Understanding the Circle of Fifths and Its Diatonic Ratios

[raolduke]

The circle of fifths operates around relative diatonic scales- FCGDAEB
The reasoning is because it sounds good to the human ear at a ration of 3:2. How do you go about this?

 

[HallsofIvy]

The first thing you would do is look at the frequencies of each of those notes. What frequency change is a "fifth"?

 

[Crosson]

What frequency change is a "fifth"?

A "fifth" describes the ratio of two frequencies; when the lower tone is followed by the higher tone this is called an "interval", in this case a fifth.

The reasoning is because it sounds good to the human ear at a ration of 3:2. How do you go about this?

If you mean "how do you explain this asthetic preference?", I would say to seek the answer in philosophy.

 

[MeJennifer]

The circle of fifths operates around relative diatonic scales- FCGDAEB
The reasoning is because it sounds good to the human ear at a ration of 3:2. How do you go about this?

First of all it is not a circle. Mathematically it cannot be.

Actually most tonal systems are composed of a combination of thirds and fifths.

In the just intonation scale with C as the root the F, C, G and D are indeed pure fifths. However the A is the third above the F not the fifth above the D, also the E and B are thirds. Also the E-flat is the fifth above the A-flat.

In the Chinese scale the A is actually the fifth above the D.

Of course in equal temperament scales there are no real thirds and fifths everything is basically out of tune, the 12 tone values are simply averaged and their values are irrational.

You could also compose a tonal system which includes sevenths, seventh sound like "blue notes". The eleventh is heared in Mongolian music (it is almost exactly between the F and F-sharp of the just intonation scale). Thirteenths are sometimes heared in Arabic music, but it is very hard to distinguish from quarter tones. Quarter tones are common in Arabic music.

If we start from C the pure fifths with their ratios are C (1/1), G (3/2), D (9/8), A(27/16), E (81/64), B (243/128). The F is (4/3).

Compare the A above with the A in just intonation which is A (5:3), the E in just intonation is (5:4) and the B (15:8).

The equal temperament tones are all irrational numbers, so there are no ratios.

 

[Crosson]

First of all it is not a circle. Mathematically it cannot be.

A pitch class is a set of all pitches that are a whole number of octaves apart, e.g. the pitch class C consists of the Cs in all octaves. Then the directed graph with these pitch classes as vertices, and with an edge from x to y if y is the musical fifth of x, is a cyclic graph i.e. its natural planar embedding is a circle.

I could think of many other ways to justify the title "circle of fifths", so it is far from the case that it "mathematically could not be".

 

[light_bulb]

http://en.wikipedia.org/wiki/Werckmeister_temperament

have fun.

 

[MeJennifer]

A pitch class is a set of all pitches that are a whole number of octaves apart, e.g. the pitch class C consists of the Cs in all octaves. Then the directed graph with these pitch classes as vertices, and with an edge from x to y if y is the musical fifth of x, is a cyclic graph i.e. its natural planar embedding is a circle.

I could think of many other ways to justify the title "circle of fifths", so it is far from the case that it "mathematically could not be".

Crosson, it is most definitely not a cyclic graph.

You can stack an infinite number of fifths on top of each other but you will never get an exact multiple of the primary tone. Look up Pythagorian comma or Mercator's comma to see how to fudge it into a cyclic graph.

If you don't believe me then try solving:

$$\frac{(3/2)^n}{2^m} = 1$$

for $n, m \ne 0$

 

[Crosson]

Crosson, it is most definitely not a cyclic graph.

Thank you for the lesson, I was not aware of this as a guitar player --- no wonder I am out of tune no matter what .

 

[uart]

Crosson, it is most definitely not a cyclic graph.

You can stack an infinite number of fifths on top of each other but you will never get an exact multiple of the primary tone. Look up Pythagorian comma or Mercator's comma to see how to fudge it into a cyclic graph.

If you don't believe me then try solving:

$$\frac{(3/2)^n}{2^m} = 1$$

for $n, m \ne 0$

But it's not really (3/2) in that equation is it. It is actually

$$2^{7/12}$$

in our 12 semitone geometric system.

 

[MeJennifer]

By the way if you want to hear pure sevenths, listen to gospel singer Mahalia Jackson.

 

[HallsofIvy]

Thank you for the lesson, I was not aware of this as a guitar player --- no wonder I am out of tune no matter what .

Because a guitar has frets, it, like a piano, is a tempered instrument. That is, it is tuned so that the there is no difference between a sharp and b flat. If you played a violin (which "has no frets") you would distinguish between them and recognize that the "circle of fifths" doesn't quite match up.

 

[MeJennifer]

If you played a violin (which "has no frets") you would distinguish between them and recognize that the "circle of fifths" doesn't quite match up.

Only if the violinist plays 12 pure fifths! Most violinists would not be capable of doing that. They primarily use the equal temperament and just intonation scales. Some specialists would be capable of using the various baroque tunings and obviously a good violinist would be capable of adjusting a note if the music, even temporarily, modulates into another key, but all these things won't help in playing a "circle" of pure fifths.

For instance if the player plays the complete twelve-tone sequence starting from C-G-D-A-E -B...and on in the just intonation scale he would only reproduce 3 pure fifths from that "circle", he would only play F (as the fifth below), G and D as perfect fifths. All the other 9 tones would not be pure fifhts from that "circle", these tones would be a mixture of thirds and relative fifths. Obviously if he plays such a sequence in an equal temperament scale he will actually play zero pure fifths.

The western just intonation scale is based on both fifths and thirds. And of course, as in the case of octaves never matching up with stacked fifths, fifths cannot ever match up with stacked thirds.

Music with pythagorean tuning, which is a tuning based on perfect fifths only, is rare, we have to go back to the 13th century to see examples. Pythagorean tunings have no thirds, so if you play in the key of C the chord C-E-G will sound very awkward.