The Difference Between The Sharps And Flats

Here is a YouTube video I put up recently describing the difference between sharps and flats:

http://www.youtube.c...h?v=jCDMkmVMA2I

It requires a little prerequisite theory knowledge, though I hope it's accessible. Here is the video transcript:

"What is the difference between F sharp and G flat?" It's a question asked often enough, but what comes along less often is a satisfactory answer. It is my hope that by the end of this video you will have not only an understanding of WHY there's a difference, but an understanding of how to use that difference to make, play, and talk about music more effectively.

Let's start with a staple of music theory: the spiral of fifths. It shows that, starting from the naturals and using a chain of fifths, you can arrive at every note we could ever use in western music: The sharps, the flats, the double sharps, the double flats, and so on. Also, notice that we can derive any diatonic scale from a continuous, unbroken chain of seven fifths.

For one example, the G major diatonic scale can be generated from a chain of fifths from C to F#, which rearranges to G-A-B-C-D-E-F#.

Neither of those facts is coincidence. In fact it is by this chain of fifths that we DEFINE the notes that EXIST in western music, and it is by any 7 notes connected by an unbroken chain of fifths that we DEFINE our diatonic scales.

So most simply, the difference between F# and Gb is each's different location in the spiral of fifths, and therefor the scales to which they do and do not belong. If you're using the G major scale, the right note is F#, and if you're using the Db major scale, Gb is the correct note. But let's delve a little further and ask "So what?" If they're the same pitch, why define things that way?

First, know that intervals are measured in a unit called "cents" where there are 1200 in every octave, 100 cents per half step, and therefore 700 cents per fifth, at least in standard western tuning.

Let's prove that F# equals Gb specifically because the fifth is 700 cents.
From F# to Gb in fifths is F#-B-E-A-D-G-C-F-Bb-Eb-Ab-Db-Gb, or 12 fifths.
Starting with Gb at 0 cents, and adding
12 fifths * 700 cents/fifth gives you 8400 cents.
Dropping that note into the original octave by subtracting 6 octaves, 8400 cents - 1200 cents per octave times 6 octaves gives you 8400 cents minus 8400 cents, which comes out to 0 cents. Because 0 cents is a unison, F# is equal to Gb.

The G major scale and I IV V I progression using these pitches sound like this:
~~~~~~~~~~~~~~
And the same pattern but replacing Gb for F# sounds exactly the same.

The size of the fifth can change however, and when it is not exactly 700 cents then F# and Gb actually have DIFFERENT PITCHES. Motivations for adjusting the size of the fifth in the first place include better harmonic purity ("more in-tune") or melodic clarity (with a starker contrast between major and minor). Still, the over all musical structures stay pretty much the same.

If we flat the fifth a nearly imperceptible 5 cents to 695 cents:
From F# to Gb in fifths is still F#-B-E-A-D-G-C-F-Bb-Eb-Ab-Db-Gb, or 12 of these slightly flatter fifths.
12 fifths * 695 cents/fifth = 8340 cents
Dropping that note down 6 octaves gives 8340 cents - 8400 cents = -60 cents. That means that, with a fifth of 695 cents, F# differs from Gb by 60 cents!

The G major scale and I IV V I progression using these new pitches sound like this:
~~~~~~~~~~~~~~~
Different nuances from before, but recognizably the "same thing".
The same pattern but replacing Gb for F# sounds like this:
~~~~~~~~~~~~~~

It's completely different, and quite wrong for that "major scale" sound we all know and love.

Why would it turn out that F# sounds so much better in this context than Gb? Well, the decision to define our scales by unbroken chains of fifths is not arbitrary. Scales created by continuous unbroken chains of one given interval have perceptually relevant properties in music, where intervals sizes are regular (with only two sizes per interval class, aka major and minor), and where the different step sizes are distributed evenly throughout the scale. (For example, the diatonic scale has the minor seconds as far apart as possible between the major seconds.) Such scales are called "Moment of Symmetry" (or MOS) scales. Replacing a note like F# with another note like Gb, though they may be close in pitch, disrupts this regularity, and voids the definition of MOS.

You imply this difference between the sharps and flats even when using standard western tuning with a fifth of 700 cents, where F# and Gb are the same pitch. *This*, ladies and gentlemen, is the difference between the sharps and the flats. They differ in their location in the spiral of fifths, which describes their difference in function, which is audible in tunings where the fifth is not 700 cents.

Curious about playing and composing music in different tunings? Check out some of these links!

I'm unable to add my 5 cents to this thread .

A bit confusing to read this when you haven't specified which tuning you're using. There's an obvious difference when it's not the "well-tempered tuning" but one of the other, usually the natural tuning.

Uh, ok.

XD I'll come back to understand this in a few years... preferably when I can add and subtract and multiply better.XD

rom F# to Gb in fifths is F#-B-E-A-D-G-C-F-Bb-Eb-Ab-Db-Gb, or 12 fifths.
Starting with Gb at 0 cents, and adding
12 fifths * 700 cents/fifth gives you 8400 cents.
Dropping that note into the original octave by subtracting 6 octaves, 8400 cents - 1200 cents per octave times 6 octaves gives you 8400 cents minus 8400 cents, which comes out to 0 cents. Because 0 cents is a unison, F# is equal to Gb

Ah, 1200 X 6 = 7200. Don't you mean 1200 X 7?

Also as commented, depends on the tuning you are using. In some tunings F# and G flat are not equivalent and therefore make keys based on these tones quite unusable.

Ah, 1200 X 6 = 7200. Don't you mean 1200 X 7?

Quite right, I've corrected that in the video using annotations. Thanks!

Also as commented, depends on the tuning you are using. In some tunings F# and G flat are not equivalent and therefore make keys based on these tones quite unusable.

Keys based on those notes don't become unusable just because our enharmonics diverge. You just can't play a song that uses both on an interface like a piano that has only one key for the two of them.

I think it's confusing that you never define which tuning you're using, which would be the well-tempered tuning. But then you talk about a 5th at 695 cents (which would mean 695 cents from our starting note I assume). But you don't mention which tuning this is. Is it just someone you made up for the purpose of the example? Also, why do you subtract your new number "8340" with the one you derived using the well-tempered fifth "8400" ? The circle of fifths is only consonant as long as you operate within the well-tempered tuning, so how do you justify still using it when you have left the well-tempered tuning.

Actually on meantone instruments there are separate keys for D# and E flat. If two such enharmonic notes in more modern tunings can be separate in older tunings, then I stand by my assertion you got to state outright your tuning system.

I think it's confusing that you never define which tuning you're using, which would be the well-tempered tuning.

The tunings in the video are not well tempered. Well tempered tunings use only 12 unequally spaced notes, where the pureness of some keys is maximized at the expense of others.

The tunings I use are "meantone tunings", where all the interval sizes are regular and are defined by an (infinitely long) chain of fifths. That is, all the major seconds are the same size, all the perfect fifths, all the minor sixths, etc. If you choose a twelve note subset of the infinite chain of fifths you do end up with "wolf fifths", but that's only because they are something like C# to Ab, or a doubly diminished sixth, NOT a fifth.

Because all the fifths are the same size, when you define its size the pitches of all other notes fall right into place (because their pitches are defined by their location in the circle of these regularly tempered fifths). So I *do* define which tuning I'm using, merely by the statement of the size of the fifth:
http://en.wikipedia....one_temperament

Not sure if I follow the rest completely but I'll do my best to reply.

But then you talk about a 5th at 695 cents (which would mean 695 cents from our starting note I assume). But you don't mention which tuning this is. Is it just someone you made up for the purpose of the example?

The tuning with a fifth of 695 cents is about "1/3 comma meantone" to give it a name but, as I mentioned above, just by specifying the size of the fifth, the other notes are defined by their location in the circle of fifths. The reason you'd flat the fifth like that is then you end up with very pure thirds, and in fact a perfectly tuned minor third. Though the fifth is slightly flatter, over all the tuning is much "better" (or has less error).

Also, why do you subtract your new number "8340" with the one you derived using the well-tempered fifth "8400" ?

I subtracted the number 8400 because it is 7 octaves. What I mean is, I stacked up all those fifths, and then to compare it to the original note I started at I had to bring the new note down to the original octave. Originally it turned out that 12 fifths equalled 7 octaves (8400-8400=0), but the second time it turned out that 12 fifths did NOT equal 7 octaves (8340-8400=-60). So the 8400 came from octaves, not from the original fifth I used.

The circle of fifths is only consonant as long as you operate within the well-tempered tuning, so how do you justify still using it when you have left the well-tempered tuning.

This is very much so not true. Using the circle of fifths to define the rest of your notes provides you with good accuracy (consonance) over a wide range of values, as I showed in the video. Flatting the fifth to 695 cents gives you MUCH purer tuning than standard western tuning, and if you sharp it to 702 cents and then use C-Fb-G as your consonant triad it is *extremely* close to pure.

-J