Advantage of 7-note based theory over 12-note alternative
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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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.
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.
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?"
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:
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:
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:
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:
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.)
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.
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.) www.academia.edu/35382108/Chapter_1_DIATONIC_THEORY https://www.academia.edu/35400186/Chapter_2_WELL-FORMED_SCALES www.academia.edu/10482229/Scratching_the_scale_labyrinth
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.
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.
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.
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..
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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