logic for, say, a computer program, you'll inevitably bump into the modulo 12 stuff. But for learning or making actual music it's all backwards, the horse and cart the wrong way around. – piiperi Reinstate Monica Jun 29 at 21:52
  • @EricDuminil Not really. If you think about modulo 12 arithmetic as a series of notes on the keyboard, then you're memorizing new "words" for each number. The pattern for the "words" seems a bit arbitrary as to where the sharps and flats lie (i.e. why F to Bb rather than E to Ab?) (though I know it's not arbitrary, the circle of fifths doesn't necessarily tell you why?). Also, if you treat sharps and flats differently, then it's not really mod 12, though it could be seen as a simpler rule where F to Bb always adds a flat and B to F# always adds a sharp. – awe lotta Jun 30 at 1:08

11 Answers 11

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Not all music theory is based around 7-note scales, but the 7-note diatonic scale basically 'caught on' and became popular due to a number of subjectively useful properties it has. Most of its modes facilitate many opportunities for consonant harmony, chord building around triads, have notes that are close enough for easy melodic construction, and so on, while also giving some opportunity for interesting tensions and discords, and - also importantly - being fairly simple (7 notes is quite easy to get your head around!)

So yep, a lot of 'standard'/'western' music theory is based around that scale.

It seems very redundant to have both sharps and flats

It enables you to give each note in any diatonic scale a distinct letter name, and a distinct line on the staff.

All pitch classes seem as fundamentally important, why are 5 of them second class citizens and not assigned a proper letter?

Well, if you begin by assuming use of the diatonic scale, you can see why 7 of the 12 notes are more important - because they're in that scale.

At this point you might be thinking "but there's more to life than the diatonic scale!", and sure, there is. But here's a thing: Much of the reason that we have the chromatic (12 note) scale - and in particular 12-TET - is that it is a clever pattern in which 12 different diatonic scales fit together. More often than not, people use the chromatic scale to make music that can be viewed as still being based around broadly diatonic ideas, but with the added flexibility that 12-TET gives in terms of allowing modulations, chords from 'outside the key' still sounding good, and so on.

Of course there is value in being to look at things from different viewpoints, and for some use cases, people do use terminology that gets away from the diatonic scale: we have pitch class sets, the chromatic staff, and so on. You could certainly imagine a parallel universe in which these ideas had gained a bit more currency. It may even happen in the future if music theoreticians (or product engineers!) build a 12-tone viewpoint of music that seems to offer particularly useful and important insights that a diatonic perspective doesn't.

Of course notation and analysis suited to the 12-TET chromatic scale would still be scale-specific - it wouldn't qualify as some kind of 'pure' model for music. After all, one could reasonably ask: "there are infinite possible pitches - what's so special about these 12?"

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    Yeah, I get that in the diatonic scale 7 notes are more important, but it's not as if we were always playing in the same key... Whereas with 12-TET you can play a very large proportion of the music that exists without extra-convoluted notation. Anyway, thanks for pointing me to the chromatic staff, its notation makes a lot more sense to my brain! – Alexbib Jun 28 at 8:38
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    @Alexbib You're not alone - from musicnotation.org, “The need for a new notation, or a radical improvement of the old, is greater than it seems, and the number of ingenious minds that have tackled the problem is greater than one might think.” — Arnold Schoenberg. Personally, as I moved away from being a 'reading' violinist to an 'earing' guitarist, I found myself thinking of intervals more in terms of numbers of semitones than the traditional interval names. – topo Reinstate Monica Jun 28 at 8:47
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    To answer the question this answer asks, there is a definite reason for 12 notes. It boils down to there being a limited number of options where you can create simple ratio based values and have nearly equidistant spacing between notes. The next stop is 53, followed by 306. No one wants that many notes per octave. youtube.com/watch?v=IT9CPoe5LnM is a great explanation of this. – Azendale Jun 28 at 21:38
  • @Azendale in that video he starts by considering what options you have to 'fill out' the octave without equal temperament, then he goes on to pick 12, and explain what it means to temper it. But if you begin the process with the possibility of equal tempering, can't you pick any root of 2 as your step size? To look at it another way, if you want the notes from 12-TET and some more, don't 24-TET and 36-TET achieve this? – topo Reinstate Monica Jun 29 at 6:50
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    @Azendale 12, 53, 306, etc. are how you get "near equal temperament" with stacking the perfect fifth (aka, the ratio 2:3). But this is only a subset of the possible equal temperaments that approximate simple ratios closely. You could also manually test each equal division of the octave, or stack different intervals, such as 4:5 or 3:5, which gives you instead (near) 31-EDO and 19-EDO respectively, which also contain many other useable intervals; (quarter-comma)-meantone tuning has practically the same fifths as 31-EDO. – awe lotta Jun 30 at 1:19
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Until 12-TET was invented a 7 note system (A-G) made more sense. The tuning, with ratios 1, 9/8, 5/4, 4/3, 3/2, 5/3, 15/8 and 2 from the first note of the scale, gave flexibilty, scope for melodic invention, scope for harmonies that sounded good, and it worked well with instruments like the trumpet where some of these ratios are part of the physics of how they work. The system could be extended into sharps and into flats.

There were known issues with the system, because the sharps and flats did not work together. In particular A-flat and G-sharp were so different they could not be used to replace each other, so a keyboard could not play an A major scale and an E-flat major scale without being re-tuned.

The 12-TET tuning system approximates the traditional system well enough most of the time, and it gives far more flexibility for composers. A number of composers have tried to break out of the traditional scale system, but their attempts have not gained general popularity.

Perhaps it is a self-sustaining system where children hear scale-based music and learn to like it, so that is what their children hear, too.

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    "made more sense". Indeed still makes more sense. "A-flat and G-sharp were so different they could not be used to replace each other" True, but they mean different things, and it is only a mathematical coincidence that they are so close that anyone might think that either pitch might serve as a replacement for the other pitch. – Rosie F Jun 27 at 6:44
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    @RosieF that same coincidence is why 12-TET works at all. – Peter Jun 27 at 6:50
  • @RosieF "only a mathematical coincidence" - maybe, but it's a mathematical coincidence that (to many people's ears) does actually work, which means they genuinely can be thought of as the same note, and the same pitch. (And I do mean can be, rather than have to be). – topo Reinstate Monica Jun 27 at 8:59
  • @RosieF I would say there is no reason that 12-TET shouldn't be where we start. Then add a notation for micro pitch shifting/hinting for when we want to be as accurate as differentiating A flat and G sharp. Systems that simply express the basics and can still gracefully upgrade to handle the complicated are a beautiful thing. Pitch shifting/hinting could then be also used for microtonal music, giving a system that can do more, while still being simple for the beginner. – Azendale Jun 28 at 22:53
  • @Azendale But this is exactly what we do have. Guido's system provided notation for hexachords on G, C and F; this (with B and B flat) gracefully upgraded to handle some 7-note scales; further flats and sharps upgraded it to handle further 7-note scales and chromatic alterations. – Rosie F Jun 29 at 5:56
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There are some good answers here, but I would like to answer two of your points in a way that nobody here has used.

It seems very redundant to have both sharps and flats (not to mention double flats and double sharps)

At the first glance it is so. However by discarding the other accidentals you rob yourself of a lot of features that make reading scores considerably easier.

Here's a C major scale:

enter image description here

Notice how the dots are nicely lined up. Each dot is one line or space above the previous one, and each line and space is occupied by exactly one dot (within the scale). That makes the scales very easy to spot. I also feel that it's very natural to represent scales like this.

Now consider F major. Traditionally, you write it as shown on the left. Since each degree of the scale has its own line/space, you can introduce key signatures that can target each note separately. So you can write the scale in the way shown on the right too:

enter image description here

If you forbid the usage of flats, suddenly it's very hard to do this. You could write an A# instead of the Bb, but that will put two dots on the same space and the next line will be empty, so the nice properties are lost. The only way to keep the nice properties and write the scale without using flats is this:

enter image description here

I certainly prefer the traditional way.

In fact, there are lots of similar features in the traditional notation. There are quite some patterns that make reading simpler: for instance, if you're in A minor, the dominant chord is E major, written as E-G#-B. Now you use the same pattern in the other keys too, so in B flat minor the dominant chord is F-A♮-C (you had too many flats, so instead of a sharp you use a natural), and in G# minor you would use D# major, written as D#-F𝄪-A#. Each time you used a different accidental for the middle note, but it is always "one semitone sharper than the rest of the key". (By the way, the reason for using the double accidentals is just upholding these patterns even in keys with lots of sharps or flats.) If you forbid using some accidentals, this breaks for some keys. (Also the chords would need to change their "shape" on the staff in some of the keys which would make them harder to read.)

Here's an image to make it more clear, hopefully:

enter image description here

In the first bar, there is a very simple chord progression in A minor. In the second bar, I have written the same progression, but transposed to D♯ minor. You see that if I use a double sharp, it looks just like the original. However if I forbid using double sharps, I need to write what is in the third bar. You can certainly see that the chord highlighted in red now looks different (it's no longer a nice stack of three notes), even though it's the same chord, so in this way we have only made it more confusing. To get rid of this confusion, we use double sharps. (Similarly for double flats in other situations.)

Is there an advantage to memorizing unpleasant things like the circle of 5ths instead of just doing mod 12 arithmetic?

Yes. There is a decisive advantage. Suppose you have two different major keys. Now let's define the distance d(A,B) of those two keys as the number of notes in which they differ (not taking the enharmonic equivalents into the account, so for the purpose of this definition, A# = B♭ etc.)

For instance, C major has the notes C, D, E, F, G, A and B, and D major has notes D, E, F#, G, A, B, C#. They share 5 notes and differ in two, so d(C major, D major) = 2. However, the C# major scale has the notes C#, D#, E#, F#, G#, A#, B#, so it share two notes with C major (E#/F and B#/C) and d(C major, C# major) = 5.

I think that this notion of distance is quite natural. (This is very useful. For instance if you are in a certain key, you want to harmonize melodies mostly using the "nearby" chords in this sense.)

And now the important thing: on the circle of fifths, the adjacent keys have always d = 1. So d(A, B) = the number of steps you need to take on the circle of fifths to get from A to B (taking the shorter way). I think that this makes the circle immediately useful and well worth remembering. (And by the way, the circle measures the distance for the minor keys in just the same way.)

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  • From a genuinely 12-tone perspective, perhaps the idea of a 'key' being based around a 7-note scale would not exist, and therefore nor would the concept of 'distance' in the sense you've explained it..? – topo Reinstate Monica Jun 28 at 8:28
  • @topoReinstateMonica: That's true – Ramillies Jun 28 at 12:13
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    The lines in a staff also make it easier to recognize common chords (e.g. CEG or ACE). – Eric Duminil Jun 28 at 19:43
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    @EricDuminil that is true, but it it seems to be a "fuzzy" recognition that can gloss over the major vs minor. Where there is more space in a 12 note system which makes it slightly harder to recognize chords, but a major or minor chords have distinct shapes and are always consistently shaped, since notes are not being "modified" by a key signature. – Azendale Jun 28 at 21:45
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    @Azendale: that's of course true. I find the advantage of the traditional system in the fact that tracking diatonic notes is not too hard and the system makes it painfully obvious when some note is out of key (accidentals appear — at least when a correct key signature is used). A 12-note system would also use >50% more vertical space, which is not negligible (considering that most instruments can play both well below and well above their staff). – Ramillies Jun 30 at 11:35
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It's a big question, hopefully with not-so-big answers.

For starters, the grand stave developed as the simplest way to portray where the notes can be put so people can translate them into playable music. Seven letters work well, diatonically, as by the time we get to eight, the cycle repeats. And each letter has its own place, on a line or space. Non confusing, in reality.

Sharps/flats? As we move away from the C D E F G A B found in key C, certain letter names are o.k., except that they don't represent a white key on piano any more. For instance, in key E, the note G doesn't work as well as G♯, which while it's in a different place on piano, it has the same place on the stave. If we ten called it G♯ as the tonic in G♯ major, it affects all the notes and complicates things unnecessarily. Callin it A♭ makes things far simpler. That paragraph may take a bit of unravelling.

Intervals? Again, because any note could have at least two names, the naming of intervals has to be a little involved, and it's not possible to name an interval heard accurately. It has an academic factor which involes knowing what the notes actually are. Yes, with your idea of maybe only sharps or flats, that could be simplified, but further down the line it makes things more complex! Life is full of compromises!

Cirsle of fifths? Unpleasant? Don't get that. It's contrived, maybe, but it's a useful tool in music, and even if you're not aware of it, you use it anyway. Actually knwing it can make theory and playing easier. Look at any letter name. Call that chord I. its neighbours are IV and V - the mainstay of most Western diatonic music, for starters.

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I think the OP is right, 7-note based music theory is unnecessarily complicated and convoluted compared to the 12-note alternative. It is like this due to historical legacy. This is similar with how natural languages have grammar that is often irregular and full of exceptions to rules for historical reasons. Yet, once you have learned the language, it will start to feel natural despite its irregular structure. Most native speakers of a language would be opposed to reforming their language just to make it easier for foreigners to learn. In the same way, most people who have learned 7-note based music theory and have been using it for a long time are strongly opposed to switching to a different system because the 7-note based system has started to feel natural to them despite its flaws. There are artificial languages like Esperanto that have more logical grammar than natural languages but they have not really caught on. The same goes for alternatives to 7-note based music theory: they may be theoretically better but have not caught on much. So the main advantage of the 7-note based system is simply that it is already widely used.

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    @personal_cloud I'm not sure, but I think everything can be represented properly in 12-based theory, including triads and the circle-of-fifths, they just become a bit more explicitly about mathematical relations, so maybe less intuitive for some people. – Alexbib Jun 28 at 8:27
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    Life experience has shown that many of the standards and systems we "suffer" under were not picked because they are the most logical or best, but because they were first, and switching from them is too hard until everyone else does. So the US keeps not using the metric system, almost nobody speaks Esperanto, most people use Qwerty instead of Dvorak or Colemak, and the internet still mainly uses IPv4 instead of IPv6. Personally, I have found using sheet music printed in 12 tone notation helps connect sight reading with hearing music, which is a massive positive for me. – Azendale Jun 28 at 22:07
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I would like to give a really elementary perspective.

If you know what music is, but don't have a lot of experience making it except maybe by singing along, then it seems like the simplest thing to do is make an instrument with all the notes uniformly spaced, and the simplest notation would be some kind of graph where each note had its own row.

But not all combinations of notes make equal sense together. This is at least partially cultural, but some of it has to do with the physics of how the sound waves interact. For instance, if one note is a vibration that's twice as fast as another, then people in lots of cultures think of them as, in some sense, "the same note". We say they are an octave apart and give them the same letter. If one note vibrates 1.5 times as fast as another, people often think they sound good together, and we call that a "perfect fifth".

Because of this, if you write a melody that sounds good to Western ears, there will usually be one note that is sort of the "main note", and most of the other notes will come from a 7-note scale starting with that main note (which is called the "tonic"). In other words, the major scale is a set of notes that sound a certain way together, and that set is so important that it's built into the notation rather than treating all the notes in an evenhanded way.

So the instruments and notation all evolved in such a way that the notes that are "most natural" for the piece you are playing don't require any special notation, but you can use other notes by putting a sharp or flat right in front of the note on the page (that is, an accidental). This ends up being a convenience for a musician, once you develop some experience.

There is one complication. If you take an octave and divide it into twelve equally-spaced steps, none of the notes is (for instance) exactly 1.5 times the frequency of the tonic. The closest one is about 1.498 (according to Wikipedia), which is pretty good. This kind of tuning is called 12-tone equal temperament, or 12-TET, which others have mentioned. Centuries ago, instruments would be tuned so that a 5th was a true perfect 5th, but then you would have to re-tune instruments to play in a different key.

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    The reference to Bach is a common misconception. Bach wrote *The Well-Tempered Clavier" for a well-tempered instrument, not an equal-tempered one. en.wikipedia.org/wiki/Well_temperament The point of the work is not that all the keys sound the same, the point is that all the keys sound different, and each prelude and fugue is especially suited for the intonation of that key. – brendan Jun 28 at 20:52
  • Thanks, @brendan, I think I probably never understood that correctly. I'll remove that part. – Mark Foskey Jun 28 at 23:37
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This question is very interesting. It touches the fundaments not only of all music symbolic representations, but also the theoretical system, the tone resources, note repertoire, intervals, triads and chords and the of notation, reading and playing.

I can imagine a 12 tet notation system that is more comfortable than the traditional grand staff - I‘ ve even developed such a system by myself 40 years ago. It was something like a horizontal piano roll that we know today from Youtube: there were 5 lines (2 and 3 with a standard space between the lines and double space between the 2 groups) representing the black keys , the notes for the white keys are notated in the space between the lines. So the sharps resp. flats were notated on the lines, d between the 2 lines, g and a between the 3 lines, the semitone steps (ef and bc) in the double space between the 2 and 3. This system fitted well for notating (and reading!) 12 tone music.

About 30 years ago I had my 1st atari ST 1024 computer and was working with the notator program. There was a grid editor where the note lengths and the pitch was represented in a grid system, maybe this was something you have in your mind.

A mathematician invented a program called Presto, you could draw with the mouse lines and circles which the program computed in tones. (It’s the software of which Karajan said, he could have played the whole night with it - me too!)

Yes, you are not alone. But don’t forget the notation system and the entire music theory of western music is the result of a development of thousands of years, and it has not only been influenced by the greek tetrachords and scales, natural tones and over tone series but also by the instruments and the way we play them. We still could know the tablature for organs and lutes, we still use guitar tab, and ... imagine the setting of the buttons of an accordeon! (I don’t know how this works.) maybe this would be an approach to another system?

Anyway, the theoretical system of western music and it’s notation, the function of the tones and chords, the harmonic analysis all this alone is an artwork by itself, apart the great compositions written based in this system, that could never be interpreted and understood without this bases of relations of keys, chords, functions, circle of fifths.

Maybe everything had been said in this language when Schoenberg started writing his TET music.

But Bartok, Hindemith, Gershwin, Shostakovich, Bernstein, Rutter (many others) and Jazz make me assume something different.

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  • I did something similar, but I used twelve lines per octave placed to represent the edges of keys, so there were five pairs of closely spaced lines, separated by a wide gap, and two lines that had medium gaps on either side. IMHO, it might have been nice for actual piano rolls to be marked in such a fashion since it work work nicely with the uniform hole spacing they used, but still make it visually clear what notes were represented. – supercat Jun 29 at 19:04
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Just for context, I'm a maths nerd and I have to agree it does all seem completely arbitrary: to me music seems like set theory. I have spent considerable amounts of time talking to musicians and not understanding why they construct their notation/music in the way they do: particularly as they often disagree with each other.

So my final understanding is that it's ultimately about making 'good sounds.' I reckon 'good' has two aspects: one is arbitrary and cultural i.e. 'this is how we have always made good sounds, they have these meanings and they work around these scales (i.e. tone sets, often with between 5 and 8 elements) on these chord movements (these sub-sets of the main set played simultaneously, in this order.) '

The other aspect of ‘good’ is probably related to physics. A perfect 5th is so near to the root note, (to my ear) that it sometimes sounds like an overtone of the string I'm playing on the guitar (especially with distortion); so practically speaking it isn't even part of a chord, merely a fatter tone, with no musical colour. What I mean is that some intervals are simpler, and occur more commonly in nature (in terms of frequency ratios) and so they are favoured more often. But the order in which these intervals are considered 'good' isn’t purely due to the simplicity of the frequency ratio and is also partially determined my cultural meaning. For example, the Gypsy-Spanish music I love seems to prefer a semitone and a minor third – rather than the ‘harmonically simpler’ major third and tone.

How you stack these intervals into an octave, and the tone/semitone runs you use to fill out those 'harmonies' into a scale seem completely arbitrary (but you are constrained it you want to have a rich set theory – ‘classical music’ is one of those, I think.) You could also split the octave into more intervals than 12 (24 springs easily to mind) and you would also have a perfect 5th, 4ths major thirds etc. or maybe, divide two octaves into one complex scale if you wished (or 7 – but at some point the constraints of human memory play a role.)

So to me these are cultural set-theory games, but they do often seem to play either with the tension between what is considered 'consonant' and 'dissonant', with the latter often resolving to the former, or they enjoy repetition, perhaps in some dance/meditative sense (okay, I’m ignoring dynamics for now.) I think that whatever musical culture precedes you will make more sense to you and those tone-sets/scales will also have a particular meaning (e.g. the western ‘minor is sad’.) Again, Gypsy-Spanish music, twists many classical music theory constraints, but sounds fantastic to my ear.

As for notation – well just take a look at writing for arbitrary notation – anything that works well, would be my guess, just so long as we could read it easily. Actually, now that I think about it, that is a massive constraint; what we can process in real-time. Most humans couldn’t hear, remember, read or play even a fraction of the possibilities of music. So maybe that cuts the tone-set down to five (pentatonic) plus a few extra notes (perhaps one or two quarter tones for extra colour.) This means that attempting to create a notation for 12 (never-mind 24) tone music might not work. So maybe seven feels about right..

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The point of the 7-tone scale is that it reflects compositional practice over the past 1000 years or so. Early theory (and for that matter, early music like Gregorian Chant) used only 7 notes (actually 8 as B could change to B♭ under some circumstances.) In Western theory, the 12-note chromatic scale came later than the 12 note diatonic stuff. That's the historical answer.

There is a (hand-waving) mathematical argument explaining the interest in a 7-note scale. If one takes 7 perfect fifths (ratio 3/2) then they line up nicely as F to E (one can take 12 perfect fifths and line up F to F if desired too.) One gets a scale with 6 perfect fifths and 1 diminished fifth. By positioning the diminished fifth in different places, one gets 7 different patterns; the chromatic scale (12 note) gives only 1 pattern.

The cycle of fifths exists in any 7 or 12 tone system (in common use.) However, the 7-tone patterns are different from one another as well as occurring on different pitches.

A couple of references I found (while looking for something else.) https://www.academia.edu/35382108/Chapter_1_DIATONIC_THEORY https://www.academia.edu/35400186/Chapter_2_WELL-FORMED_SCALES https://www.academia.edu/10482229/Scratching_the_scale_labyrinth

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    The commonly made Pythagorean 7-perfect-fifths argument doesn't really make that much sense because a) this doesn't really explain why you'd stop at 7 b) tonal music since the Renaissance has been 5-limit, not 3-limit. – leftaroundabout Jun 27 at 0:21
  • "mathematical argument explaining the interest in a 7-note scale" True, but the importance of the 7-note scale still exists even if nobody argues the mathematics. Intervals between pitches whose frequencies are in the ratios of small integers sound nice when played together or one after the other. – Rosie F Jun 27 at 6:47
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    As far as I can tell, there's no mathematical explanation why the major scale should have 7 notes. It's only because of historical and cultural reasons. – Eric Duminil Jun 28 at 19:38
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Some excellent questions here.

It seems very redundant to have both sharps and flats (not to mention double flats and double sharps)

Sharps and flats are a holdover from pythagorean tuning. In this method, the "circle of fifths" is actually more of a spiral of fifths - stacking fifths yields a sequence of sharps, while traversing the spiral in the other direction (stacking fourths) yields a sequence of flats. Pythagorean theory is interesting because it actually can yield an infinite set of notes (or at least a very large finite set). It's true that in equal temperament, this spiral is "flattened" so that we go from an infinite set to a set with just twelve members.

Is there an advantage to memorizing unpleasant things like the circle of 5ths instead of just doing mod 12 arithmetic?

All pitch classes seem as fundamentally important, why are 5 of them second class citizens and not assigned a proper letter?

(I'm going to use the words "scale" and "set" here interchangeably.)

The natural major scale and its associated modes is very important in Western theory. There's an easy programmatic way to build a natural major set from the twelve-tone set, assuming that the twelve-tone set is cyclically ordered in a specific way (the circle of fifths). Take any note, and stack fifths until you have seven notes. That is a natural major set (ordered in the Lydian mode). (Note that even though we use equal temperament, this method is still rooted in Pythagorean philosophy.)

The natural major scale represents an adjacent ordering of seven pitch classes on the circle of fifths. The pentatonic scale represents an adjacent ordering of five pitch classes on the circle of fifths (in the natural major case, it's the set of five "unused" notes). The same method of stacking fifths, as such, also works to construct a pentatonic scale.

I can't really speak to the lettering, as they do seem somewhat arbitrary. (Essentially, why are the white keys white and the black keys black? Even with only seven note names, I'm not sure why the "sharps and flats" seemed to get the shaft.) My guess is that someone started with what we know as F and built a natural major set out of that, and it because the "default".

Why not name intervals by their actual distance (let's say 4 semitones, for example), instead of having to see what the base note is to figure out if you should call it a doubly augmented second, a major third, a diminished fourth, etc?

I believe this again goes back to Pythagorean tuning, which depends on a base note to determine where the other notes lie. Unfortunately, equal temperament leaves this making a lot less sense as it's largely unnecessary outside of strict classic musical theory analysis standpoints. (Due to enharmonic equivalency influencing music theory, I think we're now seeing branches of music theory that break away from classical theory, which is interesting.)

That said, I believe there are systems that do what you are talking about - they eschew ciassic naming conventions, notation, and interval categorization in favor of a system that more accurately reflects the state of equal temperament tuning. However, I don't believe that these systems have been well accepted in musical parlance - by composers, theorists, and performers alike - and that's why we don't see them. Essentially, despite some of its drawbacks, the western music systems we have are perpetuated in the name of tradition, and they will continue to be.

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Your question had two parts. One about 7-note scales, the other about sharps and flats. Clearly the 7-note scale is driven by actual use of that scale, as other answers have noted. But the reason for sharps and flats is structural, having to do only with abstract mathematical properties of translations on subgrids.

You need subgrid structure. Sure, mod 12 arithmetic is fine. But 12 is a lot of points to think about or see. Think of trying to read a ruler that only marks full inches and 1/12 inches, with all the tick marks between the inches looking the same. Hard to read, right?

So you want some kind of subgrid. Regular subgrids (that include the octave) are based on 2, 3, 4, or 6 pitch classes. Maybe the best is 6. Let's call them 0 1 2 3 4 5 (= C D E F# G# A#). Suppose we try to get rid of the "flat" concept, which you complained was redundant. Pentatonic scales would be:

  • 0 1 2 3# 4#
  • 0# 1# 2# 4 5
  • 1 2 3 4# 5#

etc.

Now you see an issue here: do the base numbers go 0,1,2,3,4, or 0,1,2,4,5? So we need a flat concept, so then we'd have:

  • 0 1 2 3# 4#
  • 1b 2b 3b 4 5
  • 1 2 3 4# 5#

etc.

The sharps/flats issue comes up with any scale that goes off a regular grid, not just pentatonic scales. Any kind of harmonious music will go off any regular grid (since the overtone series itself quickly goes off all regular grids). And so, if you want to play in any key, then whatever regular subgrid you choose will need sharps and flats.

(For irregular subgrids like the 7-note scale, sharps and flats are also required in order to keep the base numbering consistent in all keys. To see this, take a scale with both on-grid and off-grid notes. Shift it up by semitones, and note that the on-grid and off-grid notes change base symbol at different points unless you have redundant modifiers).

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