What is considered a wide third?

I have a question mentioned in my music theory class. It is as follows: In this temperament, we have eight pure thirds and four wide thirds. The four wide thirds are C#-F, G#-C, F#-Bb, and [my answer]. I know a couple possibilities are A-C#, E-G#, B-Eb, and C-F#, sure, but I have no idea what is implied by a wide third as a opposed to a pure third. Could you explain how I would find wide thirds?

• Since this seems to be a homework question, I won't answer but instead will offer some hints: do you know what the red fractions are around the outside of the circle, or the blue decimals inside it? Do you know how many steps sound the circle you have to go to get from one note to the third above or below it? Nov 22 '21 at 3:10
• I'll note that C-F# is an augmented 4th, not a 3rd of any kind. Nov 22 '21 at 12:08

In just intonation, a pure third (a pure major third, that is) has a frequency ratio of 5:4 between the higher and lower pitches.1

Consider a just tuning based on A4 = 440Hz.

A4 = 440Hz
C#5 = 440Hz * 5/4 = 550Hz
F5 = 550Hz * 5/4 = 687.5Hz
A5 = 687.5Hz * 5/4 = 859.375Hz

However, we expect just octaves to have a 2:1 frequency ratio, so the expected frequency of A5 is 880Hz. That means that (at least) one of our thirds has to be made "wider" — that is, the ratio has to be larger — in order for the octave to be in tune.

Suppose using the above example, we tune C#5 and F5 in 5:4 ratios as shown, but then tune A5 in a 2:1 ration to A4. That would mean that the ratio between F5 and A5 would be 880/687.5, which is 1.28; whereas, 5:4 = 1.25. Thus, a "wider" third.

Put another way:

x Hz * 5/4 * 5/4 * 5/4 < 2x Hz

1 Other ratios are sometimes used, but the principle is the same.

• The temperament in the question is based on the need to reconcile four perfect fifths with one major third, not to reconcile three major thirds with one octave. Nov 22 '21 at 3:49
• @phoog The question asked is "what is a 'wide third'". I gave an example. Answering the homework question is up to the OP. Nov 22 '21 at 3:50
• Should it be "ratio between F5 and A5"?
– Emil
Nov 22 '21 at 7:08
• @Emil Fixed. Thanks. Nov 22 '21 at 7:09