# Double Sharps in Just Intonation. The mathematics?

So I'm experimenting around and I'm creating a small little thing in C#### minor just because. I understand the mathematics of C#### in Pythagorean and Equal Temperment music systems but how do you represent multiple accidentals in the Just Intonation system? Like Gbbb is mathematically different than E. Thank you!

Unfortunately, it's a bit more complicated than that. There is no single "Just Intonation System"; instead, there are multiple systems which can be said to be just, by virtue of the fact that they use just intervals (i.e. integer frequency ratios). The problem is in determining which ratios you want to use.

One such Just system is the Pythagorean system, which only uses Just perfect fifths (a ratio of 3/2, or ~702 cents) and just octaves to generate all the notes of the scale. To find the mathematical ratio for any note, you just count how far it is from your reference note, along the line of fifths, and multiply by that power of (3/2). You can also multiply/divide by arbitrary powers of 2 (raise or lower by octaves) to get the resulting ratio into the desired octave.

...B♭♭ - F♭ - C♭ - G♭ - D♭ - A♭ - E♭ - B♭ - F - C - G - D - A - E - B - F♯ - C♯ - G♯ - D♯ - A♯ - E♯ - B♯...

So, for example, B♭♭ is 12 fifths below A, so these notes differ by a ratio of 2k * (3/2)-12 (where k is whatever factor is needed to make the ratio close to 1). The result ends up being the Pythagorean Comma.

Another Just system is Quarter Comma Meantone, which only uses Just major thirds (a ratio of 5/4, or ~386 cents) and just octaves to generate all the notes of the scale. To find the ratio for notes in this scale, you can count how many fifths it is on the line of fifths, and multiply by that power of 51/4. As usual, octave shifts are permitted. You can also construct a line of thirds, each of which is a ratio of (5/4) from the previous, but it will only have a quarter of the notes. The other notes are equally distributed in between (hence the fifths being the fourth root of five: four stacked fifths is octave-equivalent to a third).

...F♭ - A♭ - C - E- G♯ - B♯...

What you might be referring to as "just intonation" is what wikipedia calls "5-limit tuning". This uses a combination of Just perfect fifths (3/2) and Just major thirds (5/4), as well as octaves (2/1), to define notes. If the previous two systems can be depicted with one-dimensional lines, then this system can be depicted with a 2D lattice known as the Tonnetz. Moving one step to the right along this graph represents multiplication by (3/2), while moving up one step represents multiplication by (5/4).

`````` F  --{C} -- G  -- D  -- A -- E -- B -- F#
|     |     |     |     |    |    |    |
[Db]-- Ab -- Eb -- Bb -- F --(C)-- G -- D
|     |     |     |     |    |    |    |
Bbb-- Fb -- Cb -- Gb --[Db]- Ab-- Eb-- Bb
|     |     |     |     |    |    |    |
Gbb-- Dbb-- Abb-- Ebb-- Bbb- Fb-- Cb-- Gb--[Db]
``````

The problem is that there are many ways to get to any given note, and the resulting ratios from these different routes don't agree. In fact, every note will show up once per row, but in each row, it will have a different pitch. (Note: I'm going to be a bit sloppy by not explicitly counting octave shifts (factors of two) and just using 2k as a placeholder, as I did above. These k's are not necessarily all equal).

For example, from the parenthesized (C), you can get to another {C} by going down four fifths and then up a major third. The ratio between these C's is therefore: 2k * 3-4 * 5. Since this number cannot equal 1, the two C's must have different frequencies. In fact, the result is 81/80, which is called the Syntonic Comma. A chord progression which starts with one definition of (C), and ends up with a different definition of {C} has drifted pitch by a comma, despite singing only just intervals. This type of progression is called a Comma Pump.

Similarly, I've noted multiple instances of D♭, which have ratios (relative to (C)) of:

• 2k * 3-5
• 2k * 3-1 * 5-1
• 2k * 33 * 5-2

Again, because these numbers are different, the resulting notes are different (each by a syntonic comma). I will leave it as an exercise to extend this system to Gbbb or C####, and determine the multiple sets of interval combinations which produce these notes. Which one is "right"? All of them. Or perhaps none of them. In this kind of system, every time you use a note, you need to look at your context to determine how the note is being used, in order to figure out which version of the note should be used.

• Thank you! Yes I am referring to 5-limit-tuning. It is pretty simple in the other tuning systems you listed. It gets even worse when it comes to like 17-limit tuning and things. So you just set up a big table? I get the fact that it sorta depends on when you need the note upon if it's right, but do you mean based upon key or in general? Like is there a correct C## in C major? Because I assume that for my C#### minor, I'd be transposing from A minor directly into that. – Neo Scott Jun 23 '15 at 21:24
• I don't know that there's any way to define a single one as being right, other than defining the specific intervallic relationship you want. In this case, there's no simple intervallic relationship between A and C####, in fact, the later is much more likely to be interpreted as an out of tune version of E, which does have a simple relationship (a fifth above) rather than a triply-augmented third. I can't think of any context in which the later interval even makes sense. – Caleb Hines Jun 24 '15 at 3:40
• However, if you want to find a version of this note without reproducing the entire chart, you can find the nearest simple enharmonic (in this case, a fifth higher, at E = (3/2)), then create a chain of major thirds (up or down) to get to the desired spelling: E-G#-B#-D##-F###-A###-C#### = (5/4)^6. Then you can get to other versions of that note by multiplying (or dividing) by the syntonic comma (81/80) as many times as desired to put it in the desired row. The final result is: A * (3/2) * (5/4)^6 * (3^4/(5*2^4))^k (for any integer k). – Caleb Hines Jun 24 '15 at 3:51