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More stuff about the circle of 5ths
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Subject: More stuff about the circle of 5ths From: GUEST,josepp Date: 24 Mar 12 - 01:35 PM Yeah, more stuff about the goddamn circle of 5ths so shut yer bloody, fookin' pieholes! The circle of 5ths has tremendous uses and I lay no claim that I know every way in which it is or can be used. I'm sure some people here have forgotten more ways to use it than I'll ever learn but I like to surf the net and find blogs, websites and such where people find their own uses and cool relationships just by screwing around with it. I'll repeat a few things I've posted previously that, for no reason, sparked HUGE arguments and angry responses. Hopefully, we can avoid the ranting that seems to typify so much of online communication these days. First thing--how to draw a circle of 5ths without memorizing a thing. 1. Draw a circle and make it as accurate as you can--a lopsided bloop won't do. Use a compass or trace around a plate or something. 2. Now put 12 marks on the circle as though it were a clockface. One mark where each number would be. Again, be as precise as you can. If it helps, the marks should be 30 degrees of arc apart from one another. 3. Now, you can choose any letter from A to G and start at any mark you want. But since all the circles of 5ths that you see have C at the 12 o'clock position, we'll do the same. So place "C" at the 12 o'clock position. 4. Now, draw a line (it's better if the line is imaginary but whatever makes you happier) from the 12 o'clock position to the 6 o'clock position. What note goes at 6 o'clock? We don't know, empirically speaking. What we can do, though, is mark the two notes that flank C which are B and Db (that's D-flat). These will go on either side of the 6 o'clock position and just go clockwise, lower note to higher note--B at the 5 o'clock position and Db at the 7 o'clock position. So fill those two spots in with those notes. 5. Now you have two more notes to work with. Pick one--let's just pick B at the 5 o'clock position. Again, go to the opposite side of the circle, which is the 11 o'clock position. We don't know which note goes there so we'll determine what two notes flank B which are Bb and C. Now C is already filled in at the 12 o'clock position so fill in Bb at the 10 o'clock position. 6. Now just keep going around the circle like this until you fill in all 12 marks. Now you may ask why we use Bb in step 5 instead A# (A-sharp). We want to use only flats BUT you can use sharps. It's just easier to keep the "black key" notes flats. The exception is at the 6 o'clock position where F# is often preferred over Gb and we'll go over examples. But enharmonic equivalents are just different designations for the same note and either can be used if you are so inclined. But when you are done, it should look something like this: http://www.google.com/imgres?hl=en&safe=off&biw=1041&bih=398&gbv=2&tbm=isch&tbnid=kqLfcDp94cyXYM:&imgrefurl=http://www.hearandpl [this URL sets of malware warnings, so I'm going to unlink it - visit this URL at your own risk - Joe Offer] Now, you notice this circle has minors written on it. The notes were filled in represent major scales but every major scale has an equivalent minor. How do we know what it is? Drop back a minor 3rd interval (or 3 half-steps) from the major scale note designation. Take C major: 3 half-steps back is A (C down to B to Bb to A). So the equivalent minor scale of C major is A minor. Using the circle of 5ths, we can find the minor by moving clockwise up 3 letters. Look at the circle and demonstrate for yourself. Another way find the minor is to for a 90-degree angle clockwise on the chart. It helps to have a keyboard in front of you to count up and down the half-steps. Lastly, I want to make clear that I am talking STANDARD Western music theory here. I don't care about quarter-tones and all that crap. Draw the circle as outlined above many times until it comes easily to you. The circle of 5ths doesn't do you a lot of good if you don't know it by heart. Also all the little ways of using it that I'll talk about won't make much sense if you don't know the layout of the circle. |
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Subject: RE: More stuff about the circle of 5ths From: Big Al Whittle Date: 24 Mar 12 - 01:40 PM Its easy for you to say that. |
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Subject: RE: More stuff about the circle of 5ths From: Barbara Date: 24 Mar 12 - 01:40 PM Clicking on the above link for the circle set off my malware protection system. JoeClones take note. Blessings, Barbara |
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Subject: RE: More stuff about the circle of 5ths From: Jeri Date: 24 Mar 12 - 01:48 PM Here it is, I think: from www.hearandplay.com Josepp's like was to a link found with a Google search, which involves a re-direct. |
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Subject: RE: More stuff about the circle of 5ths From: Barbara Date: 24 Mar 12 - 01:57 PM Yep, no problems this time. Thanks, Jeri. Blessings, Barbara |
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Subject: RE: More stuff about the circle of 5ths From: GUEST,josepp Date: 24 Mar 12 - 03:10 PM On the link I gave to the circle (which may set off some malware alarms but it's an ok link), you may have noted that it has the number of sharps and flats noted in parentheses. What does that mean? Most every scale has certain notes that will either sharped or flatted. The reason is because if we don't sharp or flat the notes, the scale will be out of tune. That doesn't mean that note will ALWAYS be played that way in that scale. Sometimes, it will be played natural and will be specially designated for that purpose but it is otherwise to be played flat or sharp depending on the scale. The scale degrees are simply numbered 1-8 with 1 and 8 being a full octave (12 half-steps). So the D major scale has two sharps according to that chart. Why? Let's lay it out: D (1), E (2), F# (3), G (4), A (5), B (6), C# (7) and D' (8). Notice that there are half-steps between 3 and 4 and also between 7 and 8. The rest are whole steps. That's how a major scale is laid out. So in D major, there are two sharps—F# and C#. Now the equivalent minor of D major is B minor (remember, we move up three note designations clockwise on the circle). Why do we tie it to D major? Because it uses the same notes. The difference is that while a major scale has half-steps at 3 and 4 and also at 7 and 8, the minor scale has half-steps at 2 and 3 and also at 5 and 6. Let's lay it out: B (1), C# (2), D (3), E (4), F# (5), G (6), A (7) and B' (8). Notice that B minor has the exact same notes sharped in its scale—F# and C#. Now you may ask why we go from B to C# in the above example. Why not to C? Because at 1 and 2, we need a whole step. B to C is a half-step. If you look at a keyboard, you'll see that between B and C there is no intervening black key. The same is true of E and F. A white key and black key next to each other is a half-step. But if there is no black key, then the two white keys at the point are a half-step apart. All the other white keys are whole steps apart. Let's try another scale: Eb major which has three flats according to that chart. Which notes in that scale are flatted? Eb (1), F (2), G (3), Ab (4), Bb (5), C (6), D (7) and Eb' (8). So there are our flats in Eb major—Eb, Ab, Db. Now we count up three notes on the circle clockwise to get the equivalent minor of Eb major and we see it is C minor. It should have the same notes as Eb major. Let's see: C (1), D (2), Eb (3), F (4), G (5), Ab (6), Bb (7) and C' (8). Once again, the same flats—Bb, Eb and Ab. So to keep from having to write flats in front of every B, E and A in a piece, we use a key signature. The order of the sharps in the key signature is FCGDAEB while the order of the flats in the key signature is BEADGCF so one is simply the reverse of the other. The thing to keep in mind is that if, say, D is flatted in the key signature then so are B, E and A. D will never be flatted by itself. The only note that can be flatted by itself in the key signature is B and this order must always be followed. Same goes for the sharps. If A is sharped in the key signature, then so are F, C, G and D. Only F can be sharped by iself. The reason is that, once again, the scale would be out of key if we didn't do this. |
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Subject: RE: More stuff about the circle of 5ths From: GUEST,josepp Date: 24 Mar 12 - 03:15 PM Circle of 5ths showing key signatures: http://playtheaxe.com/theory/lesson09/circle_of_fifths.gif |
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Subject: RE: More stuff about the circle of 5ths From: GUEST,josepp Date: 24 Mar 12 - 03:27 PM Sorry, something got screwed up in my post: "Let's try another scale: Eb major which has three flats according to that chart. Which notes in that scale are flatted? Eb (1), F (2), G (3), Ab (4), Bb (5), C (6), D (7) and Eb' (8). So there are our flats in Eb major—Eb, Ab, Db." Of course that should read Bb, Eb and Ab. Sorry for any confusion. |
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Subject: RE: More stuff about the circle of 5ths From: GUEST,josepp Date: 24 Mar 12 - 03:52 PM So how can we use the circle to notate the key signature? We know that C major has no key signature and that G on the clockwise side has one sharp and that F on the anticlockwise side has one flat and that it goes in sequence, e.g. D major has two sharps, A major has three sharps, Bb major has two flats, Eb major has three flats and so on. I showed how to figure out which notes are sharped and flatted for each scale by plotting it out. But can we shortcut the process by using the Circle? Yes, we can! To determine the flats, drop back two notes on the circle counterclockwise. G major gets one sharp so drop back two spaces on the circle to F major and that is the note that gets sharped in G major. Or you can draw a line from G to the center of the circle and go back 60 degrees counterclockwise. Moving to D major, which has two sharps, drop back 60 degrees to C and that is the next sharp so D major has two sharps—F and C. And so on. Remember that the sharps and flats accumulate so add the ones previous. For flats, we look at the note on the opposite side of the circle. So the note opposite F is B so that is the note that gets flatted in F major. Or we can drop back (or move forward) 180 degrees to find the flats. The next note of the circle is Bb and will get two flats. We already know that one will be B and the other is 180 from Bb which is E so Bb major is flatted at B and E. And so on. Remember sharps and flats accumulate and always follow a specific order and it bears repeating: The order of the sharps in the key signature is FCGDAEB while the order of the flats in the key signature is BEADGCF so one is simply the reverse of the other. And I would like to add that you can go around the circle in either direction but the clockwise direction is counting in 5ths. The counterclockwise direction is counting in 4ths. 4ths and 5ths are simply inverses of each other and 99% of all music is dependent on them. As one writer put it--the 4th and 5th are the twin pillars of music (he must have been a Freemason). |
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Subject: RE: More stuff about the circle of 5ths From: Don Firth Date: 24 Mar 12 - 04:32 PM I bought my first guitar ($9.95 plus $5.00 for a fiberboard case) back in 1952 or '53. I was very lucky with this guitar in a number of ways. The first was that the bloody thing was actually playable! The intonation was on, the frets were accurately placed, and the action was low and soft. It did, however, sound a bit like it was made of apple-crate wood, which it was. The salesman threw in a free pick, which I never used. The second lucky thing was that the salesman tossed in a free copy of Nick Manoloff's Spanish Guitar Method, which included a handy-dandy patented Nick Manoloff Chord Wheel. CLICKY (Click on the images and they increase in size. The second image is the backside of the bottom disk of the chord wheel, for quick reference.) Two cardboard disks attached with a metal eyelet, with "windows" in the top one. Dial a key and it tells you what chords are available in that key. I got curious, so I peeked behind the top disk (they were suffiently flexible) and lo! I got my first lesson in music theory. The chords were arranged in what I learned later was the "Circle of Fifths." It not only gives you chord families, it also shows how the keys are related to each other. It merits a great deal of serious study. I have absorbed all this, reinforced by two years of music theory study at the U. of Washington, and another two years at the Cornish College of the Arts (a sort of conservatory). The Chord Wheel and the Circle of Fifths are not the be-all and end-all of music theory, but knowing the Circle of Fifths is a tremendous help in learning chords and chord relationships, and it's a very important aspect of music theory. I no longer need to use the Chord Wheel (it's all in my head), but I still have it around here somewhere. Don Firth |
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Subject: RE: More stuff about the circle of 5ths From: Jack Campin Date: 24 Mar 12 - 06:02 PM Drawing accurate circles is irrelevant to the concept, and there is VERY little music where it matters that the circle closes. A ladder of fifths is more appropriate for traditional music - also it doesn't require that you use equal temperament. |
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Subject: RE: More stuff about the circle of 5ths From: Don Firth Date: 24 Mar 12 - 06:13 PM Just out of curiosity, Jack, why is a ladder of fifths more appropriate for traditional music than the more commonly used circle of fifths? Don Firth |
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Subject: RE: More stuff about the circle of 5ths From: theleveller Date: 24 Mar 12 - 06:23 PM My malware says there's a trojan in that link. |
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Subject: RE: More stuff about the circle of 5ths From: GUEST,josepp Date: 24 Mar 12 - 07:34 PM Which one? There's several links on here. It would slightly helpful to let us know which one. |
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Subject: RE: More stuff about the circle of 5ths From: GUEST,josepp Date: 24 Mar 12 - 07:48 PM As far as chord wheels go, the circle is tailor-made for sliding the halves around. Good look doing that with a ladder. The circle is helpful with chords, harmonies and bass lines and all for the same reason which we'll get to. Right now, I want to lay a bit of groundwork so that people who have to play catch-up have something to catch up to. |
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Subject: RE: More stuff about the circle of 5ths From: Jack Campin Date: 24 Mar 12 - 08:23 PM The ladder is more appropriate because traditional music doesn't modulate far, never goes round the circle, rarely uses the extreme keys and never presupposes equal temperament, which has to be assumed for the circle to close. The ladder design is tailor-made for making slide rules. Something a bit like it is described in Athanasius Kircher's Musurgia Universalis of 1655 - a set of calculational "bones" designed for harmonizing tunes. |
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Subject: RE: More stuff about the circle of 5ths From: GUEST,Josepp Date: 24 Mar 12 - 08:54 PM Example? |
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Subject: RE: More stuff about the circle of 5ths From: Jack Campin Date: 24 Mar 12 - 09:09 PM There are 745 pages in the book and no search box for the online copies I've seen. |
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Subject: RE: More stuff about the circle of 5ths From: GUEST,Chord Chunker Date: 24 Mar 12 - 09:29 PM Does traditional music modulate keys at all, Jack? Certainly there are modern arrangements of traditional melodies that use modulations, and the chordal accompaniment to a melody with a Lydian scale could sound like it was modulating keys when harmonizing the #4th(using a D chord when playing in C Lydian),but a lot of traditional music is modal and doesn't even shift to a dominant, let alone moving to another key. Feel free to put your 2p in josepp, or anyone-I just have a question, no answer;-) |
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Subject: RE: More stuff about the circle of 5ths From: GUEST,josepp Date: 24 Mar 12 - 09:36 PM Some might wonder, "Since any major and its equivalent minor have the same key signature, how do we know when we look at a piece of sheet music whether it is major or minor?" Well, often we can't just by looking at it. We need to hear the tonal center. Major scales generally lend themselves to happy, grand or sweet melodies. Minor scales generally sound sad, melancholy, haunting. But these descriptions can be deceiving because some songs are not overtly happy or sad sounding. I didn't realize that "Walk On By" by Dionne Warwick was in a minor scale until I absently picked out the scale on a bass guitar just to amuse myself and realized that it was a minor scale. That's why a lot of classical pieces were titled like "Etude in D minor" and like that so that the scale mode was apparent. |
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Subject: RE: More stuff about the circle of 5ths From: Don Firth Date: 24 Mar 12 - 10:07 PM I can't see any advantage of a linear arrangement of keys and related chords over a circular (or cyclical) arrangement, just because the music being played may have limited perameters. The circular ('Circle of Fifths") arrangement shows the true relationship of keys and chords to music in general. A linear arrangement implies limitations that don't exist. More confusion for the innocents. Don Firth |
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Subject: RE: More stuff about the circle of 5ths From: JohnInKansas Date: 24 Mar 12 - 10:31 PM The ladder design is tailor-made for making slide rules ????? Actually a circle is a much better choice for slide rules than a ladder. The real reason most people use them as slip-sticks is that the straight one is easier to carry and a little less likely to get bent, but definitely not because it works better. John |
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Subject: RE: More stuff about the circle of 5ths From: Jack Campin Date: 25 Mar 12 - 06:25 AM The circular ('Circle of Fifths") arrangement shows the true relationship of keys and chords to music in general. It is only true that D flat is enharmonic to C sharp if you are playing stuff like Schoenberg's First Chamber Symphony (which does actually require that). The circular arrangement is making an assertion about tonal relationships that only applies to certain types of music. Does traditional music modulate keys at all, Jack? Just in the Shetland repertoire, "Da Slockit Licht", "St Anne's Reel" and "Miss Susan Cooper". In the Scottish danceband repertoire there are a few dozen commonly played tunes that use the trick of modulating to the dominant and subdominant for whole sections - "The Bluebell Polka" includes an extra trick with a brief move to the dominant of the dominant. |
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Subject: RE: More stuff about the circle of 5ths From: Don Firth Date: 25 Mar 12 - 02:01 PM From a volume on music theory, discussing the matter of pure intonation versus equal temperament: If we take the note Middle A, which by standard convention has a frequency of 440 Hz and we multiply that by the 12th root of two which is 1.05946, we will get a frequency of 466.16376151. That's many more decimal places than we really needIt goes on to say that if one were to tune a fixed pitch instrument, such as a piano or organ—or a guitar—to pure intonation, it would be perfectly in tune in only one key, although one could play it in the two neighboring keys [on the Circle of Fifths] and it would be near enough in tune so that most people would not notice that it is not perfectly in tune. To most musicians, the only time pure intonation might be a matter of importance is in unaccompanied vocal music in which close harmonies are desired, such as a capella choirs and barbershop quartets. Most people simple do not hear the difference. In fact, most well trained musicians do not hear the difference. Most trained singers cannot make the delicate distinctions, especially if they have a slight natural vibrato in their voices, which most people do. Four years of formal musical training in a university music department and in a music conservatory, plus years of private lessons in classical guitar, voice, and composition, and the subject has only come up a couple of times. And then, more as a historical curiousity rather than a matter of any serious concern. Don Firth |
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Subject: RE: More stuff about the circle of 5ths From: GUEST,josepp Date: 25 Mar 12 - 04:23 PM /////It is only true that D flat is enharmonic to C sharp if you are playing stuff like Schoenberg's First Chamber Symphony (which does actually require that). The circular arrangement is making an assertion about tonal relationships that only applies to certain types of music.///// Urm...I would say that the vast majority of music requires that Db be enharmonic to C#. Anything you're going to use a piano or a guitar on is going to require that. I can't comment on Shetland music or anything Scottish--I have zero knowledge of non-American music outside of classical--but every music theory class and lesson I ever took assumes C# and Db are enharmonically equivalent. Anything that doesn't, well, sorry, but I did make clear in the opening post that we're not interested in that stuff here. |
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Subject: RE: More stuff about the circle of 5ths From: Jack Campin Date: 25 Mar 12 - 04:54 PM the only time pure intonation might be a matter of importance is in unaccompanied vocal music in which close harmonies are desired, such as a capella choirs and barbershop quartets. It's essential to the sound of the nyckelharpa, the hammered dulcimer, the Cajun accordion, and almost all diatonic harmonicas (which are probably the commonest musical instrument on earth). These are not subtle effects: it stands out a mile if you don't have the predominant intervals tuned pure. the vast majority of music requires that Db be enharmonic to C#. Anything you're going to use a piano or a guitar on is going to require that And essentially zero percent of traditional music was conceived for the piano or for melodic playing on the guitar. Just show me *one* traditional tune from anywhere and any age that uses enharmonicity. |
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Subject: RE: More stuff about the circle of 5ths From: Don Firth Date: 25 Mar 12 - 05:36 PM There is a large Scandinavian community in Seattle, and I know a couple of people who play the nyckelharpa. There are also a number of people around who play the hammered dulcimer. No Cajun accordions that I'm aware of, but lots of diatonic harmonicas and a couple of chromatic ones. NONE of them are concerned with pure intonation, and there are a few self-taught folks who wouldn't know what you're talking about. It does not affect their ability to play and enjoy some very listenable music. Jack, the point of this thread, as Josepp outlined it in the opening post, is to provide people who are a bit "iffy" on matters of which chords go with which keys--i.e., relative beginners or those without formal training—with an easy way of working it out. In his next-to-last paragraph he made it quite plain that dickering over "quarter-tones" was not to be part of the discussion. I've done a great deal of teaching. And I know that one of the things that can really screw up a relative beginning student is to lumber them with a bunch of unnecessary historical minutiae before they fully understand the basics. Let's not try to confuse the issue, okay? Don Firth P. S. I doubt very seriously that anyone, even with fairly acute hearing, is going to even notice a discrepancy of 38/10,000s of a cycle per second. You're familiar with the fairy tale about "The Princess and the Pea?" The young woman was so refined and sensitive that she could feel a pea through a whole stack of twenty mattresses. The fact that she was so sensitive was proof that she was a genuine princess. Can you imaging living with someone who is that sensitive? |
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Subject: RE: More stuff about the circle of 5ths From: Jack Campin Date: 25 Mar 12 - 06:27 PM NONE of them are concerned with pure intonation If you don't tune the sympathetic strings to pure intervals they don't work. The difference in power is enormous. If the thirds on a mouth organ are tuned to equally tempered pitches it sounds crappy and feeble. Which is why they aren't made that way. one of the things that can really screw up a relative beginning student is to lumber them with a bunch of unnecessary historical minutiae before they fully understand the basics You and josepp are the ones introducing unnecessary theoretical clutter. The idea of closing the circle of fifths by 12-tone equal temperament has exactly as much rationale in traditional music as closing it by 19-tone, 31-tone or 53-tone ET, i.e. none whatever. For folk idioms, you simply don't need to know ET ever existed. It's a piece of 19th century art music theory which is great for understanding Wagner, Schoenberg and Duke Ellington, and completely pointless for anything far outside that range of idioms. (I just flipped through Rameau's "Treatise on Harmony" of 1722, the most influential music theory book of the 18th century; no mention of it there, which is not surprising since the first piece of music to use it in a nontrivial way wasn't written till 100 years later). It doesn't make you play any better, doesn't add any understanding of the music you're dealing with, and it seems it blocks off your perceptions so you can't even hear the sound of a mouth organ right. On the other hand it does make some sense for people to understand ET if they're playing instruments that are such a disastrous misfit to traditional melody as the guitar - simply so they'll know when to just STFU with the out-of-tune noises that blur the overall harmony. |
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Subject: RE: More stuff about the circle of 5ths From: Don Firth Date: 25 Mar 12 - 08:50 PM More confusion for the innocents. As I say, I've been at this awhile. I'm not making it up. The Circle of Fifths is a standard teaching technique used in prestigious music schools such as Berklee, Juilliard, Cornish, and others. Also, many classic guitar technique books include a few pages about the Circle of Fifths, including, of course, a full explanation of chord and key relationships. As to objections to the guitar or other small, portable fixed-pitch instruments being used to accompany traditional songs, I'm afraid taking issue with that is a bit like King Canute ordering the tide to recede. Don Firth |
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Subject: RE: More stuff about the circle of 5ths From: GUEST,josepp Date: 25 Mar 12 - 09:53 PM ////The idea of closing the circle of fifths by 12-tone equal temperament has exactly as much rationale in traditional music as closing it by 19-tone, 31-tone or 53-tone ET, i.e. none whatever.//// It sure does. They've built 31-tone keyboards and they've discovered that while it sweetens intervals currently a bit dissonant in 12-tone, it made other intervals even more dissonant, In fact, the perfect 5th becomes more dissonant in 19-tone and since it plays such an important role in music, we don't want to worsen it. Moreover, the keys had to be narrower otherwise the keyboard would be too long to play which makes it harder to play. They tried switching the positions of the black and white keys which would have been a good solution except keyboardists simply didn't like it. 19-tone is cumbersome and 31-tone would be even more so. They're harder to play, harder to build, harder to tune. 12-tone seems to be the limit that the human mind and body has a natural aptitude for. It allows us to modulate the diatonic scale through twelve keys and gives enough physical space to do it in. It works very well and the secret to understanding how to use it most effectively use it is the circle of 5ths. It's a beautiful scheme. ////For folk idioms, you simply don't need to know ET ever existed.//// That's like saying to understand alchemy you don't need to know a thing about chemistry. Or to understand the humors and the biles you don't need to know a thing about modern medicine. The truth is, your understanding of alchemy is deepened greatly with an understanding of chemistry. Understanding modern medicine helps us to appreciate what the old study of humors and biles was aiming for. ////It's a piece of 19th century art music theory which is great for understanding Wagner, Schoenberg and Duke Ellington, [and virtually any other artist you can possibly name--Josepp] and completely pointless for anything far outside that range of idioms. //// Actually, 12-TET was aspired to in Bach's time, they just didn't have a way to gauge it precisely. And the man credited with coming up with the value of the square root of 2 as the measure of a half-step was a Chinese theoretician named Zhu Zaiyu in 1584 proving that 12-TET was being searched for long before the 19th century and outside of Europe. Going back to Ancient Greece, the god of music was Apollo who was also called Pythias. In Greek isopsephia, where each letter has a numerical value, Apollon adds up to 1059 and Pythias to 1061. The value of the square root of 2 is approximately 1.0595. Coincidence, no doubt. ////It doesn't make you play any better,//// Neither does taking lessons but I'd rather take lessons than learn on my own. ////doesn't add any understanding of the music you're dealing with,//// I flat out disagree with that. It adds a great deal of understanding of the music you're dealing with. It is the very secret of how to utilize that music in the most effective way. ////and it seems it blocks off your perceptions so you can't even hear the sound of a mouth organ right//// I agree with Don here. You're talking differences too minute for most people to care. |
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Subject: RE: More stuff about the circle of 5ths From: GUEST,josepp Date: 25 Mar 12 - 11:05 PM For those, who don't get what Jack Campin is getting at (and maybe I don't either), he's talking about music that existed before the technology of precisely measuring out the square root of 2 as the value of the semitone. Without that value, 12-TET (or 12-tone equal temperament) can't exist. In Bach's day, they used meantone and Kirnburger I, II III and so on. The musical scale is not a circle in its raw form, we have to "temper" it into a circle. Here's some background that may prove helpful: Dodecatonic Scales So if we want to be able to play diatonic scales with total modulation through various keys, we need to add the 7-note scale to the 5-note scale for a 12-note scale: C G D A E B F# C# G# D# A# F. Now we can play all pentatonic and diatonic scales. To get full modulation of the diatonic through all 12 keys would again require equal temperament. So we add another 5th, B#, which is the enharmonic equivalent of C. But since B# and C are the same tone, this scale must already be tempered, right? Wrong. In an octave, C major in this case, the intervals and their ideal ratios are as follows: 1. C – Fundamental (1) 2. C# – Minor 2nd (15/16) 3. D – Major 2nd (8/9) 4. D# – Minor 3rd (5/6) 5. E – Major 3rd (4/5) 6. F – Perfect 4th (3/4) 7. F# – Tritone (5/7) 8. G – Perfect 5th (2/3) 9. G# – Minor 6th (5/8) 10. A – Major 6th (3/5) 11. A# – Minor 7th (5/9) 12. B – Major 7th (8/15) 13. C' – Octave (1/2) If we could arrange these ratios throughout the octave, we'd have the prefect scale but that cannot happen. The true values are not precise but have some small amount of overrun. Pythagoras encountered it long ago in his attempts to construct a perfect scale and it became known as the Pythagorean comma. Mathematicians today prefer to call it a syntonic comma. The Syntonic Comma Now, if we form a circle of the C major octave where the full octave is 360 degrees exactly, then C=0 and 360, D=320 (360 x 8/9), E=288 (360 x 4/5), F=270 (360 x ¾), G=240 (360 x 2/3), A=216, B=192, and C'=180. From D to F is a minor 3rd (3 half-steps) with a ratio of 320/270 or 6.4/5.4 even though the true ratio should 6/5 or 324/270. So the actual difference is 324/320 or 81/80. That ratio is called the syntonic comma. From C to G is a perfect 5th of 360/240 or 3/2. However, if we were to measure a perfect 5th in the next octave from D to A or 320/216 ratio, we notice that it is an 80/54 ratio. A true perfect 5th would be 81/54 or 3/2 and so, again, we end up with a discrepancy of 81/80 (oddly, the reciprocal of this number is 0.987654321). The next octave after that would yield the major 6th (F-D) and the perfect 4th (A-D) also off by the syntonic comma. The comma is very noticeable and must be dispersed in some manner. So even in a perfect 12-note scale, we must temper it to keep the octave at an exact 2:1 ratio because of the syntonic comma. Methods of Dispersing the Comma Method I One way to disperse the comma is through adjusting the major 3rds in the octave. In a 12-tone octave, there are three major 3rds (4 half-steps x 3 = 12 half-steps). A true major 3rd has a 5:4 ratio, that is, if you shorten a string by 4/5 but retain the same tension, the string will play a major 3rd interval higher. If we start at middle C, our three major 3rds would be C-E, E-G#, Ab-C (remember that Ab is the enharmonic equivalent of G#). Since the octave interval must always be an exact 2:1 ratio, the Ab-C major 3rd will be a bit flat. Why? Since a major 3rd is 4/5, then we would cube that ratio to obtain 64/125 for the full octave. But a full octave is 64/128 (1:2 ratio). Since 64/125 represents a longer string length than 1/2, the major 3rds will be noticeably flatter than in a true octave since a string's length is proportional to the pitch. The discrepancy is the ratio of 125/128 or 0.9765625, which is called a diesis. Each major 3rd interval in the octave must be sharpened slightly by a third of the diesis or about 0.3255208333. Method II We may also measure the minor 3rds in an octave of which there will be four. Using C major as an example and starting at middle C, our minor 3rds will be C-Eb, Eb-F#, F#-A and A-C'. Since a true minor 3rd would have 5/6 ratio, then the total value of an octave in minor 3rds is 5/6 raised to the power of 4 or 625/1296. The actual value of a full octave is 648/1296 or 1/2. Since the string length of an octave of minor 3rds is somewhat shorter than a true octave resulting in a higher pitch, the minor 3rds will be a bit sharp and must be uniformly flattened. So, the ratio of 648/625 or 1.0368 tells us the total difference in tone and so each minor 3rd interval must be flattened by a quarter of 1.0368 which is 0.2592. While other ratios are called a comma or diesis, this 648:625 ratio has never been named for some reason. Method III Suppose we measure the 5ths in an octave. There's only one 5th in an octave. Two 5ths will pass out of the octave. So, if we start measuring 5ths, we have to find a way to keep the tones within the octave. Once the 5th is out of the octave, its value must be halved to keep it within the octave. Starting at C, for example, the first 5th interval ends at G and we know that the ratio is 3/2 (or 2/3). The next 5th takes us to D and so we would square 3/2 to obtain 9/4 but that passes out of the octave (is greater than 2 and an octave must be exactly 2/1). So we multiply 9/4 by 1/2 to obtain 9/8, which is less than 2 and so is within the octave. Next, we jump up to A which is mathematically obtained by multiplying 9/8 by 3/2 or 27/16 which is within the octave. If we keep going through 5ths until we pass through all 12 semitones (after A, we go to E, B, F#, C#, G#, D#, A#, F and C) we end up with a final value of 262144/531441. The true octave, however, would yield 262144/524288. Again, the actual length of the string would be somewhat shorter than the true octave string length and so would be sharp. Our differential is the ratio of 531441/524288 and is called the ditonic comma (not "diatonic"). The value of the ratio is approximately 1.01364327. So we would flatten our 5ths by 1.01364327/12 or about 0.08447. This is the method mentioned earlier for tempering the 12-tone scale. |
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Subject: RE: More stuff about the circle of 5ths From: JohnInKansas Date: 26 Mar 12 - 03:59 AM For minutiae with which to confuse the novice, it may be noted that if you go around the circle in the direction of ascending notes it's a circle of fifths; but if you go around it "backwards" in the direction of descending notes it's a circle of fourths. You can lose some students with the conundrum of how the distance between the end points of a fouth down plus a fifth up equals two fourths down or two fifths up, depending on which eye is crossed when you do it. By the time you explain that one, the semester's over and everyone quit the band and is trying out for the "futball" team, or just gave up and downed the fifth a said f***k the fourth. John |
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Subject: RE: More stuff about the circle of 5ths From: GUEST Date: 26 Mar 12 - 04:31 AM Jospepp, it's the Twelfth root of two, not the square root. Square root of two is 1.41421..... Twelfth root of two is the magic fret spacing factor 1.0595 (ish). it's the number you have to multiply by itself 12 times to make two. Space your frets, or pitches by that factor and by the time you have done it twelve times and you are half way between the nut and the bridge, and/or at double the original pitch. That's why it works for a twelve tone equal temperament. 24th root of two would give you a quarter tone scale, A scale based on the square root of two would only contain two notes in the octave! If you are going to base your music theory on mathematics, you must get the maths right! Personally, I find all the circle of fifths concept something of a case of creating a mnemonic which is harder to remember than the thing it is meant to help you to learn, which is the scale relationships of notes and chords in the different keys. Cheers Dave |
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Subject: RE: More stuff about the circle of 5ths From: Jack Campin Date: 26 Mar 12 - 06:11 AM he's talking about music that existed before the technology of precisely measuring out the square root of 2 as the value of the semitone. Music that doesn't use that mathematics didn't go away after it was invented. And it took a long time before anybody could use the mathematical ideal of ET at all - the technology to measure irrational frequency ratios in practical situations didn't exist. The idea of closing the circle of fifths by 12-tone equal temperament has exactly as much rationale in traditional music as closing it by 19-tone, 31-tone or 53-tone ET, i.e. none whatever. It sure does. They've built 31-tone keyboards and they've discovered that while it sweetens intervals currently a bit dissonant in 12-tone, it made other intervals even more dissonant, In fact, the perfect 5th becomes more dissonant in 19-tone and since it plays such an important role in music, we don't want to worsen it. 19-tone ET is a very close match to one of the meantone temperaments commonly used in Baroque music. You don't need to have all those notes on your keyboard to use it; if you're staying on the flat side of the spectrum you can leave the sharps out, or put up with a badly out of tune enharmonic if it only occurs rarely. 53-tone ET is the standard framework used for Turkish art music, but again no instrument ever has all 53 available at once. and it seems it blocks off your perceptions so you can't even hear the sound of a mouth organ right I agree with Don here. You're talking differences too minute for most people to care. These differences are not subtle. There is a reason why diatonic mouth organs sound much richer and louder when playing diatonic music than chromatic ones; the pure thirds ring out far more clearly than the muddied ET ones you get on a chromatic instrument. Which is why you don't do much chordal playing on a chromatic. It seems the first commercially available ET instrument was the Broadwood piano as tuned by A.J. Hipkins in 1846, with Chopin's style in mind. And it would have been some time before there were enough piano tuners trained in his methods to keep a piano in ET after it left the factory, if it was a long way away. And the first use of a modulation right round the "circle", as far as I can see, is in a late Beethoven quartet from the 1820s. Concertina makers only started advertising ET tuning for some of their products in the late 19th century. ET is much more historically recent than most people think and has only been the a technology composers could count on since Schoenberg's lifetime. |
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Subject: RE: More stuff about the circle of 5ths From: Jack Campin Date: 26 Mar 12 - 06:56 AM A correction to that last post. While 19th century piano tuners thought they were tuning to equal temperament, they actually weren't, and the place where it went wrong was precisely at the "fifth" between G# and Eb, where the "circle" was supposed to close. Willis Miller's dissertation The guitar might have been more accurately ET at the time, but was a pretty rare instrument then and couldn't have been in tune with the piano. He also points out that Rameau changed his mind and started advocating an inaccurate and unimplementable version of equal temperament after writing the "Treatise on Harmony". It was no surprise that this later work was completely ignored. This sounds nice, doesn't it? Chopin in unequal temperament |
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Subject: RE: More stuff about the circle of 5ths From: Will Fly Date: 26 Mar 12 - 09:01 AM instruments that are such a disastrous misfit to traditional melody as the guitar What nonsense. |
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Subject: RE: More stuff about the circle of 5ths From: GUEST,leeneia Date: 26 Mar 12 - 11:53 AM Thanks, Will. |
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Subject: RE: More stuff about the circle of 5ths From: GUEST,josepp Date: 26 Mar 12 - 11:55 AM Thanks, Dave. It is, indeed, the 12th root of 2. Jeesh, I was definitely up too late last night. |
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Subject: RE: More stuff about the circle of 5ths From: GUEST,josepp Date: 26 Mar 12 - 12:23 PM Jack, I'm tracking Don's statement about the Princess and the Pea. You're definitely a "purist" shall we say. I fail utterly to understand your statement about the guitar being disastrous to traditional melody. The guitar is popular for a reason and not because it sounds disastrous. You can play anything you want on a guitar if you're good and nobody is going to hear any big difference. A purist as you will dismiss it simply because it's a guitar. Your earlier statements about traditional music not modulating and rarely even making it to the dominant only proves to me, as I interpret it, that they were aware of the limitations of the scale they were using. They knew spanning the octave was going to go out of tune and knew it was not a desirable thing. Again, I believe they would have tuned in 12-TET had it been available to them. |
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Subject: RE: More stuff about the circle of 5ths From: Don Firth Date: 26 Mar 12 - 01:02 PM You know, sports fans, if you just want to sing folk songs and ballads (or put tunes to dirty limericks), and accompany them on the guitar, ukulele,or autoharp, you don't really need to know any of this stuff. Just tune your instrument to a halfway decent pitch-pipe or an electronic tuner (preferable), do your thing, and let the nit-pickers go ahead and argue in the corner. My first instruction on the guitar was from my girlfriend at the time, Claire, who had inherited her grandmother's old Washburn "Ladies' Model" parlor guitar (made 1898) and was teaching herself out of Guckert's Chords for Guitar without Notes or Teacher. Claire could read music, but she used the chord book because she was a college student and didn't have any spare money to spend on lessons. When I got my guitar (Regal "El Cheapo" model) I also bought the Guckert's chord book and the salesman tossed in the Nick Manoloff book and the Chord Wheel for free. THAT was a real help. A couple of years later, when I knew maybe forty songs (learned from Lomax's Folk Song U.S.A., Carl Sandburg's American Songbag, and Cecil Sharp's 100 English Folk Songs, along with a growing record collection) and people started hiring me to sing, I figured, "If I'm going to do this professionally, I'd better learn something about music." Which is when I started taking lessons, and later enrolled in the U. of Washington's School of Music. Maybe I haven't set the world on fire, but I've made a halfway decent living from singing traditional folk songs and ballads since the mid-1950's, and thoroughly enjoyed myself in the process. And all with my guitar (I've had several very good Spanish-made classical guitars since I retired the $9.95 Regal apple crate) tuned in equal temperament. Nobody in my audiences complained about my guitar not being tuned in pure intonation in all that time!* Don Firth *By the way, since none of the guitars I've owned have movable frets, I'm not really sure how I could do that. |
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Subject: RE: More stuff about the circle of 5ths From: dick greenhaus Date: 26 Mar 12 - 01:27 PM AS a point of possible interest, a fine old banjo picker (Rufus Crisp by name) once explained to me that he had filed the frets down flush to the fingerboard on his banjo because, otherwise, he fouldn't play the "right: notes. On t'other hand, I really can't envision musicians who play larger instruments (like pianos or even guitars) carrying a separate instrument for each key they wished to play in. There are practical considerations. |
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Subject: RE: More stuff about the circle of 5ths From: Don Firth Date: 26 Mar 12 - 02:22 PM Good point, Dick. I have two classical guitars, one flamenco guitar, and two small travel guitars. But I'm not about to tune them to pure intonation in different keys and then lug all five of them to every gig! By the way, it's a matter of programming necessity—at least to me—that my guitar(s) be playable in a number of different keys without having to spend time in retuning. If you hear someone sing several songs in a row, all in the same key, a sort of subliminal feeling of boredom, or at least "sameness" can begin to set in. I knew a really fine singer some years ago who was absolutely terrific in a song circle or at a party. And then I heard him in a full-blown concert. He sang the same really great songs and in a fine, husky tenor voice, but there was something sort of "lack-luster" about the program. Then, I realized something that I had never noticed before. He did everything in the keys of G and Em. No variation. With someone who is limited this way, you're hearing the same set of notes over and over again. To avoid this, I plan my sets so I avoid singing two songs in a row in the same key. I also try to vary the tempo and mood of the songs. This is just simply good programming. Something I learned in school: how to program a recital so it contains variety and you don't put your audience to sleep. As to the guitar itself in relation to singing, including traditional material, since many centuries ago, bards, minstrels, and troubadours (who may very well have originally authored some of the songs whose "folk processed" descendants we sing today) almost invariably accompanied themselves on harps of various kinds, lutes, other plucked string instruments—including early guitars, such as the Renaissance guitar [that's not me, by the way]—(a bit larger than a baritone ukulele, with four "courses," or double strings, save the top string, which is single; tuned like the top four strings of a modern guitar). The use of a small, easily portable instrument to accompany the singing voice is a tradition that is centuries old. And there has always been a great deal of overlap between that tradition and traditional songs in general. So trying to say that accompanying traditional songs with such an instrument is some kind of modern abomination simply ignores the facts of history. Don Firth |
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Subject: RE: More stuff about the circle of 5ths From: Big Al Whittle Date: 26 Mar 12 - 02:31 PM I have joined the circle of fifths and they have elected me an elder. You lot are hereby excommunicated. |
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Subject: RE: More stuff about the circle of 5ths From: Will Fly Date: 26 Mar 12 - 02:35 PM Pour me a 5th, Al - got a thirst on... |
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Subject: RE: More stuff about the circle of 5ths From: Don Firth Date: 26 Mar 12 - 02:52 PM If I drink a whole fifth, I'm liable to wind up inverted. An inverted fifth is a fourth. You've heard of the Firth of Forth, haven't you? Don Firth |
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Subject: RE: More stuff about the circle of 5ths From: Jack Campin Date: 26 Mar 12 - 03:00 PM Your earlier statements about traditional music not modulating and rarely even making it to the dominant only proves to me, as I interpret it, that they were aware of the limitations of the scale they were using. They knew spanning the octave was going to go out of tune and knew it was not a desirable thing. Where did that come from? There is quite a bit of traditional music that goes way beyond the octave. First-position fiddle does two octaves and a third, flutes and whistles about the same using only basic technique. Range doesn't have anything to do with intonation. One reason for the limited use of tonal modulation in traditional music is that it's very hard to sing wildly modulating tunes without an accompaniment providing a pitch reference. Listening to somebody at a singaround trying to do a 1930s Tin Pan Alley or Stephen Sondheim number with even that degree of chromaticism as an unaccompanied solo is not always a pleasurable experience, and there is no way something like the Liebestod from Tristan or the Song of the Wood Dove from Gurrelieder could have been transmitted in oral tradition. Not all traditional music is essentially connected to vocal melody, but most of Western Europe's is. Again, I believe they would have tuned in 12-TET had it been available to them. In case you hadn't noticed, traditional music has not died out, and traditional fiddlers (anywhere, any idiom) don't care about the circle of fifths. And mouth organ manufacturers manage to shift millions of instruments tuned in some sort of meantone every year. You know, sports fans, if you just want to sing folk songs and ballads (or put tunes to dirty limericks), and accompany them on the guitar, ukulele, or autoharp, you don't really need to know any of this stuff. Just tune your instrument to a halfway decent pitch-pipe or an electronic tuner (preferable), do your thing, and let the nit-pickers go ahead and argue in the corner. What you can perform that way is the core of traditional music, and it helps to respect it. |
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Subject: RE: More stuff about the circle of 5ths From: Don Firth Date: 26 Mar 12 - 03:22 PM I respect it, Jack. But I also have a large measure of sympathy for beginners, and I don't feel it's doing them much of a service to load them down with a massive weight of nit-pickery that will only confuse them when all they need to know at this point in their development is "What chords can I use to accompany this song?" There are other, more appropriate places where you can display your erudition. Don Firth |
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Subject: RE: More stuff about the circle of 5ths From: GUEST Date: 26 Mar 12 - 04:54 PM /////First-position fiddle does two octaves and a third, flutes and whistles about the same using only basic technique. Range doesn't have anything to do with intonation//// To be honest, I have not paid very close attentio0n to what you write because I don't understand it. I don't know anything about the styles of music you are referring to. I don't even know what "traditional" actually means--apparently, whatever you say it means. But was the octave they used 2:1? If so, then it is tempered. An untempered octave is something like 2.0273:1 if memory serves me correctly. If they were tempering, then they were attempting, with at least some degree of success, 12-TET. ////In case you hadn't noticed, traditional music has not died out, and traditional fiddlers (anywhere, any idiom) don't care about the circle of fifths. And mouth organ manufacturers manage to shift millions of instruments tuned in some sort of meantone every yea/// Since they are trying to stay true to this "tradition" you speak of--I suppose they wouldn't care about the circle of 5ths in the same sense that the Amish don't care about muscle cars or smartphones. But since I and just about everybody here don't have any desire to live as anachronisms, why are you discussing it? In short, I don't have even the tiniest idea of what you are talking about. And I'm hoping that you will make the connection that all the people who have an interest in the subject matter of this thread feel no differently than I do. |
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Subject: RE: More stuff about the circle of 5ths From: GUEST,josepp Date: 26 Mar 12 - 04:58 PM The last post is mine. Don't get me wrong, Jack. I respect your erudition in this field. Obviously, you feel strongly about it but you need to start your own thread if you wish to discuss it. This thread is about the circle of 5ths and its uses. If you have no use for it, you have no reason to be here. |
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Subject: RE: More stuff about the circle of 5ths From: Don Firth Date: 26 Mar 12 - 05:29 PM To introduce the matter of "pure temperament" into this thread is: A. Off the subject; and B. Needlessly confusing to a beginner. If someone wishes to learn to drive a stick-shift automobile and they want to know how to shift gears, this is NOT the time to lumber them with a lot of related but irrelevant and unhelpful information about the parts, disassembly, and nomenclature of the manual transmission. Don Firth |
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