Easy way to draw a circle of 5ths

Subject: Easy way to draw a circle of 5ths
From: josepp
Date: 25 Apr 11 - 08:20 PM

Real simple. You don't have to memorize nuttin'. Even if you've never drawn one in you're life, you can do it right the first time. Just follow these ridiculously easy steps:

1. Draw a circle--and it doesn't have to be perfect so don't take all day, huh?

2. Then bisect the circle from the 12 o'clock position to the 6 o'clock position.

3. Then quadrate the circle with another line going from 9 o'clock to 3 o'clock. Then draw more lines from 7 o'clock to 1 o'clock, 10 o'clock to 5 o'clock and so on. When you're done, the circle should be divided into 12 equal slices.

4. It's customary to put C at the 12 o'clock position (although you don't have to). So put a C there.

5. Follow the C line to the 6 o'olock position. What goes there? We don't know yet. We're more concerned with what goes at the 5 o'clock and 7 o'clock positions. And that's easy--it's the notes that flank C which are B and Db. It's customary to keep the flats on the left side of the circle so put the Db at 7 o'clock and the B at 5 o'clock.

6. Now you have two more notes to work with. Pick the B (although you don't have to) and follow its line to the 10 o'clock position. What notes flank that position? The two notes that flank B which are C (already filled in) and Bb. So put Bb at the 10 o'clock position.

7. What about the Db? Follow its line to the 1 o'clock position. What notes flank Db? D and C. C is already filled in so put D at the 2 o'clock position.

8. Just keep doing this until all 12 keys are filled in. It will look something like this:

Circle of 5ths

Done.

Subject: RE: Easy way to draw a circle of 5ths
From: The Fooles Troupe
Date: 25 Apr 11 - 08:33 PM

Except that you have omitted some of the note names because of where you stopped going around the circle. Keep going around long enough and you will eventually end up with a B#.



:-P


:-)

Subject: RE: Easy way to draw a circle of 5ths
From: Geoff the Duck
Date: 26 Apr 11 - 06:13 AM

Anyone who says they are going to tell you something easy, than starts with Bisect and Quadrate (which in my dictionary means a square or rectangle and doesn't mean anything to do with dividing a circle) obviously doesn't understand the word EASY!
Quack!
GtD.

Subject: RE: Easy way to draw a circle of 5ths
From: Jack Campin
Date: 26 Apr 11 - 06:46 AM

The Reid Musical Instrument Museum in Edinburgh has a 19th century boxwood slide rule on display specially made for calculating pitches in just intonation, meantone and equal temperament. I haven't made one yet, but I have an idea in my head for how to make something much more sophisticated.

What I would like to do is compare just intonation, equal temperament, and a whole bunch of other tuning schemes, like 19-tone equal temperament (approximates one of the Baroque meantone schemes), 31-tone ET (another Renaissance/Baroque scheme), 53-tone equal temperament (the system used in Turkish art music theory), the 22-shruti unequal temperament systems of Indian music, the 5-tone equally tempered Indonesian slendro scale, Harry Partch's 41-tone unequal temperament and so on.

Raw material: the transparent uncoated disks you get as the cover for packs of CDRs.

Mark using a protractor, by geometric constructions like josepp's, or (for the tricky ones like 53-tone ET) take two disks, guess, rotate, interpolate, copy, guess again... bisection interpolation converges very fast, so this should only take about three go-rounds to produce 53 equal divisions of the circle to within the limit of human visual perception.

Stack the disks up and you have a calculator that answers questions like: roughly how flat or sharp you have to play when trying to play in a 7-tone equally tempered African scale using an equally tempered Western flute and microtonally tweaked fingerings?

Subject: RE: Easy way to draw a circle of 5ths
From: The Fooles Troupe
Date: 26 Apr 11 - 08:21 AM

Stirling Idea Jack!

Personally though, I'd recommend, a cup of tea, a Bex, and a good lie down!

:-)

Subject: RE: Easy way to draw a circle of 5ths
From: josepp
Date: 26 Apr 11 - 12:28 PM

/////Bisect and Quadrate (which in my dictionary means a square or rectangle and doesn't mean anything to do with dividing a circle)/////

Bisect simply means to divide into two equal halves. I used quadrate as a verb although it is properly an adjective which means to be square as in 90 degrees. I used it as in "to make square" as in 90 degrees. Your usage of quadrate--a cublical volume or square area--is a noun which I clearly did not use.

Sorry if that boggled your little mind.

Subject: RE: Easy way to draw a circle of 5ths
From: Jack Campin
Date: 26 Apr 11 - 12:36 PM

Actually what you want to do is divide the circle in 6 and then divide each of those sectors in half. Much more efficient. Any elementary text on Euclidean geometry will explain it.

BUT. Accuracy is irrelevant for that one. No use you'll ever put the diagram to requires that the marks be evenly spaced. Whereas for the calculator I described, precision DOES matter.

Subject: RE: Easy way to draw a circle of 5ths
From: Doug Chadwick
Date: 26 Apr 11 - 12:38 PM

.....

2. Then bisect the circle from the 12 o'clock position to the 6 o'clock position.

3. Then quadrate the circle with another line going from 9 o'clock to 3 o'clock. Then draw more lines from 7 o'clock to 1 o'clock, 10 o'clock to 5 o'clock and so on. When you're done, the circle should be divided into 12 equal slices.

.....

Why not say "Draw a clockface". I think most people would understand that.

DC

Subject: RE: Easy way to draw a circle of 5ths
From: Jack Campin
Date: 26 Apr 11 - 12:44 PM

Or just download one. A google image search for "circle of fifths" gives you lots of pretty pictures.

Subject: RE: Easy way to draw a circle of 5ths
From: Will Fly
Date: 26 Apr 11 - 12:52 PM

Indeed - including the picture that Josepp has made a clicky for...

Subject: RE: Easy way to draw a circle of 5ths
From: Mysha
Date: 26 Apr 11 - 02:55 PM

Hi,

My three-step simple (use pencil and eraser):
- Draw a clock face:
- Write C at 12 O'clock; write C# at 1 O'clock; write D at 2 O'clock; ...; write B at 11 O'clock.
- Exchange what you wrote at 1 O'clock with that (opposite) at 7 O'clock, likewise 3 and 9; likewise 5 and 11.

It doesn't breed understanding, but for just drawing it works.

Bye,
                                                                  Mysha

Subject: RE: Easy way to draw a circle of 5ths
From: GUEST,999
Date: 26 Apr 11 - 03:34 PM

Easier still.

Subject: RE: Easy way to draw a circle of 5ths
From: josepp
Date: 26 Apr 11 - 06:54 PM

///Why not say "Draw a clockface". I think most people would understand that.////

You obviously understood it's a clockface from what I wrote, that's why I can't understand the other fellow's attitude.

Subject: RE: Easy way to draw a circle of 5ths
From: josepp
Date: 26 Apr 11 - 07:00 PM

////Or just download one.////

When I took exams in music courses, I couldn't download one without being accused of cheating--which in fact it would have been. I had to draw one from memory. People were devising all sorts of silly schemes: "Clark Gables's Dog Ate Ellen's Big FiSh...."

No need for remember something you'll eventually forget. With my method, you can reproduce a circle of 5ths on request no matter how much time has passed since you last did it. And that's a tad more impressive than shrugging and saying, "Sorry, I don't have my computer here to download one."

Subject: RE: Easy way to draw a circle of 5ths
From: Will Fly
Date: 26 Apr 11 - 07:17 PM

To be honest, it's just one of those things that, when you've learned it, like the intervals in a scale or the notes on the staves, you never forget it. It's just there - part of playing music regularly - and I don't need to draw a diagram or download one to remember it.

Where a diagram is useful is if you're explaining it to someone else for the first time - but I just draw it logically, like a clock.

Subject: RE: Easy way to draw a circle of 5ths
From: Doug Chadwick
Date: 26 Apr 11 - 07:18 PM

You obviously understood it's a clockface from what I wrote, that's why I can't understand the other fellow's attitude.

I knew where I was heading. I'm with Geoff - you made something easy sound unnecessarily complicated.

DC

Subject: RE: Easy way to draw a circle of 5ths
From: josepp
Date: 26 Apr 11 - 07:51 PM

Suppose you want to find the relative minor of each major key using the circle. We know that the relative minor is always a minor 3rd down but there's an easier way. Suppose you're in E major on the circle. Take a straight line to the center of the circle then go 90 degrees clockwise to Db (4 o'clock to 7 o'clock). This works with any scale on the circle.

Why not not just count up three positions? Sure. You can do that. But the circle actually shows you the mathematical relationship between notes and scales.

Subject: RE: Easy way to draw a circle of 5ths
From: josepp
Date: 26 Apr 11 - 08:00 PM

////I knew where I was heading. I'm with Geoff - you made something easy sound unnecessarily complicated.////

Complicated for idiots.

Subject: RE: Easy way to draw a circle of 5ths
From: Dave MacKenzie
Date: 26 Apr 11 - 08:03 PM

"the mathematical relationship between notes and scales"

Not quite. Last time I looked, the only time powers of two equalled powers of three was when both are zero.

Subject: RE: Easy way to draw a circle of 5ths
From: Deckman
Date: 26 Apr 11 - 11:42 PM

The last time someone told me it was "real simple" ... I WOKE UP MARRIED!

Subject: RE: Easy way to draw a circle of 5ths
From: JohnB
Date: 27 Apr 11 - 11:52 PM

So! I have a pie cut up into twelve pieces, what do I do with it now?
JohnB.

Subject: RE: Easy way to draw a circle of 5ths
From: The Fooles Troupe
Date: 28 Apr 11 - 02:21 AM

Stick it in your....




































Oven and warm it up, so it tastes nicer.....

Subject: RE: Easy way to draw a circle of 5ths
From: Will Fly
Date: 28 Apr 11 - 04:04 AM

A diagram of the cycle of 5ths is useful as a teaching aid - particularly if you can show someone that a lot of basic jazz chord progressions (for example) are based on the principle of moving from the tonic chord to a point on the circle, and then working back round the circle to the tonic chord. "Sweet Georgia Brown", "Ain't She Sweet" and a thousand others of their ilk fit this style.

However, once the learning phase is over, the important thing is to understand those chord relationships on the instrument - particularly if you're playing jazz.

Example: One of the tunes in the rep of the jazz outfit I played in, many years ago now, was "All The Things You Are" (Jerome Kern) which - if you know it - you'll remember drags you kicking and screaming through the circle of 5th in no uncertain terms! The 'standard' key for that tune is Ab - and that's what we played it in. One evening, a visiting tenor player (the late Danny Moss) sat in with us and chose that tune to play - in Eb! As custom dictated, the guest had the choice of tune and key.

Now, you could attempt the transposition - on the fly - by visualising the cycle of 5ths... But, actually, it's much better to know the relationship of chords in a progression on your instrument - in my case, guitar. I always visualise chord progressions as a series of geometric paths across the fretboard across, up and down the fretboard. Instant transposition then becomes a different starting point on the geometric path. Sounds complex, but experience makes it OK after time.

If you're interested, here's the usual (simplified) chords for "All The Things You Are" in Ab:

Fm / / / Bbm / / / Eb / / / Ab / / /
Db / / / G7 / / / C   / / / C   / / /
Cm / / / Fm / / / Bb7 / / / Eb / / /
Ab / / / D7 / / / G   / / / G   / / /
Am / / / D7 / / / G   / / / G   / / /
Am / / / B7 / / / E   / / / E+ / / /
Fm / / / Bbm / / / Eb / / / Ab+ / / /
Db / / / Dbm / / / Ab / / / Abdim / / /
Db / / / Eb9 / / / Ab / / / Bbm / C7 /

As you can see, you get taken through the chords in a beautiful way. Now - without opening your book - transpose and play instantly on the guitar... in Eb.

Subject: RE: Easy way to draw a circle of 5ths
From: Will Fly
Date: 28 Apr 11 - 04:29 AM

Forgot to say, here's a contemplative version of it I made about two years ago - in F!

All The Things You Are

Subject: RE: Easy way to draw a circle of 5ths
From: Tootler
Date: 28 Apr 11 - 05:02 AM

I once made a similar thing to help with transposing tunes. It consisted of two disks with a split pin connecting them at the centre. Both disks were marked with all twelve keys (you could arrange them as the circle of fifths if you wish). You simply lined up the keynote of original key the piece was in and the new key and read off the notes you should use. I found it very useful as I was always making mistakes when transposing and it helped to reduce errors.

Subject: RE: Easy way to draw a circle of 5ths
From: Will Fly
Date: 28 Apr 11 - 05:13 AM

When I started off on guitar, I used to use two identical strips of paper - just moved one along to the key I needed and then read off the chords/notes!

Subject: RE: Easy way to draw a circle of 5ths
From: GUEST,Grishka
Date: 28 Apr 11 - 10:12 AM

Here (last post) is my paperless transposing device.

Subject: RE: Easy way to draw a circle of 5ths
From: GUEST,Desi C
Date: 28 Apr 11 - 01:59 PM

But what is it for?? I don't get it!

Subject: RE: Easy way to draw a circle of 5ths
From: GUEST,Grishka
Date: 28 Apr 11 - 06:17 PM

Desi, probably you don't need it at all.

A linear transposer calculates transposed note names (also of chords) and / or positions of capos (transposing devices for guitars and similar instruments).

Circles of fifths sometimes serve the same purpose, whence the connection. More importantly, they are used to visualize harmonic relationships, since the main harmonic functions, to wit tonic, dominant, and subdominant, of a key are neighbours there.

Subject: RE: Easy way to draw a circle of 5ths
From: The Fooles Troupe
Date: 28 Apr 11 - 11:54 PM

Write out one scale, the starting one to transpose from, on a piece of paper, left to right.

Below it, not of note write out the target scale.

That's the way I was taught it, and what helped me pass all my music theory exams. Don't need to remember anything special, just how to 'do' scales.

Subject: RE: Easy way to draw a circle of 5ths
From: The Fooles Troupe
Date: 28 Apr 11 - 11:55 PM

That's 'note for note

Subject: RE: Easy way to draw a circle of 5ths
From: josepp
Date: 29 Apr 11 - 06:53 PM

We could also draw lines to connect chords note to note along the perimeter of the Circle and we now have the geometrical shapes of the chords—majors are triangles, 7ths are trapezoids, some suspendeds form perfect squares, etc. The Circle of 5ths enables us to unite music and geometry.

Subject: RE: Easy way to draw a circle of 5ths
From: Songster Bob
Date: 29 Apr 11 - 11:30 PM

The suggestion above of putting C at 12 o'clock, and C# at 1 o'clock, etc., misses the point something grand. C should be flanked by the 5th (clockwise) and 4th (anticlockwise) -- G at 1 o'clock, and F at 11. Then from the G, go 5 to D (2 o'clock), then A (3 o'clock), etc. If you follow it around to 6, you get B. Then use the anticlockwise count by 4ths from C to F (11 o'clock) to Bb (10 o'clock), Eb (9), Ab (8), and Db (7).

The result is a chart that gives you the I, IV, and V chords in three adjacent locations, and the relative minor for any chord is always 4 numbers clockwise from the starting position. To do jazzy "circle of fifths" numbers, start anywhere, go clockwise three* or four** numbers, and move back (widdershins) in one-number increments to the starting position.

And, as it happens, if you start with C at zero, the number of each note is the number of sharps (clockwise) or flats (anti...) in the key. So G is 1 in the sharp direction, and F is one in the flats. G has one sharp, and F one flat. D is two sharps, Eb is three flats, etc.

So it's a circle (clockface) of 5ths (going clockwise).

Bob

* Don't Let Your Deal Go Down, for example
** Five Foot Two, for example

Subject: RE: Easy way to draw a circle of 5ths
From: The Fooles Troupe
Date: 29 Apr 11 - 11:37 PM

And Bob, that IV I V pattern is the secret of the Stradella Bass for Piano Accordions. It's also been implemented thus on other instruments, as well as for the left hand on those small 'chord organs' with a piano keyboard.

Subject: RE: Easy way to draw a circle of 5ths
From: Mysha
Date: 30 Apr 11 - 03:01 AM

Hi,

You're welcome.

Bye,
                                                                  Mysha

Subject: RE: Easy way to draw a circle of 5ths
From: josepp
Date: 30 Apr 11 - 05:40 PM

You can also determine what key a piece is in empirically with a Circle of 5ths.

Suppose you have 3 flats in the signature. By the rules of music theory, you know that the order of the flats from left to right have to be B E A. First, we go to the Circle and bisect it such that B E A are flatted within a half-circle. In this case, we would draw the line from D to Ab. There are only three flats in the key signature and those have to be it and those note designations are now contained within a half-circle going clockwise and are the only flatted notes in it. Now we obtain the key by rotating clockwise to the next note on the circle from Ab, which is Eb. So the key signature with three flats is Eb major (or C minor).

Likewise, the other half-circle also forms a scale whose key is obtained by the same clockwise rotation. The other half-circle contains three sharps—F# C# G# (Ab)—and the key signature on the chart with three sharps is A, which is also clockwise from D.

What about no sharps or flats? We would draw a line from F to B, and isolate the unflatted and unsharped notes within a half-circle. C is clockwise from F so the key is C major. The other half-circle is the flatted side and so we use only the flatted note designations and there are six within the half-circle starting at Cb (B). Going clockwise from Cb to the next note on the circle is Gb and so that is the name of key signature with six flats in the scale.

Wouldn't it be easy to just memorize the keys? Of course, it is. That's not the point. The point is, that this information is contained with the Circle of 5ths if you know how to read it.

Subject: RE: Easy way to draw a circle of 5ths
From: josepp
Date: 30 Apr 11 - 05:51 PM

////6. Now you have two more notes to work with. Pick the B (although you don't have to) and follow its line to the 10 o'clock position. What notes flank that position? The two notes that flank B which are C (already filled in) and Bb. So put Bb at the 10 o'clock position.////

I wrote this in the opening post but it is in error. No doubt, some of you caught it. If you follow B across to the opposite side of the circle, you end up at the 11 o'clock position and not the 10 o'clock position. Sorry if that caused any confusion.

Subject: RE: Easy way to draw a circle of 5ths
From: josepp
Date: 30 Apr 11 - 06:02 PM

/////But what is it for?? I don't get it!////

I'm not sure who you're asking, Desi, but if it's me, the Circle of 5ths has many uses. It shows you all your available scales lined up such that all their relationships are visually demonstrated. It's fun to go through one and find the hidden relationships. As you can see from reading the posts here, there are as many ways to use it as they are musicians in the world.

When a scheme as this is truly useful, is when if can be employed by a technology that didn't exist until long after the scheme was developed. Here's an example:

How DJs use the Circle for mixing

The method of employing the Circle of 5ths here is totally different from what a classical musician might employ it for but it's equally useful either way. The Circle of 5ths is a thing of true beauty.

Subject: RE: Easy way to draw a circle of 5ths
From: Jack Campin
Date: 30 Apr 11 - 06:13 PM

A variant of that can be used for the layout of multiple diatonic moothies on a spindle. A common arrangement is to have six of them, laid out so the key of neighbouring moothies are a fifth apart: Bb, F, C, G, D, A.

Iain Grant

I know another player who carries around a set of 12 diatonic moothies on a spindle like that, one for every key in the chromatic scale. Weighs a ton.

Subject: RE: Easy way to draw a circle of 5ths
From: josepp
Date: 30 Apr 11 - 06:55 PM

How can you construct the proper diatonic scales from each note on the circle?

There's a number of ways. One way is to use this pattern: major 3rd, minor 3rd, major 3rd, minor 3rd, minor 3rd, major 3rd, minor 3rd.

For example, let's use C. Starting at C we go up a major 3rd (or 4 half-steps) to E. Then go up a minor 3rd (or 3 half-steps) to G. Then go up another major 3rd to B. Then another minor 3rd to D. Then another minor 3rd to F. Then a major 3rd to A. Then a minor 3rd to C'. So the order is:

C E G B D F A C'

Now arrange these notes starting at C in a circle in alphabetical order: C (C') D E F G A B and draw lines starting from C to each note in the order obtained as shown above. You get a 7-pointed star! And this works with any key and whether major or minor.

Pick another key--say Ab. The order we obtain is:

Ab C Eb G Bb Db F Ab'

Now arrange them in a circle in alphabetical order and then draw a line from one note to the next in the sequence above--a 7-pointed star.

Another demonstration of how music and geometry are related.

Subject: RE: Easy way to draw a circle of 5ths
From: josepp
Date: 30 Apr 11 - 07:13 PM

////Ab C Eb G Bb Db F Ab'

Now arrange them in a circle in alphabetical order and then draw a line from one note to the next in the sequence above--a 7-pointed star.///

Or you can make a circle of the notes in the above sequence instead alphabetically and then connect the notes alphabetically with lines and you'll get an even cooler looking 7-pointed star.

Try it!

Subject: RE: Easy way to draw a circle of 5ths
From: The Fooles Troupe
Date: 01 May 11 - 05:21 AM

Quote
How can you construct the proper diatonic scales from each note on the circle?

There's a number of ways. One way is to use this pattern: major 3rd, minor 3rd, major 3rd, minor 3rd, minor 3rd, major 3rd, minor 3rd.
Unquote

My God, child, do you comb your hair with a pin?

Major Scale

Tone, Tone, Semi-Tone, Joining Tone, Tone, Tone, Semi-Tone.

The others are similar, but have different patterns. Send me money and I will give you these arcane secrets.

Subject: RE: Easy way to draw a circle of 5ths
From: The Fooles Troupe
Date: 01 May 11 - 05:28 AM

From the link

"This is the ring we'll be using the most. "

Hmmm, I'm very tempted to say something rude ... but no, I will be good ....

Subject: RE: Easy way to draw a circle of 5ths
From: McGrath of Harlow
Date: 01 May 11 - 08:44 AM

It shows you all your available scales lined up such that all their relationships are visually demonstrated.

Speaking pedantically, it doesn't - only the standard major and minor. There are any number of other scales available.

Subject: RE: Easy way to draw a circle of 5ths
From: josepp
Date: 01 May 11 - 11:11 AM

////Speaking pedantically, it doesn't - only the standard major and minor. There are any number of other scales available.////

Speaking pedentically, you mean modes and not scales.

Subject: RE: Easy way to draw a circle of 5ths
From: josepp
Date: 01 May 11 - 11:35 AM

////Major Scale

Tone, Tone, Semi-Tone, Joining Tone, Tone, Tone, Semi-Tone.////

You can't draw the 7-pointed star from it. That's significant because the circle unites music and geometry.

Here's the beauty of it:
We obtain the C major scale my way in the following order:

C E G B D F A

Notice that it actually forms a chord. CEG is C major. Add the B next and you get a Cmaj7. Add the D and that's a Cmaj9. But you can start at the E and adding the G and B and EGB is the E major. Add the B and EGBD is the E minor 7. Start at G and GBD is the G major and so on. And this works for all the various diatonic scales on the Circle.

What's also noteworthy is that the sequence contains how notes are laid out on manuscript paper. FACE represents the spaces between the lines on the treble staff and EGBDF represents the staff lines. Needless to say, it also describes the layout of the bass staff: ACEG GBDFA respectively.

but what's even cooler is that no matter what scale on the Circle you plot out with my method, the order of the notes follows that layout if we ignore the accidentals for the moment. For example, I mentioned the Ab flat scale:

Ab C Eb G Bb Db F

Ignore the accidentals and there it is again ACE GBDF

Pick another scale--say F:

F A C E

Subject: RE: Easy way to draw a circle of 5ths
From: josepp
Date: 01 May 11 - 11:40 AM

Sorry, I pressed the submit button by mistake.

Let's try it again:

Pick another scale--say F:

F A C E G Bb D F'

And there it is again: FACE EGBDF

It will always work out that way. It will always show the layout on staff paper and form a 7-pointed star. Doesn't matter if you use the equivalent minors and it doesn't matter if you use enharmonic equivalents. It always produces the same pattern.

There is all kinds of cool info locked up inside the Circle of 5ths if you know how to find it.

Subject: RE: Easy way to draw a circle of 5ths
From: JohnInKansas
Date: 01 May 11 - 12:55 PM

There are a couple of oversimplifications in argumens above:

1. A couple of references were made to "divide the circle into twelve equal arcs." There are 13 notes in the chromatic scale so the division must be triskadecate, not duodecate.

2. In conventional "theory" there's no such thing as a "circle of fifths. Each "movement clockwise" adds one sharp to the key signature, or cancels one flat. Each movement counterclockwise adds one flat or cancels one sharp. For a "theoretically pure" graphic, you must draw a spiral of fifths so that the "keys" of C#, C## etc are properly shown. The notion that you can "fall back an octave and substitute the enharmonic notation" doesn't work accurately except for equi-tempered tunings.

For an equi-tempered scale, a circle suffices, but in just temperament you DO NOT arrive back at the same "C" (or any other) pitch after each transit of the "circle." It is that difference that needs to be shown/understood, and an accurately constructed "spiral of fifths" will show it quite clearly.

Once the concept of the spiral is understood, it can be rendered immaterial whether the octave is encompassed in exactly one turn around the (non-existent) circle, and you can use any "scale" (unit of mensuration) to represent the frequency ratio of adjacent "notes" in some arbitrarily selected "unit of arc/angle" (creating a sort of logarithmic spiral of pitches). Drawing any two "kinds of scales" on opposite sides of the same "curl" of the spiral will accurately show the disagreements between the tunings.

If you can discard the notion that a "circle" has some mystical significance, you can easily create a linear (straight line) slide rule for comparisons between tunings, although for a "complete set" of scales encompassing suffient octaves to show the differences clearly it may be a rather long stick.

John

Subject: RE: Easy way to draw a circle of 5ths
From: josepp
Date: 01 May 11 - 01:12 PM

////. A couple of references were made to "divide the circle into twelve equal arcs." There are 13 notes in the chromatic scale so the division must be triskadecate, not duodecate.////

The Circle of 5ths arranges the major/minor scales not the chromatic notes. Pedantically speaking.

///2. In conventional "theory" there's no such thing as a "circle of fifths. Each "movement clockwise" adds one sharp to the key signature, or cancels one flat. Each movement counterclockwise adds one flat or cancels one sharp. For a "theoretically pure" graphic, you must draw a spiral of fifths so that the "keys" of C#, C## etc are properly shown. The notion that you can "fall back an octave and substitute the enharmonic notation" doesn't work accurately except for equi-tempered tunings.////

Yes, it is a spiral. So is a true octave but try playing your favorite folk song in that tuning at a coffeehouse.

////If you can discard the notion that a "circle" has some mystical significance, you can easily create a linear (straight line) slide rule for comparisons between tunings, although for a "complete set" of scales encompassing suffient octaves to show the differences clearly it may be a rather long stick.///

Yeah, yeah, I've been to wikipedia too.

Subject: RE: Easy way to draw a circle of 5ths
From: josepp
Date: 01 May 11 - 01:25 PM

///For an equi-tempered scale, a circle suffices, but in just temperament you DO NOT arrive back at the same "C" (or any other) pitch after each transit of the "circle." It is that difference that needs to be shown/understood, and an accurately constructed "spiral of fifths" will show it quite clearly.///

For just tuning. Who cares? You talk like just tuning is more correct than equal tempered. You have all kinds of tunings besides those two: True tuning, modal tuning, twelve fifths tuning. Since I'm talking about the Circle of 5ths, it should be painfully obvious that I have no interest in these other tunings. You can keep talking about just tuning if that floats your boat but it really has nothing to do with anything thus far being discussed and rightfully belongs on another thread.

Subject: RE: Easy way to draw a circle of 5ths
From: Jack Campin
Date: 01 May 11 - 07:57 PM

The point is that the ET circle of fifths relates to other temperaments, and in fact the only use for a precise construction is if you're going to calculate such comparisons. So it is perfectly reasonable to be interested in both at once. (As I pointed out in my first post in this thread - that boxwood slide rule dated from the 1880s and there are tables equivalent to it in Helmholtz's book, this is not a new idea).

There is no such thing as a "modal tuning". I presume "twelve fifths tuning" is a subliterate description of Pythagorean intonation.

Just intonation has lots of well-established uses in folk idioms worldwide. The commonest musical instrument in the world uses just intervals.

Subject: RE: Easy way to draw a circle of 5ths
From: Smokey.
Date: 01 May 11 - 09:53 PM

"TWELVE TRUE-FIFTHS TUNING: Maria Renold (1917-2003) was a professional viola-player who, playing in string quartets, struggled with tunings for decades, until in 1962 she hit on this tuning, which some just call the Twelve Fifths Tuning. The point of it however is that in today's equal-tempered system, none of the 5ths are true natural 5ths as found in the overtone series, whose frequencies are exactly in the proportion 3/2. That is because a piano-tuner tunes each note a 5th above the previous note, in what is called the circle of 5ths. However, if he uses natural 5ths, he will find on returning after 12 steps to the beginning note, he has overshot a little bit. Hence he makes each 5th slightly smaller to make them fit. Thus we never hear the pure, ringing 5th on a keyboard instrument. Choirs and string players constantly adjust a little to make it possible.

Briefly, in this tuning, you tune the white notes on the piano—the 7 notes from F to B—in natural 5ths. You then find the exact middle of the C octave—F#/Gb—and tune the black notes—the next 5 notes—from there, again in natural 5ths. You thus have two series of natural 5ths, linked by a new interval, called a "formed 5th" between B-F#, and Bb-F.

This interval however satisfies because it is based on one of the three systems underlying all real scales in all cultures—the overtone series, the undertone series, or equal divisions; or in mathematical terms—arithmetic, harmonic or geometric proportion, respectively. Apparently, intervals derived from valid intervals are also valid. Thus, the equal division of the octave in half created these two valid bridging 5ths. Ultimately, our inner experience is the judge.

There is a further refinement, however, she discovered later, and which, when added, makes the tuning what its fans call "Renold II". That is, remarkably, that one can slightly expand the 5ths—and hence the octaves—slightly past the mathematical interval and find a satisfying interval that still works.

This tuning is based on an indication by Rudolf Steiner, on whom she based all her work."

Subject: RE: Easy way to draw a circle of 5ths
From: The Fooles Troupe
Date: 01 May 11 - 11:32 PM

There is so much gibberish and mystical nonsense in this thread, it perhaps deserves the BS label!

"Since I'm talking about the Circle of 5ths, it should be painfully obvious that I have no interest in these other tunings"

You may think you know everything, and may have read Wiki - but this statement reveals that you have no understanding! If you need to ask WHY, that proves you know naught!

Quote
Tone, Tone, Semi-Tone, Joining Tone, Tone, Tone, Semi-Tone.////

You can't draw the 7-pointed star from it. That's significant because the circle unites music and geometry.
Unquote

You have a love of mysticism - but musicians do not need mystical gibberish to make nice noises. Some merely need to open their mouth, but some should perhaps keep it shut.

JiK
"A couple of references were made to "divide the circle into twelve equal arcs." There are 13 notes in the chromatic scale so the division must be triskadecate, not duodecate."

John - I thought you knew about computing science - you have committed the 'fencepost error' - 13 notes (posts) have 12 intervals (panels) in a straight line, but the same number ()12 in a circular fence! AND by convention, the octave is the same as the fundamental.

Bloody hell, I'm off to get more peanuts!

Subject: RE: Easy way to draw a circle of 5ths
From: Don Firth
Date: 01 May 11 - 11:42 PM

As someone formally trained in music theory, I find much of this thread, particularly the attempted forays into mysticism, unduly complicated.

Confusion for the innocents!!

Don Firth

Subject: RE: Easy way to draw a circle of 5ths
From: The Fooles Troupe
Date: 01 May 11 - 11:50 PM

Clue: The reason it's called a circle of 5ths has nothing to do with Bb, et al.... It's built of 5ths....

WHY?

Because scales (Western Music only!) are built of Tetrachords..

Subject: RE: Easy way to draw a circle of 5ths
From: The Fooles Troupe
Date: 02 May 11 - 02:29 AM

"As someone formally trained in music theory"

Ditto Don. But since we are not trained in Mysticism, obviously we know nothing.... :-)

Subject: RE: Easy way to draw a circle of 5ths
From: The Fooles Troupe
Date: 02 May 11 - 08:40 AM

For those of us who know the Theory and done a fair bit of Practice...

This Circle is probably more relevantly Pragmatic

Subject: RE: Easy way to draw a circle of 5ths
From: Don Firth
Date: 02 May 11 - 04:08 PM

Hey, josepp!

CLICKY

Subject: RE: Easy way to draw a circle of 5ths
From: Smokey.
Date: 02 May 11 - 09:33 PM

Interesting how it all ties in with astrology and the twelve disciples - I'm surprised no-one's noticed that. See, I went to an English grammar school and the curriculum at that time included mysticism, pedantry and sarcasm.