WARNING! If you have little theory, little interest in theory, or are close to losing the will to live, then I wouldn't bother with this article.
This may be of interest to a very small number of people, and may have no practical value whatsoever, but it interests and, since writing it, amuses me, and so I'm going to go ahead with it. It came about because of my spending time with ear training. Not only does that involve recognizing intervals, chords, etc, but also having to name their notes something we guitarists sometimes skip over because of the nature of the fret board.
Keyboard players are fortunate in that they can't get away with thinking in G while they play in Gb, and so they have to learn the proper names for things. As I'm doing most of the ear training at a keyboard, I can't indulge in my usual lazy guitar player habits, so I'm having to name things properly.
More important than that is the fact that songs written in F# simply don't sound the same in C, and so we really should get used to giving things their correct labels. That means no more thinking of chords in terms of shapes. Let's name them properly, including the notes contained within.
This is about naming all of the intervals by the notes they contain (at least those inside a single octave). So we need to know in the blink of an eye, what is a minor sixth if the bottom note is Gb etc.
By knowing your interval notes, you know your chord notes. If a major triad is made up of the root, major third, perfect fifth, then if you can name the notes of all intervals, then whatever the root is, you'll instantly know the other notes. This is very useful while improvising and/or writing.
Below are some immutable conventions for naming the intervals; Rules that never change no matter what key you're in. Whether or not they make things easier than just counting up the half steps or whatever is debatable.
The conventions within this approach to learning the note names within intervals is a bit easier if you come at it from a music notation theory perspective rather than a practical fret board pattern one. The reason being that when you learn the lettered note names from the fret board, you tend to think A, A#/Bb, B, C, C#/Db, D, D#/Eb, E, F, etc. Whereas the notation perspective is simply a case of A, B, C, D E, F, and G, with those lettered lines being assigned values such as sharp, flat, natural, and sometimes double flat and double sharp. This is a much simpler perspective when approaching anything to do with theory. It's particularly useful in this case as we don't really need to concern ourselves too much with the sharps and flats because we can just generally think of A's, B's, C's, D's etc.
Knowing standard notation isn't essential to get something out of this article, but if you think theory is important, which frankly it is, then learn it!
We're going to group intervals in terms of whether or not the two letters share the same value, sharp, flat, or natural. The notes of the A-E perfect fifth have the same value in that they're both natural. The Ab-Eb perfect fifth also has a single value and that is flat. The important thing is are the two notes the same or different? For example the B-F# perfect fifth has differing values for each note, natural and sharp. We're going to refer to the first of the two notes as the bottom note as it will be the lower, or root, of the interval. I'm assuming you know about intervals in terms of what constitutes a second, third, fourth etc.
So let's get grouping!
We're coming at this from letters, so any A-B, B-c, C-D, D-E, E-F, F-G, G-A is a second whether it be minor or major, sharp or flat. A-Bb, is an A of some sort to a B of some sort, and so it's a second of some sort. A#-B is an A of some sort to a B of some sort, and so is a second of some sort. See what I mean?
There are 7 letters, A, B, C, D, E, F, and G. These are our lower/root notes, and we'll separate the intervals based on them into two groups, sames' and differents'.
Before separating them, first let's list them all:
Notice that we've only listed the minor seconds that start with a simple natural letter name in the bottom. This is because we only need to concern ourselves with the letters and not so much with their additional flat/natural/sharp values this will be explained in a moment.
Hopefully you'll have noticed that only two of the intervals listed above are sames', and five of them are differents'. The sames are: B-C, and E-F. The reason for this may have occurred to some of you, and it's essential to this whole grouping intervals concept.
B-C and E-F are the only two naturals that are side by side a half step apart. All the other naturals, A-B, C-D, D-E, F-G, and G-A are a step apart and so have a note in between.
Before we move on to major seconds, I should probably demonstrate that the sames and differents group in the same way even if the bottom note is sharp or flat, and not just natural. I'm only going to do it once though. The rest of the time, I'm just going to put naturals in the bottom.
These are all minor seconds, and you can se they all have an A through G of some sort in the bottom and a B through A of some sort in the top:
Ab-Bbb, A-Bb, A#-B (different)
Bb-Cb, B-C, B#-C# (same)
Cb-Dbb, C-Db, C#-D (different)
Db-Ebb, D-Eb, D#-E (different)
Eb-Fb, E-F, E#-F# (same)
Fb-Gbb, F-Gb, F#-G (different)
Gb-Abb, G-Ab, G#-A (different)
Okay, so no matter what sort of additional value was added to the letter, the grouping remained the same. Two sames, B of some sort C of some sort, E of some sort F of some sort, and five differents.
To get our major second groupings, we simply need to reverse the minor seconds. So instead of two sames and five differents, there will be two differents and five sames. Guess which ones they are? TheB's-C's, and E's-F's are now different, and all the rest are sames:
We saw that the seconds grouped into two and five, sames and differents, which reversed according to whether the interval was minor or major. The reversing thing will be the same for thirds, but the groupings will be three of one and four of the other rather than two and five.
It's worth noting at this point that we don't need to think of the whole interval when grouping the sames and differents, but merely the bottom/root notes. So if we refer back to seconds, we saw that B-C and E-F were sames for minors, and differents for majors. We can however just say that the values for the intervals based on B and E were the same for minor seconds and different for the majors, and the rest sorts itself out accordingly.
So with that in mind, the groupings for minor thirds are:
A, B, D, E (same)
C, F, G (different)
As with minor and major seconds, we just reverse the groupings.
C, F, G (same)
A, B, D, E (different)
Fourths And Fifths
We saw that the groupings for seconds were two and five, and for thirds it was three and four. For fourths and fifths, it will be one and six.
A, B, C, D, E, G (same)
Reverse the groups:
A, B, C, D, E, G (different)
A, C, D, E, F, G (same)
Reverse the perfect fifth groups.
A, C, D, E, G (different)
Aug fifths and their inverted counterparts, diminished fourths, are the trouble-makers of this article. They're the only ones that don't divide into sames and diferents, but instead into differents and extra differents.
A, C, D, E, F, G (different)
B (extra different)
B-F## (extra different)
A, B, C, D, E, G (different)
F (extra different)
F-Bb (extra different)
A word at this point about inversions. Fourths and fifths are inversions of each other. G-C perfect fourth is C-G perfect fifth inverted. C-G# augmented fifth is G#-C diminished fourth inverted. Because fourths and fifths are inversions of each other, they share the same groupings, one and six.
Sixths are inversions of thirds, and they too share the same groupings, three and four.
A, B, E (same)
C, D, F, G (different)
Reverse the groups.
A, B, E (different)
C, D, F, G (same)
Sevenths are inversions of seconds, and they share the same groupings, two and five.
A, B, D, E, G (same)
C, F (different)
Reverse the groupings from the minor sevenths.
A, B, D, E, G (different)
C, F (same)
At this point you'd be forgiven for thinking this is all more trouble than it's worth. Remembering and employing The naming conventions is more hassle than just trying to memorize the interval notes themselves. Well, look at how many intervals have just been listed using only natural values on the bottom. There were 14 catagories: minor/major seconds, minor/major thirds, diminished/perfect/augmented fourths, diminished/perfect/augmented fifths, minor/major sixths, and minor/major sevenths. Each contained 7 intervals based on the seven letters. 14X7=98. Add sharps and flats to those bottom/root notes and you can multiply that by 3 to give you 294 different intervals to remember. Hope that's right. I'm really dreadful with even the simplest math.
Okay so let's try and boil all this down into something useful. Wish me luck.
We saw that we don't have to worry too much about whether the notes are flat, natural, or sharp. We just need to focus on the letters themselves. There are only seven of these so that makes life a little easier.
We saw that intervals that were inversions of each other shared the same groupings, and these are:
Fourths and Fifths: one and six
Seconds and Sevenths: two and five
Thirds and Sixths: three and four
In the case of fourths and fifths, the letters to remember are
B and F. These two are always involved in the one' part of the groupings.
For seconds and sevenths, the B/C E/F thing is the thing to note. We saw that for seconds, the intervals based on B and E were same for minor, different for major. For sevenths we look to the other end of the B/C E/F thing because it's the C and F that now make up the two side of the groupings. Seventh intervals based on C and F are same for majors, different for minors.
Up to this point, B, C, E and F have been at the centre of everything, and they're easy to remember because we all learn that they're the neighbouring naturals.
Finding something easily remembered for thirds and sixths isn't quite so easy. For thirds, C, F, and G were same for major, different for minor. We guitarists shouldn't have too much trouble remembering C, F, and G.
For sixths, C, F, and G could still be used because after all, they're thirds inverted. For instance, for minor sixths, A, B, and E are the sames. Their top notes are F, G, and C.
The trouble with that idea however is that the C, F, and G notes would be on the top, and as we build chords/intervals from the bottom up, it's not practical to invert things, so we just have to remember A, B, E instead. Lincoln perhaps? For sixth intervals, A, B, and E are sames for minor, and differents for major.
For fourths and fifths, it's one and six, and the intervals to remember are based on B and F. B and F are always different, or extra different, and this makes up the one group.
For seconds and sevenths, it's two and five, and the intervals to remember are based on B and E for seconds, and C and F for sevenths. B and E are same for minor seconds, different for majors. C and F are same for major sevenths, different for minors.
For sixths, it's three and four, and the intervals to remember ar based on C, F, and G for thirds, A, B, and E for sixths. C, F, and G are same for major thirds, different for minors. A, B, E are same for minor sixths, different for majors.
Clear as mud.
Rules For Intervalic Tonality
The next thing is to understand the rules that govern the tonality of intervals. That is to say, whether they be minor, major, diminished, perfect, or augmented, and how this applies to our sames and differents.
One Up One Down
One up one down refers to sharpening/flattening tones.
One up from double flat is flat. One up from flat is natural. One up from natural is sharp. One up from sharp is double sharp.
One down from double sharp is sharp. One down from sharp is natural. One down from natural is flat. One down from flat is double flat.
This is useful for working out what the differents' are going to be. The sames' are, of course, always the same; natural on the bottom, natural on top, flat on bottom, flat on top etc.
For ALL minor intervals, seconds, thirds, sixths, and sevenths, if the top note is different, it will ALWAYS be one down from whatever is the value of the bottom note.
For example, if the bottom is natural, the different top notes will always be flat. They will never be double flat, sharp or double sharp. If the bottom note is flat, the different top will always be double flat, never, natural, sharp or double sharp. If the bottom is sharp, the different top will always be natural, never double flat, flat, or double sharp.
So whatever the bottom note is of a minor interval, the top note will either have the same value, or be different by one down.
Minor Second Examples
We know that in the case of minor seconds the sames occur when any interval has a B or E of some sort on the bottom. The other five intervals are differents, and we now also know that they'll be different by one down.
Ab-Bbb, A-Bb, A#-B (different/one down)
Bb-Cb, B-C, B#-C# (same)
Cb-Ebb, C-Eb-, C#-E (different/one down)
Minor Third Examples
C, F, G are differents, and different by one down because it's minor.
Ab-Cb, A-C, A#-C# (same)
Cb-Ebb, C-Eb, C#-E (different/one down)
Eb-Gb, E-G, E#-G# (same)
Fb-Abb, F-Ab, F#-A (different/one down)
This same or different/one down' factor holds true for all intervals with a minor tonality.
We saw with the minors that if the top note is different, it will always be one down. In the case of the majors, it will always be one up.
Major Second Examples
Intervals based on B and E are the differents, and they're always different by one up.
Ab-Bb, A-B, A#-B# (same)
Bb-C, B-C#, B#C## (different/one up)
Cb-Db, C-E, C#-E# (same)
Eb-F, E-F#, E#-F## (different/one up)
Major Sixth Examples
A, B, E are the differents, and because it's major, the difference is always one up.
Ab-F, A-F#, A#-F## (different/one up)
Cb-Ab, C-A, C#-A# (same)
Db-Bb, D-B, D#-B# (same)
Eb-C, E-C#, E#-C## (different/one up)
Perfect Fourth Examples
If the top note of a perfect fourth is different, it will always be one down.
F is different, and so different by one down.
Ab-Db, A-D, A#-D# (same)
Bb-Eb, B-E, B#-E# (same)
Fb-Bbb, F-Bb, F#-B (different/one down)
Perfect Fifth Examples
If the top note of a perfect fifth is different, it will always be one up.
B is different, and different by one up.
Ab-Eb, A-E, A#-E# (same)
Bb-F, B-F#, B#-F## (different/one up)
Cb-Gb- C-G, C#-G# (same)
Diminished Fifth Examples
If the top note is different, it will always be one down.
B is same.
Ab-Ebb, A-Eb, A#-E (different/one down)
Bb-Fb, B-F, B#-F# (same)
Cb-Gbb, C-Gb, C#-G (different/one down)
Remember how aug fifths put a spanner in the works? Well the top note of an aug fifth is always either different/one up, or extra different/two up.
B is extra different.
Ab-E, A-E#, A# E## (different/one up)
Bb-F#, B-F##, B#-F### (extra different/two up)
note the triple sharp! Told you aug fifths were trouble-makers.
Diminished Fourth Examples
The top note will either be one down, or two down.
F is extra different.
Ab-Dbb, A-Db, A#-D (different/one down)
Bb-Ebb, B-Eb, B#-E (different/one down)
Fb-Bbbb, F-Bbb, F#-Bb (extra different/two down)
Note the triple flat! Also be aware that diminished fourths have exactly the same tonality as major thirds. However, if you invert an augmented fifth, the result must be a fourth of some sort, and not a third, and so diminished fourths they be. Tip: inversions always add up to 9 (Fourth/fifth, second/seventh, third/sixth), and majors always become minors and vice versa. Augmenteds always become diminished and vice versa. Perfects remain perfect.
Let's use this ludicrously flabby system to answer the question at the top of the article. What is a minor sixth above Gb?
First we have to know that any G to any E is a sixth of some sort. If you're still having to count intervals on your fingers, then do so. It will eventually solidify. So G-E=sixth.
Next, it's a minor sixth we're trying to work out, and for minor sixths hopefully we have remembered that A, B, and E are sames, so our G is going to be different. It's minor so it's going to be different by one down. Therefore, a minor sixth from Gb is Ebb. Simple!
I'm probably only going ahead with posting this article now for purely comic reasons. It has to be one of the most unwieldy and convoluted systems ever.
This idea was far harder to put into words than it seemed when it was just in my head. Apologies if it was confusing and useless. I worked hard to try to make this as elegant as possible, but I have a hunch that it will never be up there with the circle of fifths.
About The Author:
Chris Flatley is Professor of Music Studies at the School for the Congenitally Befuddled.