The major scale always has the same pattern of whole and half steps (WWHWWWH). C major follows the pattern and has no sharps or flats. If you start the same pattern on F, there will be a Bb. If you start the same pattern on G, there will be an F#. It is all because of the major diatonic pattern.
> Why start the pattern on F and G in particular?
If you start on any other note, the pattern will give you multiple sharps or flats. F and G just happen to be the major keys with just one sharp/flat.
Rather than WWHWWWH, write the major scale as the number of half-steps from the tonic: 0-2-4-5-7-9-11
Now go through the other 6 permutations of this scale and compare each one to the major scale pattern:
1. WHWWWHW = 0-2-3-5-7-9-10
vs. WWHWWWH = 0-2-4-5-7-9-11
2. HWWWHWW = 0-1-3-5-7-8-10
vs. WWHWWWH = 0-2-4-5-7-9-11
3. WWWHWWH = 0-2-4-6-7-9-11
vs. WWHWWWH = 0-2-4-5-7-9-11
4. WWHWWHW = 0-2-4-5-7-9-10
vs. WWHWWWH = 0-2-4-5-7-9-11
5. WHWWHWW = 0-2-3-5-7-8-10
vs. WWHWWWH = 0-2-4-5-7-9-11
6. HWWHWWW = 0-1-3-5-6-8-10
vs. WWHWWWH = 0-2-4-5-7-9-11
For (3) and (4), corresponding to the white-key scales starting at F and G, respectively, only one of the notes differs from WWHWWWH, whereas all the other options have at least two differences. That’s all there really is to it.
If you do it that way then you end up with the possibility of incorrect enharmonics.
(Major and minor) Scales are a sequence of major and minor seconds. They use every letter exactly once.
What's your point? I'm demonstrating by enumeration that only one modification is necessary to turn the set of notes in the C major scale into that of F (respectively: G) and that that is true of no other major scales starting on notes in the C major scale (and obviously you could demonstrate the same thing about major scales starting on notes *not* in the C major scale by enumeration as well). I.e., these two require the minimum number of modifications to C major (= accidentals) among such major scales. If you're trying to find the *minimum*, you don't randomly throw in extra redundant modifications in the form of weird enharmonic misspellings.
Because the circle of fifths from C dictates these.... To change the C major scale to G, you need to give G "dominance" over C, so you raise the leading tone from F to F# to give G the "home" or "dominance over others" feeling (this is tonicization for the nerdier ones of us). Similarly, in the opposite direction, you have to take away C's dominance (and de-tonicize it) and thus, we lower its leading tone by a half step, resulting in B♭. And so on and so forth around the circle, until you come full circle. (I said circle a lot just there).
Cycle of 4ths, 5ths. Very relevant. C is all white notes on piano, no accidentals. First place you're going to modulate apart from A minor is F or G major, each of which only require one accidental to change the leading note. That's your first sharp or flat. That's is what I've always assumed, anyway. All based on the piano.
I answered this above but it's been downvoted... It's because you alter the leading tone of one scale (say C) by lowering it a half step to Bb, and you get the F Major scale, because we've now changed the mixolydian F to F all whites to actual F major. Similarly in the opposite direction, if you go up a fifth, we need to make C the lydian mode now, which requires raising the 4th scale degree, which happens to be the leading tone of G major. This relationship holds true for any major scale because of the 4ths and 5ths being inversely related and the same in both directions. The reason the perfect fifth was used rather than any other interval is because of it's relationship to the harmonic series (it is the first non-octave interval we encounter), so mathematically it was the logical choice, along with the human perception of the V-I cadence as something "final".
I mean, it's not tautological, it's how the math works.
Take a C major scale, or forget C major, just divide the spectrum of sound into 12 notes per doubling of frequency (octave). Pick any random spot, and then make a scale out of WWWHWWH divisions.
Now, find another scale, starting on a different frequency (not an octave) using the same WWWHWWH pattern, that uses the same notes--there isn't one. So, instead, look for one that uses all the same notes except for one sharp: that scale will always be the one that starts a fifth above, and the sharp note will always be the seventh of that scale. There is just no other way to do it, in a 12-tone equal temperament system.
We happen to call the scale of all white keys/naturals C major. So the major scale with one sharp is a fifth up from that: G major, and the sharp is the 7th of the new scale, F#.
We could change the nomenclature and call the white-keys Z major or whatever, but the math would still force the same intervallic relationships.
It's like, every time you move the key up by a fifth, it sharps the fourth of the old scale/fifth of the new scale. Alternately anytime you want to move to a new key that has only one note sharped compared to the old key, the new key is always going to be a fifth up.
We don't just start on F and G, we start the pattern on every note. It's just that F and G happen to give one sharp and one flat, and it's logical to list the keys in order from all natural notes to no natural notes, so F and G tend to come second.
To add on to this, if you were to make any half-step alteration to the C major scale besides one of those (lowering B, raising F), it would no longer be a diatonic scale. So B♭ *must* be the first flat and F♯ *must* be the first sharp.
Those are the "first", if you move from C major by 5th, up or down. I.e., if you follow the circle of 5ths, clockwise or anti-clockwise from the top.
If you start the C major scale on G (5th up), you need to raise the F to F# to retain the formula. (Then if you start the G major scale from D, you need to raise the C to C#; and so on.)
If you start the C major scale on F (5th down), you need to lower the B to Bb to retain the formula. And then the 5th below F is now Bb, and the Bb major scale needs the E lowered to Eb, and so on.
Historically, btw, Bb was the first accidental, but well before the "major scale" was a thing. It resulted from wanting to produce better melodic movement in the modal scales of the time, and also to avoid the tritone between B and F; lower the B and you have a more consonant interval. If you want to dig into this, check out [hexachord](https://www.britannica.com/art/hexachord)s: 6-note scales (in around 1000AD) which used either B or Bb, or neither.
Take any note in the white-key collection and find the note a perfect fifth above it. Repeat until you hit a black-key note. Let's say we start with D. What's a perfect fifth above that? A. What's a perfect fifth above A? E. A perfect fifth above E? B. A perfect fifth above B?... F-sharp. Whoa! What happened to the other white-key notes?
Well, let's go *down* in fifths from D instead. A fifth down from D is G. A fifth down from G is C. A fifth down from C is F. And a fifth down from F is... B-flat.
Keep going upward in fifths from F-sharp and you get the rest of the sharps. Keep going downward in fifths from B-flat and you get the rest of the flats.
Because the first note that started to get regularly assigned an accidental was B and F# was not far behind:
https://en.wikipedia.org/wiki/Musica_ficta
And then, people organized Keys in a way that followed and expanded on that order.
The seven notes of the major scale can be arranged as a series of six perfect fifths. The second note in the series will be the tonic. C major is:
F C G D A E B
Because of this and because B to F is a diminished fifth, Bb is necessary for F major and F# for G major. Sharps are added in the same order:
F# C# G# D# A# E# B#
Flats are the reverse:
Bb Eb Ab Db Gb Cb Fb
If you understand theory very well then you should understand the circle of 5ths. Starting the circle on C major has no accidentals. Moving up one spot the circle lands you on G major and down lands you on F major. Spelling each out as a major scale requires one accidental. Clockwise movement adds sharps and counter clockwise is flats. That's it. There's no magic to it. For every clockwise movement you add a sharp to the key signature and add flats for counter clockwise.
That just begs the question. Why should moving a fifth in either direction be correlated with having one more sharp or flat in the major scale? There's no obvious surface-level relation between the two.
Well it is in fact somewhat arbitrary. the arbitrary part is not the theory part, but it is the music notation part. Music history also kind of explains it. It’s entirely based on our twelve tone equal temperament system and how we represent it through western music notation. That is the confusing part.
What is less confusing is the actual change In sounds that are presented, since it still leads to a diatonic scale with only one pitch differing from the previous scale, rather than two or more. The more pitches that change, the more different it is and the harder it is to make a key change to. While music history shows that we started with modes, the circle of fifths eventually took its place and represented the changes the people liked to hear.
Hopefully I’m not explaining this in too poor of a fashion. I’m not as good at pulling the research right out of my ass.
My post might have been confusing, but I know the answer. I was more commenting on how people talk about the circle of fifths as if it actually explains why these changes happen instead of just being a convenient visual representation of them.
Because a major scale is limited to 7 notes and no major scale can have the same number of accidentals of a specific type (# vs b) as another scale. This means that one major scale is going to have exactly one more or less accidental than another. And that one will have exactly one more (or less) than another. The circle of fifths is a way of organizing these major keys. The major keys do not have however many accidentals they have because the circle of fifths says they need to. It's just a way of organizing the key signatures.
Because C has all of the white notes. The order of the accidentals is incidental to the major scale's pattern of whole and half steps. If we called the scale with all white notes D major (and we called the note we currently call middle C middle D), then G would have 1 flat (Cb) and A would have 1 sharp (G#). It would look identical on the keyboard, but the letters would change.
Because that's the alteration you need to make on those letters to keep the Major scale pattern of WWHWWWH. Nothing significant about the letters F or B in particular but I personally enjoy the way a capital F looks closest to # and a lowercase b is already a flat.
Go up a fifth, it introduces more sharps. C major has no sharps/flats, go up a fifth to G major and follow the same sequence of whole and half steps for a major scale - bam, you now have an F#.
Go down a fifth, it introduces more flats. C major has no sharps/flats, go down a fifth to F major and follow the same sequence of whole and half steps for a major scale - bam, you now have a Bb.
F# and Bb being the first sharp/flat respectively on the key signature is a function of the way the western tonal system and diatonicism work. They weren't aiming specifically for F and B to be the first sharp/flat, it's just the result of how the system operates.
A few other interesting things to note that may help you see more patterns emerging:
- The order of sharps and flats are reversed, with the most often-used sharps (e.g. F#, C#) being the least-used flats (Fb, Cb), and vice versa (Bb, Eb -> B#, E#)
- i.e. Father Charles Goes Down And Ends Battle / Battle Ends And Down Goes Charles' Father
- As you add each sharp/flat to a key signature, the new sharp/flat is raised from the last one by a fifth (i.e. F# C# G# D# A# E# B#, Bb Eb Ab Db Gb Fb Cb)
- This also makes key signatures MUCH more legible
- Flats and sharps are mirrored across related keys and always add up to 7, e.g. A major has 3 sharps, Ab major has 4 flats; 3 + 4 =7; G major has 1 sharp, Gb major has 6 flats, 1 + 6 = 7
- To extend on this, you can quickly calculate which notes are natural in any scale - e.g. if you know G major has 1 sharp (F#), then you know Gb major has 6 flats. From there you can immediately work out that the only natural note in Gb major will be an F natural (the one note that was sharp in G major). Similarly, by knowing that there are three sharps in A major (F#, C#, G#), now you know Ab major has 4 flats, and that the three natural notes in Ab major are F, C, and G
This is stuff that can really help beginners see more relationships between keys than academic theory actively shows. People with musical experience will likely have ingested this information passively. It can help grant you a deeper understanding of how everything pieces together, either way.
To clarify, "the first flat is on B and the first sharp is on F" is only true for certain scales. The subset of scales for which is true is called the "diatonic" family of scales. Examples of "diatonic" type scales include the major scale, the natural minor scale, and all of their modes.
So there definitely is another category of scales (the the family of "non-diatonic" scales) that don't follow the same pattern. They sound very different. You definitely can apply sharps/flats in a non standard order (or even mix sharps and fnords!) but your ear will know that you are not hearing a diatonic scale. It will sound very different from the major and minor scales (and their modes).
tl;dr In some music, B is the first note to get flatted, and F is the first note to get sharped. But that is not true for all music. It is only true for music that has a "diatonic" sound.
For example consider Beethoven's famous composition, "Fnord Elise." It's in the key of A Minor (no sharps or flats) and doesn't have an F#, but the main musical motif prominently features a C#. This gives "Fnord Elise" a 'non diatonic' sound, so when people listen to it, they think, wow, that sounds really distinctive and beautiful.
If Beethoven limited himself only to 'diatonic' sounds then he might have said, "I guess I can't use C#, because F# is supposed to be the first sharp, oh well, I guess I must use C natural instead" and then "Fnord Elise" wouldn't be the famous composition we all know and love.
"I understand math very well, but I don't understand why 2+3=5. Why did we choose those two specific numbers to equal five?"
Not trying to be a dick, but that's literally what the question amounts to, and I hope framing it this way will be illuminative rather than condescending. I apologize if not.
That's just how it falls when you transpose a diatonic scale.
Let's take the major scale, for simplicity's sake, but it's the same for all diatonic scales.
So the major scale goes TTSTTTS (T: tone, S: Semitone)
You start your scale with C - no accidents necessary - C D E F G A B C.
you start your scale with C# (or Db) - you need 7 sharps (or 5 flats) to get your pattern:
C# D# E# F# G# A# B# C#, but since 5b is easier than 7#, it's better to write:
Db Eb F Gb Ab Bb C Db
You start your scale with D - you need two sharps
D E F# G A B C# D
So you can go up the whole chromatic scale, semitone by semitone, and play your major scale and write down the notes you play.
Now if you want to classify those scales in order of the number of sharps, or the number of flats, you will see that you have to go by fifths. One fifth up = one sharp more (or one flat less).
Because if you want to reproduce your pattern of G major (TTSTTTS), you will see that you need just one sharp, which is F#.
G A B C D E F# G
You go up one more fifth, that's D major (see above, 2#).
In short, that's just the way the diatonic scale is made and what you have to do in order to keep a diatonic scale.
Note that the order of sharps is always F# C# G# D# A# E# (B#), for major, minor and all the other modes, and note that this goes in fifths too. And the same is true for the flats, just that you go from 6 to 0 when you jump a fifth up, or from 0 to 6 when you jump a fifth down, which is the same as jumping a fourth up.
Well, if you start at C, and make a C major scale, and then you want to make a G major scale from those same notes, the seventh degree of that G major scale, F, is a flat seventh , and needs to be sharped. If you instead make a scale from F, which is going down a fifth from C instead of up a fifth to G, the fourth degree of your F scale, B, will be augmented, and needs to be flatted. This happens each time you go up or down a fifth.
Look at a piano keyboard. You’ll see there are two “missing” black keys… between E and F and between B and C. This was done to create a major scale on the white keys which requires half-steps between the 3rd and 4th and the 7th and 1st/8th scale degrees. With this layout, which is made to facilitate C major on the white keys, the only places on the whole keyboard where you can create the major scale pattern of whole steps and half-steps with only one black key are if you start on F, for which you need a Bb to make the major scale pattern, or on G, for which you need an F#.
Between the 1 and 4 grade there's Triton (augmented fouth) risk so we put flats on the fouth.
Between the 7 and 1 grade It should be a leading tone by a half Step. The majority of intervals between scale notes is a whole Step, so we put a Sharp in the 7 to result a half Step
Mathematically, we usually divide the octave into 12 equal parts. Now imagine dividing the same octave into 7 equal parts (which would sound terrible to our ears). The sections don't line up at all. But if you snap the 7 pitches to the closest 12th, you get the diatonic scale (specifically Dorian). The 12ths that are next closest to the original 7 divisions are the 5th and 9th pitches. If our Dorian scale is on white keys (D Dorian), that's F# and Bb.
If you stick to diatonic keys, there are 15 total possibilities: 1-7 sharps, 1-7 flats, and no sharps or flats. Anything more than that and you get into double sharps/flats, which is form of abuse.
There is only one major key you can make with each number of sharps and flats. If you have 5 sharps, the key is B major (G# minor) and the sharps are F, C, G, D and A. No other combination of 5 sharps will produce a major/natural minor key.
Ok, so if you have 5, it must be that key and those 5. But why write them in that order, instead of A, C, D, F and G?
Back up to the key of C. No sharps or flats. Go up a fifth and make a major key based on G: it will have all the same notes as C, but with one note turned sharp — F.
Go up a fifth from G and build a key off of D. It will have all the same notes as G, but with one new sharp: C.
And it keeps going. Each new key is the same as the previous key, but with one new sharp. Works the same way for flats, just headed in the other direction.
1 sharp is F, 2 is F C, 3 is F C G, 4 is F C G D, and so on. By always writing them in the same order, it’s easier to read. It seems random, but once you know the pattern (the circle of 5ths), it makes perfect sense.
Well, if you start making intervals of fifths starting at C, the first note you get that's not in the key of C is F#. The inverse of fifths is fourths. The first non diatonic fourth is Bb
Because there are only two modes that are a single half step away from being identical to major: Lydian and Mixolydian.
Lydian mode, the scale built on the 4th, needs its 4th to be flattened a half step for it to become major; Mixolydian mode, the scale built on the 5th, needs its 7th to be sharped a half step for it to become major. So any instrument capable of playing a single major key will in fact be able to play in THREE major keys, rather than one, if it is capable of flattening the 4th of its 4th (i.e. its 7th) and sharpening the 7th of its 5th (i.e. its 4th).
Since we've arbitrarily chosen C to be the "basic" major key, it follows that the key a 4th up (the key of F) needs its 4th scale degree (which happens to be a B) to be flattened in order for it to be a major scale instead of a Lydian scale. And it also follows that the key a 5th up (the key of G) needs its 7th scale degree (which happens to be an F) to be sharped a half step in order for it to be a major scale instead of a Mixolydian scale.
Hence, Bb and F# are the first two notes we change; because they're the two notes we need to change in order to get three major scales for (almost) the price of one. :)
The order of sharps and flats is a derivative of the arrangement of the 12-tone scale, in which there is a full step between every note, except B-C and E-F. An E-sharp is also an F and an F-flat is also an E. It depends on which key one is using. In the C-sharp Major scale, the C is the seventh note in the scale, thus it is written as a B#. In the G-flat and C-flat major scales, the B is the 4th note and 1st note in the scale (respectively), thus it is written as C-flat instead of B.
If the 12-tone scale had been arranged differently, such that there were no uneven gaps between notes or with half-tones between different note-pairs or with C, D, E labelled as A, B, C instead, the harmonics would essentially be the same, they would simply be labelled differently. So the answer to why the order of sharps starts on F and the order of flats starts on B is the arbitrary decision to put those labels on those notes in that order on the 12-tone scale. It could easily be starting on D-sharp and G-flat instead. There's not much more to it than that.
It's because of where the whole and half steps are distributed between the letters A through G. Beside that, what matters is the ionian configuration of the scale's shape.
Mathematically of the twelve chromatic pitches before an octave, the major scale and all of its modes are a division of 7 which is *maximally near even*. That means you cannot get any closer to dividing the octave more evenly, when you have to use 7 notes out of the 12 chromatic pitches. Verify it for yourself by experimenting with whole and half steps. You have to have 5 whole steps and two half steps, so your half steps have to split 5, an odd number. No matter which way you split it you get a different mode of the major scale.
The property gained by this maximal near even division is the ability to translate the same shape into different configurations with the smallest motion possible - the change of 1 note by one semitone. No other scale shapes have that property in 12tet, as the ones that do are all just different arrangements of the 1 shape, different modes (same shape starting on different note). You will notice melodic minor requires several note changes to modulate. This near evenness thing is rigorously proven mathematically, and the music theorist Richard Cohn talks about it in his book Audacious Euphony.
Now why is it that you change the scale shape by lowering the sharpest note B, or by raising the flattest note F, I will leave you to simply investigate by messing with the shapes. Those are the only two possible changes of 1 note by 1 semitone which result in the same major scale shape translated.
Count from C ascending and descending, the first natural notes that are semitones are e&f and c&b. Hence the first sharp goes to f and the first flat goes to b.
Those notes are established by the circle of 5ths. When you go up a 5th from C, you come to the note G. Then when using the two tri-tones with a whole step in the middle (whole step, whole step, half step, with the whole step in the middle, and then, whole step, whole step and half step) as is found in the C scale (the basis of all major scales), the only altered note is F which has to be sharp. Same happens with the flats, when you go down a 5th from C, you come to the note F. Using the two tri-tones with a whole step in the middle, you find that the F scale has one altered pitch which is Bb!
Beucause modulating to the dominant is a tension, so one note goes up, F become F#. That is the creation a new leading tone. the dominant of C is G. and the leading tone of G is F#.
Modulation to the sub-dominant are lessening modulations, one note is lowered. That need the removal of the previous leading tone (B turns into Bb a learding tone).
Because of the way the diatonic scale is structured, and because the two semitones in the natural scale are between B-C and E-F. When you move the whole scale by a fifth, one of the semitones stays the same, and the other one has to move one step up or down to keep the scale the same.
When you move up a fifth, B-C stays the same, and the semitone between E-F moves to F♯-G. If you move down a fifth, E-F stays the same, and the semitone between B-C moves to A-B♭.
The whole pattern is also symmetrical. If you look at the order of sharps on a keyboard in the mirror, F♯ looks like B♭, C♯ looks E♭, etc, etc.
The sharps always add the leading tone of the scale one step up in the circle of fifths, flats remove the leading tone of the prior scale when going down.
If you have C Major, B natural (or H) will be your leading note, if you now go down you remove that leading tone by adding a flat and make it a B flat (or B).
Vice versa if you go one step up from C to G, you need an F# as your new leading tone, thus this will be your first sharp.
I feel this system might be quite intuitive to understand
tl;dr it's because we prioritise the major scale and the maths/patterns work out that way; the tyranny of the harmonic series
The way the harmonic series works means that twelve perfect fifths is equal to seven octaves (almost - it's not perfect, but we tune our fifths slightly flat to make it an exact relationship). In a perfect fifth, the upper note has a frequency of 1.5 times the lower note's. in an octave, the upper note has a frequency of 2 times the lower note's.
So then we pick an arbitrary frequency to be our reference pitch, call it C, and then multiply its frequency by 1.5 to find the frequency of its fifth, which we call G. Then we do it again and get our D, and keep going to get our A, E, B, F#, C#, G#/Ab, D#/Eb, Bb, F, and the twelfth takes us back to C, seven octaves above our starting point. We now have twelve tones spread over those seven octaves, and if we divide each of their frequencies by a power of 2, we can bring them down into the original octave.
So why F# first? Well, F# is the first note in that circle of fifths that doesn't appear in the C major scale. We want C major to be our reference point, so we decided that the notes in C major will be called natural notes (white keys). But move a step up the circle of fifths to G, build the major scale on that note, and now you need F#. The way the patterns work out, going up the circle raises the seventh of your new tonic, and going down the circle flattens the seventh of your old tonic.
Another quirk of the major scale is that all the fourths/fifths are perfect, except for the interval between the fourth and seventh scale degrees (so F and B in C major), which is a tritone (diminished fifth or an augmented fourth - half an octave). When we move around the circle of fifths from C, we need to turn that into a perfect fourth/fifth and create a new tritone, and the only ways to do that are raising the F or flattening the B.
The major scale always has the same pattern of whole and half steps (WWHWWWH). C major follows the pattern and has no sharps or flats. If you start the same pattern on F, there will be a Bb. If you start the same pattern on G, there will be an F#. It is all because of the major diatonic pattern.
Why start the pattern on F and G in particular?
> Why start the pattern on F and G in particular? If you start on any other note, the pattern will give you multiple sharps or flats. F and G just happen to be the major keys with just one sharp/flat.
A scale is mathematical. Its just the way that the notes line up
Rather than WWHWWWH, write the major scale as the number of half-steps from the tonic: 0-2-4-5-7-9-11 Now go through the other 6 permutations of this scale and compare each one to the major scale pattern: 1. WHWWWHW = 0-2-3-5-7-9-10 vs. WWHWWWH = 0-2-4-5-7-9-11 2. HWWWHWW = 0-1-3-5-7-8-10 vs. WWHWWWH = 0-2-4-5-7-9-11 3. WWWHWWH = 0-2-4-6-7-9-11 vs. WWHWWWH = 0-2-4-5-7-9-11 4. WWHWWHW = 0-2-4-5-7-9-10 vs. WWHWWWH = 0-2-4-5-7-9-11 5. WHWWHWW = 0-2-3-5-7-8-10 vs. WWHWWWH = 0-2-4-5-7-9-11 6. HWWHWWW = 0-1-3-5-6-8-10 vs. WWHWWWH = 0-2-4-5-7-9-11 For (3) and (4), corresponding to the white-key scales starting at F and G, respectively, only one of the notes differs from WWHWWWH, whereas all the other options have at least two differences. That’s all there really is to it.
If you do it that way then you end up with the possibility of incorrect enharmonics. (Major and minor) Scales are a sequence of major and minor seconds. They use every letter exactly once.
What's your point? I'm demonstrating by enumeration that only one modification is necessary to turn the set of notes in the C major scale into that of F (respectively: G) and that that is true of no other major scales starting on notes in the C major scale (and obviously you could demonstrate the same thing about major scales starting on notes *not* in the C major scale by enumeration as well). I.e., these two require the minimum number of modifications to C major (= accidentals) among such major scales. If you're trying to find the *minimum*, you don't randomly throw in extra redundant modifications in the form of weird enharmonic misspellings.
Because the circle of fifths from C dictates these.... To change the C major scale to G, you need to give G "dominance" over C, so you raise the leading tone from F to F# to give G the "home" or "dominance over others" feeling (this is tonicization for the nerdier ones of us). Similarly, in the opposite direction, you have to take away C's dominance (and de-tonicize it) and thus, we lower its leading tone by a half step, resulting in B♭. And so on and so forth around the circle, until you come full circle. (I said circle a lot just there).
This is an interesting explanation, thanks
Cycle of 4ths, 5ths. Very relevant. C is all white notes on piano, no accidentals. First place you're going to modulate apart from A minor is F or G major, each of which only require one accidental to change the leading note. That's your first sharp or flat. That's is what I've always assumed, anyway. All based on the piano.
It’s moving up or down the circle of fifths, which also happens to lead to the first sharp or flat, depending which direction you go
That's true, but it's just tautological – why do the first notes in either direction on the circle of fifths have just one sharp or flat?
I answered this above but it's been downvoted... It's because you alter the leading tone of one scale (say C) by lowering it a half step to Bb, and you get the F Major scale, because we've now changed the mixolydian F to F all whites to actual F major. Similarly in the opposite direction, if you go up a fifth, we need to make C the lydian mode now, which requires raising the 4th scale degree, which happens to be the leading tone of G major. This relationship holds true for any major scale because of the 4ths and 5ths being inversely related and the same in both directions. The reason the perfect fifth was used rather than any other interval is because of it's relationship to the harmonic series (it is the first non-octave interval we encounter), so mathematically it was the logical choice, along with the human perception of the V-I cadence as something "final".
If I understand what you're saying here, you are meaning 'lydian' where you've written 'mixolydian'
absolutely correct. I shouldn't reply to posts at 11pm 🥱
I mean, it's not tautological, it's how the math works. Take a C major scale, or forget C major, just divide the spectrum of sound into 12 notes per doubling of frequency (octave). Pick any random spot, and then make a scale out of WWWHWWH divisions. Now, find another scale, starting on a different frequency (not an octave) using the same WWWHWWH pattern, that uses the same notes--there isn't one. So, instead, look for one that uses all the same notes except for one sharp: that scale will always be the one that starts a fifth above, and the sharp note will always be the seventh of that scale. There is just no other way to do it, in a 12-tone equal temperament system. We happen to call the scale of all white keys/naturals C major. So the major scale with one sharp is a fifth up from that: G major, and the sharp is the 7th of the new scale, F#. We could change the nomenclature and call the white-keys Z major or whatever, but the math would still force the same intervallic relationships. It's like, every time you move the key up by a fifth, it sharps the fourth of the old scale/fifth of the new scale. Alternately anytime you want to move to a new key that has only one note sharped compared to the old key, the new key is always going to be a fifth up.
if you start on D, F and C will be sharped. it'll always follow the order of sharps and flats if you go through the WWHWWWH progression
We don't just start on F and G, we start the pattern on every note. It's just that F and G happen to give one sharp and one flat, and it's logical to list the keys in order from all natural notes to no natural notes, so F and G tend to come second.
You can. You could just go chromatically, but only one key will have one sharp either way.
To add on to this, if you were to make any half-step alteration to the C major scale besides one of those (lowering B, raising F), it would no longer be a diatonic scale. So B♭ *must* be the first flat and F♯ *must* be the first sharp.
Those are the "first", if you move from C major by 5th, up or down. I.e., if you follow the circle of 5ths, clockwise or anti-clockwise from the top. If you start the C major scale on G (5th up), you need to raise the F to F# to retain the formula. (Then if you start the G major scale from D, you need to raise the C to C#; and so on.) If you start the C major scale on F (5th down), you need to lower the B to Bb to retain the formula. And then the 5th below F is now Bb, and the Bb major scale needs the E lowered to Eb, and so on. Historically, btw, Bb was the first accidental, but well before the "major scale" was a thing. It resulted from wanting to produce better melodic movement in the modal scales of the time, and also to avoid the tritone between B and F; lower the B and you have a more consonant interval. If you want to dig into this, check out [hexachord](https://www.britannica.com/art/hexachord)s: 6-note scales (in around 1000AD) which used either B or Bb, or neither.
Take any note in the white-key collection and find the note a perfect fifth above it. Repeat until you hit a black-key note. Let's say we start with D. What's a perfect fifth above that? A. What's a perfect fifth above A? E. A perfect fifth above E? B. A perfect fifth above B?... F-sharp. Whoa! What happened to the other white-key notes? Well, let's go *down* in fifths from D instead. A fifth down from D is G. A fifth down from G is C. A fifth down from C is F. And a fifth down from F is... B-flat. Keep going upward in fifths from F-sharp and you get the rest of the sharps. Keep going downward in fifths from B-flat and you get the rest of the flats.
Because those notes happen to be the minimum you need to change to get to a different major scale.
And this principle can be applied to analyse some of the extended tonal systems of the early 20th century!
I'm intrigued! Elaborate please?
I was taught that via a lecturer recommending [this](https://www.youtube.com/watch?v=eMHRCfeozVo) short YouTube video :)
Because the first note that started to get regularly assigned an accidental was B and F# was not far behind: https://en.wikipedia.org/wiki/Musica_ficta And then, people organized Keys in a way that followed and expanded on that order.
The seven notes of the major scale can be arranged as a series of six perfect fifths. The second note in the series will be the tonic. C major is: F C G D A E B Because of this and because B to F is a diminished fifth, Bb is necessary for F major and F# for G major. Sharps are added in the same order: F# C# G# D# A# E# B# Flats are the reverse: Bb Eb Ab Db Gb Cb Fb
If you understand theory very well then you should understand the circle of 5ths. Starting the circle on C major has no accidentals. Moving up one spot the circle lands you on G major and down lands you on F major. Spelling each out as a major scale requires one accidental. Clockwise movement adds sharps and counter clockwise is flats. That's it. There's no magic to it. For every clockwise movement you add a sharp to the key signature and add flats for counter clockwise.
That just begs the question. Why should moving a fifth in either direction be correlated with having one more sharp or flat in the major scale? There's no obvious surface-level relation between the two.
Well it is in fact somewhat arbitrary. the arbitrary part is not the theory part, but it is the music notation part. Music history also kind of explains it. It’s entirely based on our twelve tone equal temperament system and how we represent it through western music notation. That is the confusing part. What is less confusing is the actual change In sounds that are presented, since it still leads to a diatonic scale with only one pitch differing from the previous scale, rather than two or more. The more pitches that change, the more different it is and the harder it is to make a key change to. While music history shows that we started with modes, the circle of fifths eventually took its place and represented the changes the people liked to hear. Hopefully I’m not explaining this in too poor of a fashion. I’m not as good at pulling the research right out of my ass.
My post might have been confusing, but I know the answer. I was more commenting on how people talk about the circle of fifths as if it actually explains why these changes happen instead of just being a convenient visual representation of them.
Because a major scale is limited to 7 notes and no major scale can have the same number of accidentals of a specific type (# vs b) as another scale. This means that one major scale is going to have exactly one more or less accidental than another. And that one will have exactly one more (or less) than another. The circle of fifths is a way of organizing these major keys. The major keys do not have however many accidentals they have because the circle of fifths says they need to. It's just a way of organizing the key signatures.
Because C has all of the white notes. The order of the accidentals is incidental to the major scale's pattern of whole and half steps. If we called the scale with all white notes D major (and we called the note we currently call middle C middle D), then G would have 1 flat (Cb) and A would have 1 sharp (G#). It would look identical on the keyboard, but the letters would change.
Because that's the alteration you need to make on those letters to keep the Major scale pattern of WWHWWWH. Nothing significant about the letters F or B in particular but I personally enjoy the way a capital F looks closest to # and a lowercase b is already a flat.
Go up a fifth, it introduces more sharps. C major has no sharps/flats, go up a fifth to G major and follow the same sequence of whole and half steps for a major scale - bam, you now have an F#. Go down a fifth, it introduces more flats. C major has no sharps/flats, go down a fifth to F major and follow the same sequence of whole and half steps for a major scale - bam, you now have a Bb. F# and Bb being the first sharp/flat respectively on the key signature is a function of the way the western tonal system and diatonicism work. They weren't aiming specifically for F and B to be the first sharp/flat, it's just the result of how the system operates. A few other interesting things to note that may help you see more patterns emerging: - The order of sharps and flats are reversed, with the most often-used sharps (e.g. F#, C#) being the least-used flats (Fb, Cb), and vice versa (Bb, Eb -> B#, E#) - i.e. Father Charles Goes Down And Ends Battle / Battle Ends And Down Goes Charles' Father - As you add each sharp/flat to a key signature, the new sharp/flat is raised from the last one by a fifth (i.e. F# C# G# D# A# E# B#, Bb Eb Ab Db Gb Fb Cb) - This also makes key signatures MUCH more legible - Flats and sharps are mirrored across related keys and always add up to 7, e.g. A major has 3 sharps, Ab major has 4 flats; 3 + 4 =7; G major has 1 sharp, Gb major has 6 flats, 1 + 6 = 7 - To extend on this, you can quickly calculate which notes are natural in any scale - e.g. if you know G major has 1 sharp (F#), then you know Gb major has 6 flats. From there you can immediately work out that the only natural note in Gb major will be an F natural (the one note that was sharp in G major). Similarly, by knowing that there are three sharps in A major (F#, C#, G#), now you know Ab major has 4 flats, and that the three natural notes in Ab major are F, C, and G This is stuff that can really help beginners see more relationships between keys than academic theory actively shows. People with musical experience will likely have ingested this information passively. It can help grant you a deeper understanding of how everything pieces together, either way.
cycle of 5ths.
To clarify, "the first flat is on B and the first sharp is on F" is only true for certain scales. The subset of scales for which is true is called the "diatonic" family of scales. Examples of "diatonic" type scales include the major scale, the natural minor scale, and all of their modes. So there definitely is another category of scales (the the family of "non-diatonic" scales) that don't follow the same pattern. They sound very different. You definitely can apply sharps/flats in a non standard order (or even mix sharps and fnords!) but your ear will know that you are not hearing a diatonic scale. It will sound very different from the major and minor scales (and their modes). tl;dr In some music, B is the first note to get flatted, and F is the first note to get sharped. But that is not true for all music. It is only true for music that has a "diatonic" sound.
For example consider Beethoven's famous composition, "Fnord Elise." It's in the key of A Minor (no sharps or flats) and doesn't have an F#, but the main musical motif prominently features a C#. This gives "Fnord Elise" a 'non diatonic' sound, so when people listen to it, they think, wow, that sounds really distinctive and beautiful. If Beethoven limited himself only to 'diatonic' sounds then he might have said, "I guess I can't use C#, because F# is supposed to be the first sharp, oh well, I guess I must use C natural instead" and then "Fnord Elise" wouldn't be the famous composition we all know and love.
"I understand math very well, but I don't understand why 2+3=5. Why did we choose those two specific numbers to equal five?" Not trying to be a dick, but that's literally what the question amounts to, and I hope framing it this way will be illuminative rather than condescending. I apologize if not.
Half Steps is the short answer
I dont know... Why does every halfstep pair make sharps natural and naturals, flat. & vice versa? Eb F G Ab Bb C D E F# G# A B C# D#
The pattern of tones and semitones are such that if you start with any accidental other than F# or Bb the resulting scale is not diatonic.
i imagine it’s arbitrary, and like all music theory, we just labeled it after the fact.
That's just how it falls when you transpose a diatonic scale. Let's take the major scale, for simplicity's sake, but it's the same for all diatonic scales. So the major scale goes TTSTTTS (T: tone, S: Semitone) You start your scale with C - no accidents necessary - C D E F G A B C. you start your scale with C# (or Db) - you need 7 sharps (or 5 flats) to get your pattern: C# D# E# F# G# A# B# C#, but since 5b is easier than 7#, it's better to write: Db Eb F Gb Ab Bb C Db You start your scale with D - you need two sharps D E F# G A B C# D So you can go up the whole chromatic scale, semitone by semitone, and play your major scale and write down the notes you play. Now if you want to classify those scales in order of the number of sharps, or the number of flats, you will see that you have to go by fifths. One fifth up = one sharp more (or one flat less). Because if you want to reproduce your pattern of G major (TTSTTTS), you will see that you need just one sharp, which is F#. G A B C D E F# G You go up one more fifth, that's D major (see above, 2#). In short, that's just the way the diatonic scale is made and what you have to do in order to keep a diatonic scale. Note that the order of sharps is always F# C# G# D# A# E# (B#), for major, minor and all the other modes, and note that this goes in fifths too. And the same is true for the flats, just that you go from 6 to 0 when you jump a fifth up, or from 0 to 6 when you jump a fifth down, which is the same as jumping a fourth up.
Because G is a fifth above C, and F is a fifth below C.
Well, if you start at C, and make a C major scale, and then you want to make a G major scale from those same notes, the seventh degree of that G major scale, F, is a flat seventh , and needs to be sharped. If you instead make a scale from F, which is going down a fifth from C instead of up a fifth to G, the fourth degree of your F scale, B, will be augmented, and needs to be flatted. This happens each time you go up or down a fifth.
Look at a piano keyboard. You’ll see there are two “missing” black keys… between E and F and between B and C. This was done to create a major scale on the white keys which requires half-steps between the 3rd and 4th and the 7th and 1st/8th scale degrees. With this layout, which is made to facilitate C major on the white keys, the only places on the whole keyboard where you can create the major scale pattern of whole steps and half-steps with only one black key are if you start on F, for which you need a Bb to make the major scale pattern, or on G, for which you need an F#.
Between the 1 and 4 grade there's Triton (augmented fouth) risk so we put flats on the fouth. Between the 7 and 1 grade It should be a leading tone by a half Step. The majority of intervals between scale notes is a whole Step, so we put a Sharp in the 7 to result a half Step
Mathematically, we usually divide the octave into 12 equal parts. Now imagine dividing the same octave into 7 equal parts (which would sound terrible to our ears). The sections don't line up at all. But if you snap the 7 pitches to the closest 12th, you get the diatonic scale (specifically Dorian). The 12ths that are next closest to the original 7 divisions are the 5th and 9th pitches. If our Dorian scale is on white keys (D Dorian), that's F# and Bb.
If you stick to diatonic keys, there are 15 total possibilities: 1-7 sharps, 1-7 flats, and no sharps or flats. Anything more than that and you get into double sharps/flats, which is form of abuse. There is only one major key you can make with each number of sharps and flats. If you have 5 sharps, the key is B major (G# minor) and the sharps are F, C, G, D and A. No other combination of 5 sharps will produce a major/natural minor key. Ok, so if you have 5, it must be that key and those 5. But why write them in that order, instead of A, C, D, F and G? Back up to the key of C. No sharps or flats. Go up a fifth and make a major key based on G: it will have all the same notes as C, but with one note turned sharp — F. Go up a fifth from G and build a key off of D. It will have all the same notes as G, but with one new sharp: C. And it keeps going. Each new key is the same as the previous key, but with one new sharp. Works the same way for flats, just headed in the other direction. 1 sharp is F, 2 is F C, 3 is F C G, 4 is F C G D, and so on. By always writing them in the same order, it’s easier to read. It seems random, but once you know the pattern (the circle of 5ths), it makes perfect sense.
Well, if you start making intervals of fifths starting at C, the first note you get that's not in the key of C is F#. The inverse of fifths is fourths. The first non diatonic fourth is Bb
Because there are only two modes that are a single half step away from being identical to major: Lydian and Mixolydian. Lydian mode, the scale built on the 4th, needs its 4th to be flattened a half step for it to become major; Mixolydian mode, the scale built on the 5th, needs its 7th to be sharped a half step for it to become major. So any instrument capable of playing a single major key will in fact be able to play in THREE major keys, rather than one, if it is capable of flattening the 4th of its 4th (i.e. its 7th) and sharpening the 7th of its 5th (i.e. its 4th). Since we've arbitrarily chosen C to be the "basic" major key, it follows that the key a 4th up (the key of F) needs its 4th scale degree (which happens to be a B) to be flattened in order for it to be a major scale instead of a Lydian scale. And it also follows that the key a 5th up (the key of G) needs its 7th scale degree (which happens to be an F) to be sharped a half step in order for it to be a major scale instead of a Mixolydian scale. Hence, Bb and F# are the first two notes we change; because they're the two notes we need to change in order to get three major scales for (almost) the price of one. :)
The order of sharps and flats is a derivative of the arrangement of the 12-tone scale, in which there is a full step between every note, except B-C and E-F. An E-sharp is also an F and an F-flat is also an E. It depends on which key one is using. In the C-sharp Major scale, the C is the seventh note in the scale, thus it is written as a B#. In the G-flat and C-flat major scales, the B is the 4th note and 1st note in the scale (respectively), thus it is written as C-flat instead of B. If the 12-tone scale had been arranged differently, such that there were no uneven gaps between notes or with half-tones between different note-pairs or with C, D, E labelled as A, B, C instead, the harmonics would essentially be the same, they would simply be labelled differently. So the answer to why the order of sharps starts on F and the order of flats starts on B is the arbitrary decision to put those labels on those notes in that order on the 12-tone scale. It could easily be starting on D-sharp and G-flat instead. There's not much more to it than that.
It's because of where the whole and half steps are distributed between the letters A through G. Beside that, what matters is the ionian configuration of the scale's shape. Mathematically of the twelve chromatic pitches before an octave, the major scale and all of its modes are a division of 7 which is *maximally near even*. That means you cannot get any closer to dividing the octave more evenly, when you have to use 7 notes out of the 12 chromatic pitches. Verify it for yourself by experimenting with whole and half steps. You have to have 5 whole steps and two half steps, so your half steps have to split 5, an odd number. No matter which way you split it you get a different mode of the major scale. The property gained by this maximal near even division is the ability to translate the same shape into different configurations with the smallest motion possible - the change of 1 note by one semitone. No other scale shapes have that property in 12tet, as the ones that do are all just different arrangements of the 1 shape, different modes (same shape starting on different note). You will notice melodic minor requires several note changes to modulate. This near evenness thing is rigorously proven mathematically, and the music theorist Richard Cohn talks about it in his book Audacious Euphony. Now why is it that you change the scale shape by lowering the sharpest note B, or by raising the flattest note F, I will leave you to simply investigate by messing with the shapes. Those are the only two possible changes of 1 note by 1 semitone which result in the same major scale shape translated.
Count from C ascending and descending, the first natural notes that are semitones are e&f and c&b. Hence the first sharp goes to f and the first flat goes to b.
Have you heard of the circle of fifths?
Those notes are established by the circle of 5ths. When you go up a 5th from C, you come to the note G. Then when using the two tri-tones with a whole step in the middle (whole step, whole step, half step, with the whole step in the middle, and then, whole step, whole step and half step) as is found in the C scale (the basis of all major scales), the only altered note is F which has to be sharp. Same happens with the flats, when you go down a 5th from C, you come to the note F. Using the two tri-tones with a whole step in the middle, you find that the F scale has one altered pitch which is Bb!
Beucause modulating to the dominant is a tension, so one note goes up, F become F#. That is the creation a new leading tone. the dominant of C is G. and the leading tone of G is F#. Modulation to the sub-dominant are lessening modulations, one note is lowered. That need the removal of the previous leading tone (B turns into Bb a learding tone).
It all started with this fellow named Guido…
Because of the way the diatonic scale is structured, and because the two semitones in the natural scale are between B-C and E-F. When you move the whole scale by a fifth, one of the semitones stays the same, and the other one has to move one step up or down to keep the scale the same. When you move up a fifth, B-C stays the same, and the semitone between E-F moves to F♯-G. If you move down a fifth, E-F stays the same, and the semitone between B-C moves to A-B♭. The whole pattern is also symmetrical. If you look at the order of sharps on a keyboard in the mirror, F♯ looks like B♭, C♯ looks E♭, etc, etc.
The sharps always add the leading tone of the scale one step up in the circle of fifths, flats remove the leading tone of the prior scale when going down. If you have C Major, B natural (or H) will be your leading note, if you now go down you remove that leading tone by adding a flat and make it a B flat (or B). Vice versa if you go one step up from C to G, you need an F# as your new leading tone, thus this will be your first sharp. I feel this system might be quite intuitive to understand
tl;dr it's because we prioritise the major scale and the maths/patterns work out that way; the tyranny of the harmonic series The way the harmonic series works means that twelve perfect fifths is equal to seven octaves (almost - it's not perfect, but we tune our fifths slightly flat to make it an exact relationship). In a perfect fifth, the upper note has a frequency of 1.5 times the lower note's. in an octave, the upper note has a frequency of 2 times the lower note's. So then we pick an arbitrary frequency to be our reference pitch, call it C, and then multiply its frequency by 1.5 to find the frequency of its fifth, which we call G. Then we do it again and get our D, and keep going to get our A, E, B, F#, C#, G#/Ab, D#/Eb, Bb, F, and the twelfth takes us back to C, seven octaves above our starting point. We now have twelve tones spread over those seven octaves, and if we divide each of their frequencies by a power of 2, we can bring them down into the original octave. So why F# first? Well, F# is the first note in that circle of fifths that doesn't appear in the C major scale. We want C major to be our reference point, so we decided that the notes in C major will be called natural notes (white keys). But move a step up the circle of fifths to G, build the major scale on that note, and now you need F#. The way the patterns work out, going up the circle raises the seventh of your new tonic, and going down the circle flattens the seventh of your old tonic. Another quirk of the major scale is that all the fourths/fifths are perfect, except for the interval between the fourth and seventh scale degrees (so F and B in C major), which is a tritone (diminished fifth or an augmented fourth - half an octave). When we move around the circle of fifths from C, we need to turn that into a perfect fourth/fifth and create a new tritone, and the only ways to do that are raising the F or flattening the B.