So I'll be explaining what I manage to figure out.
First, the basics.
The Playing Field..
You may think that, on a piano, the black notes are half-way between white notes in terms of pitch, with white notes being equally separated. But you'd be wrong. On a logarithmic scale, *all* of the notes are equally spaced.
Here is a frequency analysis (logarithmic scale) of me playing the following notes on my ocarina:
D D# E F F# G G# A' A#' B' C' C#' D' D#' E'
You may notice that the frequency spacing isn't perfectly consistent. That's just because it's an ocarina. For several reasons which I won't go into, I suspect that ocarinas in general only *approximate* the actual notes, though I don't think that it's a logical necessity.
Speaking of approximations, long before the chromatic scale of exact exponential frequencies, the "just intonation" scheme was used, which means basically that frequencies progressed generally exponentially but adhered to strict ratios.
The reason for frequencies increasing exponentially is that our perception of sound is also logarithmic, so, for example, the difference between a D and a D# sounds the same as the difference between a D# and an E.
Each octave is exactly twice the frequency of the octave before it, so A` is twice A, A#' is twice A#, B' is twice B, etc. There being 12 notes per octave, A# is 21/12 times A. This doesn't mean that either the number 2 or the number 12 has any special significance to human hearing. Anything equally spaced on one logarithmic scale will be equally spaced on any other logarithmic scale. E.g., we could invent a scale where each octave has 18 notes and is 23/2 times the previous one, and play it on the exact same piano. (The first octave would contain D C C# D D# E F F# G G# A' A#' B' C' C#' D' D#' E', and the second would be the next 18 notes, etc.) Of course, we could also invent a scale where, e.g., each octave is 3x the previous and has 5 notes, but we couldn't play that on the same piano -- different spacing.
The concept of a "mode" is not that complicated in essence. What if, instead of playing Amazing Grace as "DGBGBAGEDGBADBDBDBGDEGEDGBGBAG," we played it as "EACACBAFEACBECECECAEFAFEACACBA"? Basically we took the C Major scale and translated the notes from "ABCDEFG" to "BCDEFGA'" respectively. Note that this is not quite the same song, because this way the differences we've added to the *semitones* are not the same. It goes from 3 8 12 8 10 8... to 12 1 3 5 6 8 10... Basically we're shifting the song by a certain amount, not with respect to the full chromatic scale, but with respect to a particular subset of the chromatic scale -- the 2,2,1,2,2,2,1 pattern.
But wait, there's more.. instead of framing it as keeping the scale the same but changing the starting key (which would be easier to think about), the concept of modes frames it as keeping the starting key the same but changing the scale. This has the effect of taking our basic increment pattern, 2,2,1,2,2,2,1, and *rotating it* by a certain amount. So in our altered version of Amazing Grace, instead of calling it "Mode II of C Major", we call it "Mode II of D" or "D Dorian," because D is our song's new key, and Dorian or Mode II is the name of the mode which rotates the scale from 2,2,1,2,2,2,1 to 2,1,2,2,2,1,2. So for each Major scale there are exactly 7 modes, one for each possible rotation.
Modes fell out of use for about 200 years, but I hear they're gaining in popularity again.
I hear the Locrian mode sounds funky and isn't really used.
Speaking of popularity, if you're wondering what the difference between a Major and a Minor scale is, the answer is that the Major scales are so called simply because they're the most popular. Any less popular scale is "Minor", just to make sure we get our heads around the idea that, being less popular, it's inferior and will never, ever be of the same magnitude or importance as a "Major" scale.
There are three types of Minor scales, but they're all based on the alternative increment pattern, 2,1,2,2,2,2,1. You may be interested to note that, using only 1- and 2-note increments, treating all possible rotations of the increment pattern as equal, not putting both 1's adjacent and taking exactly 7 notes, 2,2,1,2,2,2,1 and 2,1,2,2,2,2,1 make up the only two possible increment patterns.
Unlike the Major scales, the Minor scale increment pattern can be used ascendingly or descendingly. Two of the three types of Minor scales perform some extra translation on certain notes of the scale after applying the increment pattern.
In music theory there's a lot of things that are pretty much identical but, due to the complication of music theory, can be looked at in numerous possible ways. For example, a C sharp is a D flat, a D sharp sharp is an E, a whole note in 4/4 time is a half note in 2/4 time, and the "natural minor" scale is exactly the same as the Aeolian mode of the major scale. Along the same vein, a "relative minor" is a minor scale that just happens to have the exact same notes as its corresponding major scale but starts on a different key.
If any of this information is incorrect or makes you angry because it reflects ignorance regarding the history of music, don't get too mad. It doesn't surprise me. I'm just learning this stuff myself here, and I thought I might be able to clear some things up a bit for others, regarding the basic concepts.
Here is a frequency analysis (logarithmic scale) of me playing the following notes on my ocarina:
D D# E F F# G G# A' A#' B' C' C#' D' D#' E'
You may notice that the frequency spacing isn't perfectly consistent. That's just because it's an ocarina. For several reasons which I won't go into, I suspect that ocarinas in general only *approximate* the actual notes, though I don't think that it's a logical necessity.
Speaking of approximations, long before the chromatic scale of exact exponential frequencies, the "just intonation" scheme was used, which means basically that frequencies progressed generally exponentially but adhered to strict ratios.
The reason for frequencies increasing exponentially is that our perception of sound is also logarithmic, so, for example, the difference between a D and a D# sounds the same as the difference between a D# and an E.
Each octave is exactly twice the frequency of the octave before it, so A` is twice A, A#' is twice A#, B' is twice B, etc. There being 12 notes per octave, A# is 21/12 times A. This doesn't mean that either the number 2 or the number 12 has any special significance to human hearing. Anything equally spaced on one logarithmic scale will be equally spaced on any other logarithmic scale. E.g., we could invent a scale where each octave has 18 notes and is 23/2 times the previous one, and play it on the exact same piano. (The first octave would contain D C C# D D# E F F# G G# A' A#' B' C' C#' D' D#' E', and the second would be the next 18 notes, etc.) Of course, we could also invent a scale where, e.g., each octave is 3x the previous and has 5 notes, but we couldn't play that on the same piano -- different spacing.
But without our octaves being powers of 2, corresponding notes are not harmonics of each other so the notion of octaves is ineffective, which is why we use octaves of powers of 2. (On a technical level, if it weren't based on powers of 2, it probably couldn't be called an "octave," but here I use the term in the sense of one cycle of the range of note letters.)
Being "harmonic" means that one frequency is an exact integer multiple of another frequency. The advantage of this is that the underlying waveforms repeat in sync with each other. So, for example, if we had one note of 1000 hertz and another note (2 octaves up) of 4000 hertz, you could say, theoretically, that the first one is a succession of waveforms comprising four complete cycles each and the second one isa succession of waveforms comprising one cycle each. Under this perspective, the two underlying waveforms play at the exact same frequency. Either way, though, two harmonic waveforms "nuzzle into" each other, or at the very least, they can be played together without the creation of fringe patterns.
So, *why* did they choose to put some notes on black keys? Basically, the set of white keys is a set of notes that, subjectively speaking, sound good together. It's known as the "C Major" scale.
Scales
Being "harmonic" means that one frequency is an exact integer multiple of another frequency. The advantage of this is that the underlying waveforms repeat in sync with each other. So, for example, if we had one note of 1000 hertz and another note (2 octaves up) of 4000 hertz, you could say, theoretically, that the first one is a succession of waveforms comprising four complete cycles each and the second one isa succession of waveforms comprising one cycle each. Under this perspective, the two underlying waveforms play at the exact same frequency. Either way, though, two harmonic waveforms "nuzzle into" each other, or at the very least, they can be played together without the creation of fringe patterns.
So, *why* did they choose to put some notes on black keys? Basically, the set of white keys is a set of notes that, subjectively speaking, sound good together. It's known as the "C Major" scale.
Scales
A scale could refer to the entire set of notes an instrument can play -- such as the "chromatic" scale which is the one we're familiar with, or the "quarter tone scale" which has 24 notes per octave and definitely cannot be played on a piano -- or, in a more dogmatical sense, it can refer to subsets of the 12 notes in an octave used to compose a melody (such as C Major). The vast majority of "scales" of this sort contain 7 notes. There are 792 possible ways to select 7 notes out of 12 (I think? I did that in my head), but between the major scales, minor scales, and modes, you'll only see a few dozen officially defined (those that musicians think sound good).
Noting that each key on the piano is in the same geometric proportion to the key before it, we can call these increments "semitones," irrespectively of whether a note happens to fall on a white key or a black key. The C major scale thus has 7 semitones that adhere to the following increment pattern: 2,2,1,2,2,2,1. In other words, the first note is C, the second is C+2, the third is the second+2, the fourth is the third+1, etc. Thus if we number the semitones 1-12 and follow the increments, we get these notes: C1, D3, E5, F6, G8, A10, B12, C1 . The last one is redundant, of course, and is just the result of the final number in the increment pattern being used to complete the cycle (i.e., to add up to 12).
We can take this same increment pattern and start on any key we want, and it doesn't change the dynamics of the song; it just raises or lowers the overall pitch. For example, take Amazing Grace:
If our song began, "C D E F G A B," which are all the notes of the C Major Scale, our song in D Major would be (simply via moving each note up by 2) "D E F# G A B C#." Thus the 7 notes of D Major are D E F# G B C#. The same applies for any note we start on. If we started on B flat (a.k.a. A sharp), we'd call that B♭ Major or A# Major, and the notes would be (say, by adding 10 to each C Major note): A# C D D# F G A. Thus there are as many "major" scales as there are keys to start on (twelve), and they all arise from the set of intervals 2,2,1,2,2,2,1 relative to the starting key.
This would obviously be a lot less confusing if the notes weren't lettered and arranged specifically as if songs were in C Major, despite the fact that you can play a song in any key you want -- say if the notes were simply A B C D E F G H I J K L. And let alone the fact that an octave happens to start in the middle of the range, at C, instead of on A! But if there's anything I've learned about music theory, it's that it's as convoluted and confusing as possible by design. It's almost like they pretty much go way out of their way to create dogmatic thought patterns out of simple concepts. Music theory and notation are like the Newspeak of composition, except that Newspeak makes concepts simpler..
So what happens if our song happens to contain a note not contained within the scale we're using? It's not exactly against the rules; the scales are just guidelines. But with respect to the scale, such a note is called an "accidental" (because who in their right mind would do such a thing deliberately?! Sacrilege!!?), and when translating from one scale to another, you simply do the same thing you do with all the other notes: add the difference to the note number. In C major, an accidental of C#2 would translate to a D#4 in D major. An accidental of E5 in C Major would translate to an accidental of F#7 in D Major.
In Eastern music there's a lot more liberty regarding the use (or non-use) of scales.
Keys
Noting that each key on the piano is in the same geometric proportion to the key before it, we can call these increments "semitones," irrespectively of whether a note happens to fall on a white key or a black key. The C major scale thus has 7 semitones that adhere to the following increment pattern: 2,2,1,2,2,2,1. In other words, the first note is C, the second is C+2, the third is the second+2, the fourth is the third+1, etc. Thus if we number the semitones 1-12 and follow the increments, we get these notes: C1, D3, E5, F6, G8, A10, B12, C1 . The last one is redundant, of course, and is just the result of the final number in the increment pattern being used to complete the cycle (i.e., to add up to 12).
We can take this same increment pattern and start on any key we want, and it doesn't change the dynamics of the song; it just raises or lowers the overall pitch. For example, take Amazing Grace:
D3 G8 B12 G8 B12 G8 E5 D3 G8 B12 D3 B12 D3 B12 D3 B12 G8 D3 E5 G8 E5 D3 G8 B12 G8 B12 G8If we started the song at E instead, we'd be changing it from "C Major" to "D Major." Just add 2 to each semitone, giving us:
E5 A10 C#2 A10 C#2 A10 F#7 E5 A10 C#2 E5 C#2 E5 C#2 E5 C#2 A10 E5 F#7 A10 F#7 E5 A10 C#2 A10 C#2 A10
If our song began, "C D E F G A B," which are all the notes of the C Major Scale, our song in D Major would be (simply via moving each note up by 2) "D E F# G A B C#." Thus the 7 notes of D Major are D E F# G B C#. The same applies for any note we start on. If we started on B flat (a.k.a. A sharp), we'd call that B♭ Major or A# Major, and the notes would be (say, by adding 10 to each C Major note): A# C D D# F G A. Thus there are as many "major" scales as there are keys to start on (twelve), and they all arise from the set of intervals 2,2,1,2,2,2,1 relative to the starting key.
This would obviously be a lot less confusing if the notes weren't lettered and arranged specifically as if songs were in C Major, despite the fact that you can play a song in any key you want -- say if the notes were simply A B C D E F G H I J K L. And let alone the fact that an octave happens to start in the middle of the range, at C, instead of on A! But if there's anything I've learned about music theory, it's that it's as convoluted and confusing as possible by design. It's almost like they pretty much go way out of their way to create dogmatic thought patterns out of simple concepts. Music theory and notation are like the Newspeak of composition, except that Newspeak makes concepts simpler..
So what happens if our song happens to contain a note not contained within the scale we're using? It's not exactly against the rules; the scales are just guidelines. But with respect to the scale, such a note is called an "accidental" (because who in their right mind would do such a thing deliberately?! Sacrilege!!?), and when translating from one scale to another, you simply do the same thing you do with all the other notes: add the difference to the note number. In C major, an accidental of C#2 would translate to a D#4 in D major. An accidental of E5 in C Major would translate to an accidental of F#7 in D Major.
In Eastern music there's a lot more liberty regarding the use (or non-use) of scales.
Keys
The concept of a "key," as in "playing in the key of G" is pretty much identical to the concept of scales, except that the word "key" emphasizes the aspect the particular key named is the "critical note," so to speak, of the song. You can consider "key of G" and "scale of G Major" to be identical in meaning.
A "key signature" is included in a score and tells you what key it's playing. By knowing what key it's playing in you know which notes are on the scale, and hence you know which letters are to be interpreted as naturals and which letters are to be interpreted as sharps. I suppose it would be simpler just to write the notes on the lines they actually belong on, rather than having to translate based on a key signature, but whatever.. that's music theory for you.
Modes
A "key signature" is included in a score and tells you what key it's playing. By knowing what key it's playing in you know which notes are on the scale, and hence you know which letters are to be interpreted as naturals and which letters are to be interpreted as sharps. I suppose it would be simpler just to write the notes on the lines they actually belong on, rather than having to translate based on a key signature, but whatever.. that's music theory for you.
Modes
The concept of a "mode" is not that complicated in essence. What if, instead of playing Amazing Grace as "DGBGBAGEDGBADBDBDBGDEGEDGBGBAG," we played it as "EACACBAFEACBECECECAEFAFEACACBA"? Basically we took the C Major scale and translated the notes from "ABCDEFG" to "BCDEFGA'" respectively. Note that this is not quite the same song, because this way the differences we've added to the *semitones* are not the same. It goes from 3 8 12 8 10 8... to 12 1 3 5 6 8 10... Basically we're shifting the song by a certain amount, not with respect to the full chromatic scale, but with respect to a particular subset of the chromatic scale -- the 2,2,1,2,2,2,1 pattern.
But wait, there's more.. instead of framing it as keeping the scale the same but changing the starting key (which would be easier to think about), the concept of modes frames it as keeping the starting key the same but changing the scale. This has the effect of taking our basic increment pattern, 2,2,1,2,2,2,1, and *rotating it* by a certain amount. So in our altered version of Amazing Grace, instead of calling it "Mode II of C Major", we call it "Mode II of D" or "D Dorian," because D is our song's new key, and Dorian or Mode II is the name of the mode which rotates the scale from 2,2,1,2,2,2,1 to 2,1,2,2,2,1,2. So for each Major scale there are exactly 7 modes, one for each possible rotation.
Modes fell out of use for about 200 years, but I hear they're gaining in popularity again.
I hear the Locrian mode sounds funky and isn't really used.
Speaking of popularity, if you're wondering what the difference between a Major and a Minor scale is, the answer is that the Major scales are so called simply because they're the most popular. Any less popular scale is "Minor", just to make sure we get our heads around the idea that, being less popular, it's inferior and will never, ever be of the same magnitude or importance as a "Major" scale.
There are three types of Minor scales, but they're all based on the alternative increment pattern, 2,1,2,2,2,2,1. You may be interested to note that, using only 1- and 2-note increments, treating all possible rotations of the increment pattern as equal, not putting both 1's adjacent and taking exactly 7 notes, 2,2,1,2,2,2,1 and 2,1,2,2,2,2,1 make up the only two possible increment patterns.
Unlike the Major scales, the Minor scale increment pattern can be used ascendingly or descendingly. Two of the three types of Minor scales perform some extra translation on certain notes of the scale after applying the increment pattern.
In music theory there's a lot of things that are pretty much identical but, due to the complication of music theory, can be looked at in numerous possible ways. For example, a C sharp is a D flat, a D sharp sharp is an E, a whole note in 4/4 time is a half note in 2/4 time, and the "natural minor" scale is exactly the same as the Aeolian mode of the major scale. Along the same vein, a "relative minor" is a minor scale that just happens to have the exact same notes as its corresponding major scale but starts on a different key.
The Disclaimer
If any of this information is incorrect or makes you angry because it reflects ignorance regarding the history of music, don't get too mad. It doesn't surprise me. I'm just learning this stuff myself here, and I thought I might be able to clear some things up a bit for others, regarding the basic concepts.
Appendix A - Amazing Grace
Here are five different versions of Amazing Grace that I put together into an easily readable and comparable format.
Note that they *mainly* differ only in their "grace notes," which are little extra notes that people add in to just make a song fancy because life is boring.
Note also, they're not called "grace notes" just because the song is called Amazing Grace. That's just a horrible coincidence.. there must be something wrong with my karma.
Note that they *mainly* differ only in their "grace notes," which are little extra notes that people add in to just make a song fancy because life is boring.
Note also, they're not called "grace notes" just because the song is called Amazing Grace. That's just a horrible coincidence.. there must be something wrong with my karma.
