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All I know about the circle of fifths is that if one starts at C then by going up successive fifths one gets to keys with one sharp, two sharps etc. namely G,D,A,E etc. And if one goes down succesive fifths one gets to keys with one flat, two flats etc, namely F, B flat, E flat etc. (At the botton of the circle there are the enharmonic keys like D flat and C sharp.)

But why does this happen? It obviously can't just be a coincidence.

The gut reaction would be that this has something to do with the harmonic series and the fifth being the first overtone other than the octave, but this is not entirely true. It actual has more to do with the construction of the diatonic scale (all scales don't have this property). The scale consists of the intervals 1-1-½-1-1-1-½ (whole and half steps). Now if write down this sequence and then start another similar sequence on the fifth note we get:

As we can see, the notes fall in the same space, except for the second to last note in the scale starting on the fifth. Now, to look closer at why this is so. The diatonic scale can be spread out as a series of consecutive fifths (F-C-G-D-A-E-B). If you think of the scale like this as a chord consisting of 6 perfect fifths, then it should be clear that if you transpose the whole pile of fifths by one fifth in any direction, six of the notes will remain the same and only the seventh note will change. If you transpose it by one fifth up, then you should remove the lowest note in the pile (F) and add the note one fifth higher than the highest note (F#, which happens to be one semitone higher than the removed note). If you transpose it one fifth down, then you remove B and add Bb which is one fifth lower than F. Hope this made some sence.

So, as a practical matter, what does the circle of fifths teach us? Is it useful to know about it, or to think about it? Is there anything interesting about it beyond the fact of its existence?

Tomasino

I believe the circle of fifths is mostly useful when teaching or learning basic theory. By using the circle of fifths, you can figure out how many sharps or flats a scale should have and also which sharps an flats are included in the scale. Once you know the scales by heart, then I don't think it is very useful anymore. Perhaps there is some other good function to it as well, I don't know because I have never really used it in any way.

It's useful to the extent that composers used it in their works.

For example, Chopin's 24 Preludes and Shostakovich's 24 Preludes and Fugues are both organized by the circle of fifths (C major and its relative minor, then G major and its relative minor, then D major and its relative minor, etc.) That's why there are 24 in each set - that's how many it takes to get all the way through the cycle of fifths from C to C.

There are quite a few pieces of music, such as Mozart K332, which have chord progressions that move through the circle of fifths, and other works have contrapuntal lines that rise or fall by consecutive fifths.

So it's useful not just to see how many composers are incorporating this concept into their works, but also to understand what it is that they are incorporating in the first place.

I look upon it as a visual tool, a reminder of what can be done. I'm a visual learner and used it as a "cheat sheet", memory peg if you prefer, many times when I was just starting on my musical journey.

How to use it has been talked about in the other posts. I used it for a quick glance when remembering the I IV V progressions, or for how many sharps or flats in a key.

Backcycling - inserting a chord into a progression is another use. As just about everything we do goes back to that whole step, half step scenario the circle can tell you many things if you just know where to look.

http://www.ultimate-guitar.com/lessons/the_basics/the_circle_of_fifths_music_theory_for_dummies.html

http://www.google.com/search?source...10&q=how+to+use+the+circle+of+fifths

Malcolm

In the practical world, the circle of fifths is frequently used in improvisations by most jazz players.

The circle of fifths is a lot like the periodic table of elements. It's a useful way of organizing and presenting the information, but not all musicians refer to the circle in their professional work, just as not all chemists need a copy of the periodic table at hand at all times.

Great analogy.

Here's another take on the importance of the circle of fifths: It's more or less the only way of progressing through all 12 keys by a fixed interval.

For instance, suppose you start traversing the keys by major thirds. You start with C, go to E, then to G#... and back to C. You haven't hit them all.

Of course, you can also hit all the keys by traversing the circle of fifths in reverse order; that's the same as going by fourths. And you can hit all the keys by just going chromatically, like Bach does in WTC, but that's sort of a "trivial solution." But those are the only ways.

(Mathematically, there's something interesting going on here. Think of the 12 half-steps from C to C; going up a fifth corresponds to a jump of 7 half-steps; going up a fourth (or down a fifth) is 5 half-steps; going chromatically up or down is 1 and 11 half-steps, respectively. Some basic number theory tells us that in order to hit all the keys, the jump interval has to be relatively prime to 12. (Two numbers are relatively prime if they have only 1 as a common factor. So 7 and 12 are relatively prime; 9 and 12 are not, b'c they have 3 as a common factor.) Now, there are only four numbers between 1 and 12 that are relatively prime to 12: 1, 5, 7, and 11. Accordingly, to traverse all the keys, you must go in fifths, fourths, or chromatically. That's why there's basically only two schemes in the literature: chromatically as in WTC, or by fourths/fifths as in the Chopin preludes and the Shostakovich preludes and fugues.)

Great to have all these different perspectives here. I think the following have not been mentioned yet:

(1) To speak of a circle of fifths depends on the principle of well- (or equal?) temperament, which allows us to identify Gb (6 Bs) with F# (6 #s). Otherwise, we would have a line of fifths in which these two keys would not be considered identical, but 12 fifths apart.

(2) In diatonic modulation (i.e. changing the current reference key using a chord which belongs to both the original key and the new key), the circle of fifths defines the harmonic distance between two keys. If we are using only chords which are built from the material of the original key, only keys with a distance of at most 2 fifths can be reached directly. If we include the material of both major and minor of the original key (c major + c minor), then the maximal distance that can be bridged is 6 fifths (beyond that, one must use intermediate keys, or use other techniques for modulation (e.g. enharmonic/chromatic) which are no longer constrained by distance in the circle of fiths).

(3) beet3142, your explanation brought back some nostalgic memories from my algebra lessons, so let me try to continue your thoughts (it's a long time ago, so I'm a bit rusty and may have mixed up things...):

For any given interval I (e.g. a minor third) we can define a relation ~I on the set of 12 notes X={C,C#,D,D#,...B}, declaring two notes x and y I-equivalent (x ~I y) if they can be reached from each other by stacking up zero or more intervals I (e.g. zero or more minor thirds).

We have in fact defined an equivalence relation (reflexive, symmetric, transitive).

Now for any such equivalence relation ~I we can define the quotient set X/(~I) consisting of all sets of I-equivalent notes.

For example, for I="minor third", the quotient set exactly consists of the 3 different diminished seventh chords that exist:
X/(~minor third) = {{c, d#, f#, a},{c#, e, g, a#},{d, f, g#, b}}
This of course holds generally: if the different equivalence classes all have the same (here 4) number of notes, the size of the quotient set is the size of X (12 notes) divided by the size of any equivalence class (here 4) --- in other words, 12 notes / 4 equivalence classes = 3 diminished seventh chords.

Now, for I="fifth", ALL notes are actually I-equivalent, so there is only one equivalence class defining a "monster chord" consisting of all 12 tones:
X/(~fifth) = {{c,c#,...,b}}
As beet3142 mentioned, ~fifth = ~fourth = ~minor 2nd = ~major 7th, and there's no other I that induces just a single equivalence class of all 12 notes.

Question (just to see if anyone is reading this ): What if I="major third"?

Another thing: some composers have occasionally used more complex notation than enharmonically necessary, e.g. the use of C# major (7 #s) in WTC books 1 and 2.

Oddly, P&F 8 from WTC 1 has the prelude in E flat minor (6 b) and the fugue in d# minor (6 #)

Provided you are working within equal temperament, I think there are four equivalent classes for I = "Major Third":

Set 1: C E G#
Set 2: C# F A
Set 3: D F# A#
Set 4: D# G B
(and the enharmonic equivalents).

Just out of curriosity, how would you describe pivot modulation in the terms you are using for (2)? That is going from a C Major Chord, holding just E and pivoting to E major off the common tone. I finished listening to the Teaching Companies course on Beethoven's Piano Sonatas and there were many examples of this type of modulation discussed in his music.

Rich

Yes! The four augmented triads!



As you know from the Composer's Lounge, I'm just a fellow learner as well, so others feel free to correct.

Short answer: AFAIK this modulation technique is not related or constrained by the circle of fifths. Rather (again AFAIK), you are much more free in terms of what key can be reached from what other.

Longer answer: The German term for pivot modulation is tonzentrale Modulation ("center tone", "pivot" - two different images for the same idea). In this technique you fix a melody tone from the original key and keep it (with possible enharmonic identification). Below this, the harmonic progression leads chromatically into a chord of the target key, to which the fixed note belongs. This is closely related to(translated from German - same term in English?) chromatic (or enharmonic) dominant modulation in which some elements of a chord are altered chromatically, turning the chord into a dominant function whose target key can be changed by enharmonic identification. Due to the chromatic element in this technique, there is no notion of "circle of fifths distance" anymore (as opposed to diatonic modulation).

Edit: I didn't know there's a TTC course on Beethoven's sonatas. Was it good? Some TTC stuff is really great. I found "Listening to and enjoying classical music" nice but mainly for people without much previous acquaintance. The Beethoven sounds very interesting.

Circle of fifths

Pianovirus--

1. Nice explanation with equivalence classes; sounds correct to me. The only thing I'd say is that, in general, equivalence classes don't have to be the same size, so in general you can't compute the number of different classes by dividing the total by this common size. In this particular case they're all of the same size (basically because, in addition to regular equivalence class structure, we have a group structure as well, the additive group of integers mod 12).




2. The pattern I've noticed is this: For a given key, the number of flats it takes for that key signature plus the number of sharps it takes add up to 12.

For instance, Db major has 5 flats; the enharmonic key C# major (which, as you point out, Bach uses), has 7 sharps. 5+7=12.

This means that for almost every key, there's a "natural" choice in whether to do it in flats vs. sharps-- just choose the one that takes less flats or sharps. The exception, of course, is when the #flats and #sharps are both 6: F# major/Gb major and d# minor/eb minor. For these keys, there's no good reason to pick one way vs. the other way.

I think that's why Bach chooses both in WTC, at least with the minor keys. It's not hard to find other examples of compositions representing both choices. (e.g. Beethoven's 24th sonata is in F# major, but the Schubert impromptu is in Gb major.)



Trivia Question:

By the above analysis, since g# minor has 5 sharps, its enharmonic form a-flat minor has 7 flats, and therefore we should use g# minor. However, Beethoven wrote a movement of one of his piano sonatas in the wild key of a-flat minor. Which one?

pianovirus,

I thought the Beethoven Sonatas course was very good, I've listened to it in my car twice now, but need to go through it with the scores so I can see the things Robert Greenberg is talking about in greater detail. Most lectures cover 2 Sonatas, but Hammerklavier gets two lectures by itself. A total of 24 lectures. It is definitely worth listening to.

So far, I've listened to Bach and the High Baroque (except for the last few lectures), most of the Great Composers series, and Beethoven's Piano Sonatas. I'm listening to How To Listen to Great Music right now, and I pretty much agree with you - it is good but not great. (However, I think Robert Greenberg is always entertaining!)

Rich

Thanks, Rich! I'll consider buying that TTC course.



It is a work which is unconventional not only due to this tonality. It is also unconventional as a 4-movement sonata without a single movement in sonata form

While beet31425's quiz is still open, may I add another work with 7 #s - a virtuoso etude... It is a bit overshadowed by other etudes in its opus, and the whole opus is (unjustly) overshadowed by another Etude opus of the same composer. Together, these two opera are very much related to the subject of this thread

It's the second movement, "Marcia Funebre sulla morte d'un Eroe," of the Sonata Op. 26 (Sonata No. 12).

Regards,

Hmmm, would that be Alkan's Contrapunctus, Op. 35 No. 9, with the Op. 35 set ("12 Etudes in all the Major Keys") being overshadowed by Op. 39 ("12 Etudes in all the Minor Keys")?

Steven