Home
> Music Theory Questions and Answers
> Topic

Posted by despues18

circle of fifths construction and use July 27, 2017 04:01PM |
IP/Host: 96.42.46.--- Registered: 6 weeks ago Posts: 11 |

Re: circle of fifths construction and use July 27, 2017 04:27PM |
IP/Host: 76.249.226.--- Registered: 8 years ago Posts: 210 |

The circle is composed of about 12 notes because 3/2^12 = 129.746.... or about 128 = 2^7. One starts with 1, then multiplies by 3/2 (the ratio of a fifth), then divides by 2 to reduce the result to be between 1 and 2 (if necessary). If this sequence of numbers be sorted, the results are about equally spaced and can be used as 12 evenly spaced notes. The error between 3/2^12 and 2^7 must be distributed somehow which leads to the topic of temperament. Other possibilities exist, but this one "sounds nice" and this seems to the reason there are 12 notes (in what's termed the Western Music system).

It's a cycle; it can start anywhere. For various historical reasons, keyboards are laid out so that the note named C is the start of one particular pattern.

It's a cycle; it can start anywhere. For various historical reasons, keyboards are laid out so that the note named C is the start of one particular pattern.

Re: circle of fifths construction and use July 28, 2017 09:50AM |
IP/Host: 96.42.46.--- Registered: 6 weeks ago Posts: 11 |

Re: circle of fifths construction and use July 29, 2017 06:05AM |
IP/Host: 166.137.126.--- Registered: 4 months ago Posts: 29 |

It's very, very close to 128, which is 7 octaves up. If we say it's close enough, then that means we're back to the same note we started at. So if we keep going up by that 3/2 ratio each time, the cycle just repeats. We don't get any more new notes. Another way of saying it is that you can only go up by 3/2, over and over, 12 times, before you land on a note that's (almost) exactly some number of octaves above the note you started on.

The (almost) bit has frustrated mathematician musicians for centuries. The error is called the Pythagorean Comma (yeah, THAT many centuries!). How to deal with it in terms of tuning and harmony has been a topic of much debate. Today, we pretty much always use what's called an even temperament, where we build our scale by shrinking all of the intervals just a teeny little bit so that we don't land on 129.x, when we go around the cycle, but on 128 exactly. The result is that the error is spread out across all of the intervals evenly. So, all the notes are actually out of tune with each other, but juuuuust barely, so that we don't really notice (Unless we listen for it. If you're a guitar player, your probably familiar with the scenario of tweaking your tuning so that, say, your C chord sounds absolutely perfect, only to find that your F now sounds like crap. I could write pages and pages about this frustration, but I won't!)

So that's how we get from a perfect fifth to the twelve tone system. The next question is "Why 3/2?" I don't think we know exactly why it sounds good (I certainly don't) but we do know what it is that sounds good about it. It's in the way it beats. Wth a 3/2 ratio for every 3 cycles of the higher note, ther are 2 cycles of the lower note. That means the two waves go back and forth between being exactly in phase, and exactly out of phase on every other cycle of the lower note. When the two are in-phase, which is to say moving together in the same direction, they work together and sound louder together. When they are out of phase, they kinda cancel each other out and sound a little softer together. The result is audible and we call it a beat frequency, kind of a wah-wah-wah-wah effect. (Not as in wah wah pedal for guitar that's different). The interesting point with 3/2 is that since this louder-softer cycle is happening on every other cycle of the lower note, that means the pitch of the best is exactly one octave below the lower note. For some reason our brains tend to really like this sound! Same thing happens with the octave, but the beat is at exactly the same frequency as the lower note, and so can't even be heard. That's why the octave sounds so "pure" and it's why we say the upper tone is the same note. Since there's no audible beating our brain tells us the two notes are really the same.

I don't know for sure, but I believe this beating an octave below the lower tone explains why "down a fifth" movement is the single most powerful sense of resolution we have in western music. Those beats at the same note as the lower tone of the interval tell us that's the "real" note in this interval, and that the fifth above is an intruder in what would otherwise be perfect octave. There are lots of other things that make a dominant (5 chord) to tonic (1 chord) movement so powerful, but I think these only reinforce an effect that is already there...

Hopefully interesting...

-J

Edited 3 time(s). Last edit at 07/29/2017 06:26AM by jjjtttggg. (view changes)

The (almost) bit has frustrated mathematician musicians for centuries. The error is called the Pythagorean Comma (yeah, THAT many centuries!). How to deal with it in terms of tuning and harmony has been a topic of much debate. Today, we pretty much always use what's called an even temperament, where we build our scale by shrinking all of the intervals just a teeny little bit so that we don't land on 129.x, when we go around the cycle, but on 128 exactly. The result is that the error is spread out across all of the intervals evenly. So, all the notes are actually out of tune with each other, but juuuuust barely, so that we don't really notice (Unless we listen for it. If you're a guitar player, your probably familiar with the scenario of tweaking your tuning so that, say, your C chord sounds absolutely perfect, only to find that your F now sounds like crap. I could write pages and pages about this frustration, but I won't!)

So that's how we get from a perfect fifth to the twelve tone system. The next question is "Why 3/2?" I don't think we know exactly why it sounds good (I certainly don't) but we do know what it is that sounds good about it. It's in the way it beats. Wth a 3/2 ratio for every 3 cycles of the higher note, ther are 2 cycles of the lower note. That means the two waves go back and forth between being exactly in phase, and exactly out of phase on every other cycle of the lower note. When the two are in-phase, which is to say moving together in the same direction, they work together and sound louder together. When they are out of phase, they kinda cancel each other out and sound a little softer together. The result is audible and we call it a beat frequency, kind of a wah-wah-wah-wah effect. (Not as in wah wah pedal for guitar that's different). The interesting point with 3/2 is that since this louder-softer cycle is happening on every other cycle of the lower note, that means the pitch of the best is exactly one octave below the lower note. For some reason our brains tend to really like this sound! Same thing happens with the octave, but the beat is at exactly the same frequency as the lower note, and so can't even be heard. That's why the octave sounds so "pure" and it's why we say the upper tone is the same note. Since there's no audible beating our brain tells us the two notes are really the same.

I don't know for sure, but I believe this beating an octave below the lower tone explains why "down a fifth" movement is the single most powerful sense of resolution we have in western music. Those beats at the same note as the lower tone of the interval tell us that's the "real" note in this interval, and that the fifth above is an intruder in what would otherwise be perfect octave. There are lots of other things that make a dominant (5 chord) to tonic (1 chord) movement so powerful, but I think these only reinforce an effect that is already there...

Hopefully interesting...

-J

Edited 3 time(s). Last edit at 07/29/2017 06:26AM by jjjtttggg. (view changes)

Re: circle of fifths construction and use July 29, 2017 06:16AM |
IP/Host: 108.65.89.--- Registered: 8 years ago Posts: 210 |

The frequency ratio of a perfect fifth (or Pythagorean Fifth) is 3 to 2. Thus compounding 12 fifths gives the number 129.746.... Compounding 7 octaves (ratio 2 to 1) gives 128. Much of temperament is an attempt to approximate some number of octaves by some number of other intervals. The traditional guitar tuning EADGBE approximates 2 octaves by 4 fourths and a third. The latter ratio is 320/81 and two octaves would be 320/80 so things are fairly close. Three major thirds gives a ratio of 125/64 approximating one octave as do four minor thirds with a ration of 1296/625. Nothing can ever come out even.

Re: circle of fifths construction and use July 29, 2017 03:27PM |
IP/Host: 96.42.46.--- Registered: 6 weeks ago Posts: 11 |

The reason why I ask is because I understand that 12 semi tones are within a octave. I thought that the circular fifths would be reflecting the notes in a chromatic scale which I guessed summarily was just these repeating sequences of A through G notes on a scale for frequency. 3/2 the frequency successively 12 times would obviously have a large enough effect to land us outside of merely (2x or 1/2) the frequency. So now that we know we are outside of one individual octave, I have a hard time imagining what a diagram would look like if we rep.'d the frequency as a # line and all the individual notes that would constitute a key as points on that # line.

Sorry for the confusion, I'm trying to develop a better mathematical understanding of this process

Sorry for the confusion, I'm trying to develop a better mathematical understanding of this process

Re: circle of fifths construction and use July 29, 2017 08:21PM |
IP/Host: 166.137.126.--- Registered: 4 months ago Posts: 29 |

You are quite right as you multiply by 3/2 over and over again, you do go beyond an octave (with the second step, in fact). To put/keep everything together in the same octave, you just have to remember you can always divide by 2, to go down an octave and you get the same note. So, think of a keyboard instead of talking just numbers, let's say we start with middle C, i.e. C4. If we go up a fifth, that takes us to G4. Cool same octave. Now let's go up (by 3/2) again. Now we're at D5. Oops, now we're in the wrong octave. But that's ok. It's a D. Let's just go down an octave (i.e. Divide by 2). Now we're at D4 and we're good again. Up a fifth means A5. It's a 5 instead of a 4, but it's still below C5, so we're still in the same octave. Up again takes us to E5. Wrong octave again, so divide by 2 to take us down to E4.... And so on around the cycle until we eventually go up from F4 to C5 to complete the twelve steps.

Does that make sense. Mathematically, after multiplying by 3/2, if the result is greater than 2, then you divide it by 2 to get back into the first octave above the starting note.

-J

Does that make sense. Mathematically, after multiplying by 3/2, if the result is greater than 2, then you divide it by 2 to get back into the first octave above the starting note.

-J

Re: circle of fifths construction and use July 30, 2017 09:39AM |
IP/Host: 96.42.46.--- Registered: 6 weeks ago Posts: 11 |

"Does that make sense. Mathematically, after multiplying by 3/2, if the result is greater than 2, then you divide it by 2 to get back into the first octave above the starting note."

Okay so you do that for all the notes that you would find in the circle of fifths. As long as you divide by 2 @ the end of whatever progression that is required you'll be in the correctly corresponding octave despite whatever operation put you outside of the balance for one individual octave.

I found an explanation online that was very confusing but maybe not for you (When it comes to why there are 12 steps in a key),

titled: Why 12 notes to the Octave?

url includes math uwaterloo

simple google search brings the correct url up

The only interesting tidbit I would have to offer for that might be that we use the # 60 within the context of time for a very practical reason. It is the whole integer value between 1 and 100 that can be subdivided in the Highest # of whole integer subdivisions. The # 12 is one of the factors for 60, poss. due in this instance to a similar reason.

Okay so you do that for all the notes that you would find in the circle of fifths. As long as you divide by 2 @ the end of whatever progression that is required you'll be in the correctly corresponding octave despite whatever operation put you outside of the balance for one individual octave.

I found an explanation online that was very confusing but maybe not for you (When it comes to why there are 12 steps in a key),

titled: Why 12 notes to the Octave?

url includes math uwaterloo

simple google search brings the correct url up

The only interesting tidbit I would have to offer for that might be that we use the # 60 within the context of time for a very practical reason. It is the whole integer value between 1 and 100 that can be subdivided in the Highest # of whole integer subdivisions. The # 12 is one of the factors for 60, poss. due in this instance to a similar reason.

Re: circle of fifths construction and use July 30, 2017 10:50AM |
IP/Host: 166.137.126.--- Registered: 4 months ago Posts: 29 |

I can't say that the 60 and 12 thing is definitely not related, but I don see how it would be either musically or mathematically.

It really isn't terribly complicated. Two sounds having frequencies a factor of 3/2 apart sound good to the ear/brain because the beat frequency is an octave below the lower note. So, if we start with some frequency and go up by that factor, and divide by 2 whenever we go more than one octave above our starting note, we can produce a series of notes that are all within the same octave, and all different until we try to make the 13 note. When we try to make that 13th note, it comes out so close to our original note as to be effectively the same. We could keep going, but we'd just get notes that are very close to the first twelve we made.

The fact that the thirteenth note isn't EXACTLY the same as the starting point leads to lots and lots of more complicated discussion, but for all practical purposes it is the same, so the answer to the "Why 12?" question is really just that simple. "Up by 3/2" sounds good, and if you do it over and over you get exactly 12 truly distinct notes.

-J

Edited 1 time(s). Last edit at 07/30/2017 10:51AM by jjjtttggg. (view changes)

It really isn't terribly complicated. Two sounds having frequencies a factor of 3/2 apart sound good to the ear/brain because the beat frequency is an octave below the lower note. So, if we start with some frequency and go up by that factor, and divide by 2 whenever we go more than one octave above our starting note, we can produce a series of notes that are all within the same octave, and all different until we try to make the 13 note. When we try to make that 13th note, it comes out so close to our original note as to be effectively the same. We could keep going, but we'd just get notes that are very close to the first twelve we made.

The fact that the thirteenth note isn't EXACTLY the same as the starting point leads to lots and lots of more complicated discussion, but for all practical purposes it is the same, so the answer to the "Why 12?" question is really just that simple. "Up by 3/2" sounds good, and if you do it over and over you get exactly 12 truly distinct notes.

-J

Edited 1 time(s). Last edit at 07/30/2017 10:51AM by jjjtttggg. (view changes)

Re: circle of fifths construction and use July 30, 2017 11:14AM |
IP/Host: 96.42.46.--- Registered: 6 weeks ago Posts: 11 |

Re: circle of fifths construction and use August 13, 2017 06:14PM |
ModeratorIP/Host: Moderator Registered: 5 years ago Posts: 75 |

The number 84 can be divided into as many whole number pairs as can the number 60.

1x84, 2x42, 3x28, 4x21, 6x14, 7x12 = 6 pairs, versus 1x60, 2x30, 3x20, 4x15, 5x12, 6x10 = 6 pairs.

The Circle of Fourths can also generate all 12 notes in the octave, in a similar way to the Circle of Fifths.

The Circle of Fourth extends for 60 notes, 5 octaves, while the Circle of Fifths extends for 84 notes, 7 octaves.

The octave seems to be more fundamental than the Perfect Fifth, so all the Fifths have to be ajusted slightly downwards in order to reach parity with the final C.

I've often wondered why people sing using twelve divisions in the octave, rather than 10 divisions or 14 or any other number of notes. Dividing the octave by twelve seems to be the most natural and flexible way to make music.

The octave naturally creates the twelve notes of music. It should be the natural unit of organization of music as well.

1x84, 2x42, 3x28, 4x21, 6x14, 7x12 = 6 pairs, versus 1x60, 2x30, 3x20, 4x15, 5x12, 6x10 = 6 pairs.

The Circle of Fourths can also generate all 12 notes in the octave, in a similar way to the Circle of Fifths.

The Circle of Fourth extends for 60 notes, 5 octaves, while the Circle of Fifths extends for 84 notes, 7 octaves.

The octave seems to be more fundamental than the Perfect Fifth, so all the Fifths have to be ajusted slightly downwards in order to reach parity with the final C.

I've often wondered why people sing using twelve divisions in the octave, rather than 10 divisions or 14 or any other number of notes. Dividing the octave by twelve seems to be the most natural and flexible way to make music.

The octave naturally creates the twelve notes of music. It should be the natural unit of organization of music as well.

Re: circle of fifths construction and use August 13, 2017 07:52PM |
IP/Host: 96.42.46.--- Registered: 6 weeks ago Posts: 11 |

Sorry, only registered users may post in this forum.