Why Isn't There A Circle Of Sixths?
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- Опубліковано 13 бер 2022
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The circle of 5ths is cool. Like spooky cool. What else is spooky cool about music? Cue in theories about the number 12, GO. Then go grab 30% off on the courses with code MUSICTHEORY30 cornellmusicacademy.com/
The triangle of major 3rds and the box of minor 3rds are spooky
The 12 tone harmonic series is SUPER wacky in it's complete perfection. Like how in the world is it possible to have something that fits in so perfectly?
Two important points, and a half-important freebie:
• the circle of fourths is the same as the circle of fifths in the other direction and is just as useful
• the reason fifths (and fourths) work and others don't has to do with the fact that 12 is divisible by so much. 2, 3, 4 and 6 all divide 12. 5 (and 7) do no, so moving by that many half-steps you don't reach your starting point until you've taken 12 steps - hitting everything in the process.
• this is, in no way, a co-incidence. The 12tet system (which we use) is made by stacking fifths until you get beck to where you started - or close enough, anyway. Had we decided that wasn't close enough and kept going, wherever we stopped would have this property.
To put those last two thoughts together, consider that In other edos - say, 19, 22, or 31 - other circles also work. in the case of 19 (and 31 and 53 and also even 5 and 7) every circle works, because you have a prime number of steps and nothing divides that evenly. But arguably it's possible we use 12 because it does have some nice simplifying symmetries. (and because it allows us to have the asymmetric division into the major scale...)
@@diegocarrillo2360 It's that perfect because it was artificially made to be perfect! Look up the Pythagorean comma.
Baker's dozen as a musical scale?
There are 4 numbers coprime to 12 mod 12: 1, 5, 7, and 11. But 1 and 11 just give the chromatic scales. So cycles of 5 and 7, which invert to each other, are the only ones of musical interest. The intervals with 5 and 7 semitones are the approximations of the perfect fourth and perfect fifth in 12TET, and it's mere coincidence that log 3/2 base 2 is so close to 7/12. You can do similar exercises in other tunings, but it's not nearly as nice as 12TET.
What a blessing that 12TET's approximation of a perfect 5th is so close
Came to the comment section for mod12 arithmetics, and you didn't disappoint ❤
And then the Schonberg comes in and says he likes his 1's
it should be noted, though, that 1, 5, 7, and 11 just are coprime to 12. that property isn't modular in nature. performing the mod 12 operation just shows the inversions for each position in the scale very easily and intuitively. also, it's not coincidence that 2^(7/12) nearly equals 3/2. we did that ourselves. that's where we get the name "equal temperament." we tempered out both the commas of just intonation and the major differences between key signatures by ensuring every note relates to the one beneath it by the exact same ratio: 2^(1/12):1.
@@matthewbertrand4139 Being coprime to n is a mod n property. This is a special case of the Euclidean algorithm.
As for my comment about the perfect fifth being so close to 7/12 octaves, while I am aware of how 12TET came to be, my comment wasn't about how the perfect fifth covers the scale, but rather about how few notes you actually need to get a system that is so in tune. The ability to temper the Pythagoraean comma (equally) is related to the Diophantine approximation of log(3/2)/log(2), the study of which is not a trivial thing. If you look at the best small denominator approximations of this number (which can be calculated from the continued fraction expansion), you'll see that 7TET comes right before 12TET, and yet the error on 4/7 is nearly ten times as bad as that of 7/12. The next approximation is 17/29, and its error is only on the same order of magnitude as that of 7/12. You get quite a lot of bang for your buck when you use 12TET, and the fact that you get so close with so few notes is the miracle that makes Western harmony possible.
Especially in this episode i am in awe how entertaining and accessible you describe music theory
I agree. I wish I had this video when I was first learning this stuff
I want to find a way to low key send this to my old music professors :)
@@Socrataclysm Email it. They won’t find it for weeks, until they decide clean up their 28,498 unopened emails
Spoiler alert: it was always entertaining and interesting, but your teachers when you were younger were pre-occupied with teaching memorisation rather than critical thinking (remember, your most annoying, dumb peers that were desperate to do well and go to college growing up were obsessed with becoming teachers, psychologists, and biology majors: it's no coincidence that the education, mental health, and medical sectors are falling behind in general)
This made me notice an interesting connection to mathematics, specifically abstract algebra/group theory (not just the frequency ratios people usually talk about when connecting music and math). We can draw a parallel between the twelve notes and the cyclic group Z/12Z (there are other ways of writing that like C12). A cyclic group is essentially the natural numbers but looped, and you may have heard of it in the form of modular arithmetic. We say 12 is the same as 0, so 11+1=0, 10+5=3, et cetera. You can picture this by looking at a 12-hour clock - three hours after ten is one.
In cyclic groups, there is a concept called a generator, which is essentially a number where if you keep adding it to itself (or in other contexts, multiply it by every other element) you'll get every single other element. 1 is a generator, because 1+1=2, 2+1=3, et cetera. 5 is also a generator, because 5+5=10, 10+5= 3, and so on, as it takes you to every number from 0 to 11.
This parallels the musical notes because the fifth is essentially a "seven" in this cyclic group, as the fifth is seven half-steps up. Whereas the sixth is nine half-steps up, and 9 is not a generator in the cyclic group Z/12Z. 9+9=6, 6+9=3, and 3+9=0. That's why the circle of sixths only takes you to four notes.
The generators in Z/12Z are 1, 5, 7, and 11, but 5 (the fourth) just takes you to the same numbers/notes as 7 (the fifth) as it's the same thing backwards, since 5+7=0. While 11... well, those intervals don't sound great.
I'm certain I'm far from the first person to point this out, but I personally never saw it explained this way, and it's helped the circle of fifths make more sense to me, someone who knows math better than music. Cyclic groups and generators in general are a really interesting thing with tons of applications (AFAIK it does a ton in cryptography) and I'm not surprised it shows up here.
Isn't it Z/12Z rather than Z/Z12? Maybe the anglo-saxon notation is different.
@@Maxence1402a No that was a typo by me lmao thanks for pointing it out.
Bru, ok I think I got it but just in case tell me the whole thing again I wasn’t listening.
;)
No really I mostly grasped it.
One way of thinking about it that occurred to me is that most of the intervals are evenly divisible into a multiple of twelve, and 4ths, 5ths, and M7ths are the exceptions.
You might be interested in musical intervals and frequency ratios since thinking of those in combination with just intonation ending up in things like the comma pump just blow my mind
Honestly, I was kind of hoping to see you do a circle of fourths. If I'm not mistaken, it should basically be the same as the circle of fifths, but backward.
Yes. For some reason, the school I went to teaches with the circle of fourths instead of fifths. I am not sure why and I'm no good at actual music theory (we mainly used the circle for transposing and scales), but i have used the circle of fifths the same way I was taught to use the circle of fourths as long as I make sure to remember that it's backwards.
I kept waiting for it, too. :(
@@byteofwood circle of fourths is useful for memorizing secondary dominants, circle of fifths is useful for remembering accidentals/keys
@@byteofwood Fourths shows how notes change between (most) guitar strings, and fifths goes for violins and their related instruments. Maybe you had a teacher who liked playing melodies on guitar?
If I remember right it's just the circle of fifths but you go around the circle in the opposite direction
Something I noticed is that the "four chords" that get used in a lot of popular songs basically take one key on the circle of fifths, swing one step up the circle, return to the start key but relative minor, then swing down one step on the circle and repeat. So like C, G, A minor (relative minor of C), F, the I V vi IV pattern is like a pendulum swing on the circle of fifths
I V vi IV is just
here is my home,
i have to go home,
let me say hello to my sad neighbor first
and use their backyard to go home.
basically lol
I get that calling the other cycles useless helps to promote the (very real) interesting qualities of the circle of fifths, but there's tons of great music built on cycles of major thirds and minor thirds! The fact that they don't account for all twelve tones is part of what's so useful about them.
Would love to know some examples!
Sure! Check out the Sanctus from Schubert's E-flat major mass for a great bit of major-third cycling, or Coltrane's "Giant Steps"--both are built on the Eb-G-B-Eb cycle, in fact. For the minor-third cycle, there's a nice one in the medley from the Beatles' Abbey Road, right before "One Sweet Dream," and the Rebel Fanfare from the Star Wars soundtrack, and a lot of the orc music from Warcraft II (particularly the first two tracks).
There's tons more too, but those come to mind first!
I’ve been musician my whole life and already knew everything you described in this video, and yet, I couldn’t help but be entertained and fascinated by listening to you explaining all if it.
this whole idea is also cool for mathematical reasons. let's assume we view the notes in strictly alphabetical order, with A as the beginning and G#/Ab as the end. because the twelve notes in 12TET form a cycle, returning to A after you climb to Ab, you can relate any note to the A before it OR the A after it, and find that position and direction on the cycle are reversible. the eighth note, E, forms two intervals related to the ends of the octave - a perfect fifth with the bottom A, and a perfect fourth with the upper A. the sixth note, D, also forms these intervals, they just swap which A they form them with. this essentially means that a perfect fifth is the inverse of a perfect fourth in the cycle. and because of this, if you walk through the circle of fifths backwards, it becomes a circle of fourths. and this, in my opinion, totally warrants their names being "perfect" - they're the only two intervals, excluding trivial ones which just move chromatically, which can cover all 12 notes in 12TET without breaking the cycle. it's not only interesting but musically useful to understand that intervals have inverses with similar function, so i'll list all the single-octave intervals with their inverses in 12TET here. these all work both directions:
unison inverts to octave
minor second inverts to major seventh
major second inverts to minor seventh
minor third inverts to major sixth
major third inverts to minor sixth
perfect fourth inverts to perfect fifth
tritone inverts to itself
edit: fully nerded out before actually watching, he talked about interval inversions lol
Correct me if I'm wrong, but I think those are how dark harmonies work as well right?
Cue negative harmony theorism :) Interestingly the first six factors of the circle of fourths creates a quartal inversion of a m11b9b13 chord, in other words the darkest chord possible before rendering full-on Locrian dissonance. Indeed it is Phrygian, and the negative of it is that Lydian maj13#11 chord that is mentioned mid video. And as for the circle of major sevenths - much more easily rendered on a piano as a circle of minor seconds or, in other words, a chromatic scale - the circle of fifths & fourths is essentially a tritone substitution for the chromatic scale, as every other note in the circle of fifths is a tritone away from its corresponding note in the circle of semitones and major sevenths... namely if each new note is treated as a new root. Someone else can probably reiterate that last part more simply, but still: negative harmony and tritone substitution are rendered here as great examples of harmonic duality in music, namely beyond the simple binary of major and minor.
@@gillianomotoso328 i suspected that this would relate directly to negative harmony, but not that tritone substitutions would show up. that's incredible. so if i were listening to a chord progression that makes a substitution, it would be like moving to the place on the circle of semitones where the chord they subbed out should be on the circle of fifths? if so, that's 🤌 SO good
@@matthewbertrand4139 yup!
Another beautiful thing is how well the inverts divide the octave into two harmonious intervals. Like y/x and 2x/y. (For 5th and 4th: 3/2 and 4/3)
Because (y/x)(2x/y)=2/1.. just imagine math did work differently and there was no way to put a note in an octave that makes a harmonious interval with the upper and lower note.. but instead it is the opposite and a harmonious interval will always be harmonious with both octaves as only the factor 2 is introduced to the polyrhythm
My high school teacher would always reference the Circle of Fourths and I never understood how that was different from the Fifths, but this episode has helped elucidate that thank you!
Another relatively simple explanation for those comfortable with modular arithmetic: 12 is highly composite, and 12MODx has only 4 modular "primes": 1, 5, 7, 11. Translated to musical terminology, 1=half step; 5=perfect fourth; 7=perfect fifth; 11=major seventh. Each of these intervals, because they are a modular prime, "circles" every possible value. You can see the same thing happen with hours on a 12-hour circular clock. Consider what would happen if we had some other number than 12 of notes per octave, say 5: 5 itself is prime, so with 5, every interval "circles". One could consider this strictly as 5EDO or a classic pentatonic scale where one simply considers a note skip on already given notes as an interval (so the algorithm only applies to that particular scale and doesn't imply notes outside of it). For example, take C major pentatonic as C-D-E-G-A: a "1 skip" interval would result in C-E-A-D-G, "2 skip" is "C-G-D-A-E" and so on.
I was just thinking about the relations between abstract algebra and intervals in 12-TET and then I find this comment. Didn't expect that to show up here!
@@Caspitein I think it's fun to find applications of sometimes unexpected math in music. In this case though, it's so simple and commonplace that I'm always surprised how many musicians don't realize it.
It might be because the circle of fifths/fourths is just merely a fact used everywhere that the fact the circle of fifths and fourths are 5x mod 12 and 7x mod 12 where x represent how many chords you move down this circle.
But it also doesn’t help that the circle of fourths and fifths are not used all the way. You use the circle of fourths and fifths with three-five chords, so the fact that it’s a modular function is not clearly illustrated.
There is also the fact that while calling it the circle of fifths/fourths and showing it with a circle emphasizes how neatly you can transition from chord to chord and possibly the closedness of 5x mod 12 or 7x mod 12, it de-emphasizes the distance between the keys since fourths are not four keys apart and fifths are not five keys apart.
@@iantaakalla8180 …definitely going to be trying to remember fourths are 5 keys apart and fifths are 7 keys apart. That seems easier for just noodling about
I was teaching this is my trig class. 5 is the only note that doesn’t divide evenly into 12 therefore 7 and 5 half steps are the perfect intervals.
7 doesn't divide into 12.
Neither does 9, 10, or 11; but I guess they're outside the octave
@@nuberiffic No, but they are outside of the tritone which is the largest even division of the 12 tone chromatic scale. 7 can be generated by traveling backwards by 5. Same for 8 with 4, 9 with 3 and so on. So in actuality, there are only 6 intervals and only one perfect one, because depending on which direction your traveling and which octave your in tells you whether your playing a major third or an augmented 5th. And the context of the key informs you as to whether your playing an augmented 4th or a diminished 5th, they are not equivalent contextually. When traveling down it is diminished when traveling up it is augmented. 9 plus 3 is 12, therefore it really is an even division even though the proof of this involves number theory and modular arithmetic.
The inversion of a perfect fifth (seven half-steps) is a perfect fourth (five half-steps).
There is one incredible use of the "circle of major thirds/minor sixths" : Once, one of my Uke's strings broke and for some reason I decided to tune the remaining three strings to an augmented chord. I then discovered how easy it was to figure out chords and scales because I could play all of them within one key without changing my hand position. And when I wanted to change keys, I just needed to move my hand up one fret and repeat the same patterns. The only problem was that to play a chord a fifth away or fourth away became very inconvenient compared to a normal uke and that fourth string missing also meant I had to make 7th chords without fifths but adding the last string would mean I now would end up with a high string with the same note as the low string instead of having a convenient minor seventh, and let me tell you; when your open strings are an augmented chord with no way of lowering the augmented fifth, it's not a pretty look unless you like ending on a tense chord.
I'm really into open tunings on my guitar, and I try all sorts of stuff, and some are really amazing for playing like one key, and making really cool chords easy to play, but make some nearly impossible. It's so much fun tho to hear and play your instrument tuned differently, because the same muscle memory doesn't do the same things so it forces you to think a little harder and try new stuff.
Each “interval circle” has a reverse circle with the interval on the opposite end of the tritone:
m2 is just reverse maj7
maj2 is just reverse dom7
m3 is just reverse maj6
…
All the way to circle of 4 is just reverse circle of 5
🤯
Switching to the opposite interval is also equivalent to descending instead.
and the circle of tritones is the reverse of... wait for it... itself!
@@robo3007 quite (un)expected
I remember first learning the circle of 5ths while in high school and my first thought was of 'awe' and how systematically and mathematically it all fit in a perfect puzzle.
You make all these concepts I literally DREADED in theory classes in conservatory seem so accessible and fun. So nice to see that someone figured out the ideal way to learn theory isn't by endlessly transcribing figured bass to Bach sheet music.
I would like to believe(as much as I loved the video) that charles just wanted to flex how well he can play scales. Love your videos❤️
This gets a bit less mystical when you build it back up in the order it was created. I.e. rather than asking why the circle of fifths hits all twelve notes, ask why we have twelve notes to begin with. The fifth was a very natural interval before we even had the concept of intervals. The 3:2 relationship is only surpassed by the octave (2:1) in its simplicity. When you stack that interval 12 times you end up very very close to a power of 2 (129ish vs 128), which is to say the same note in another octave. 12TET spreads out that slight "error" across all the notes at the cost of the interval ratios not quite matching the original simple ratios (but close enough that most people can't tell the difference).
yeah, it's like he's blown away by how there are always 24 hours in a day, forgetting that the number 24 is arbitrary, and leap seconds exist.
@@WeirdSide I don't mind his excitement, in fact I've come to expect and appreciate it. I just wish this was something he touched on here.
Also I'm not sure if you meant it this way, but 12 is not arbitrary in this case. You could argue that making the 5th the base interval is arbitrary, but it has certain physical implications and resonances that are quite nice (e.g. it's the second interval in the overtone series after the octave). 12 is instead a mathematical quirk. It's the first power of 1.5 gets close to a power of 2. After 7 octaves it's only off by 28.5 cents. Spread that out over 12 notes, and you're only adjusting each note by a couple cents, below the threshold of hearing.
It doesn't get that close again until 41 fifths, and I don't want to have to use that many letters to describe a chromatic scale. Not to mention I don't think a 28.6 cent step is particularly useful, and you'd end up 24 octaves up. But it's only off by 19.8 cents. It gets even closer after 53 steps (3.6 cents), but you gotta go 31 octaves up.
ua-cam.com/video/CFbG-8eYKJU/v-deo.html covers this topic really well, and ua-cam.com/video/IT9CPoe5LnM/v-deo.html has a great visualization of the process.
@@JoshBrown42 yeah 12 is a great number of notes in a scale, but it's not like 12 holds some perfect mathematical property that fits all the harmonic ratios, that's what I meant. There's a reason we have 60 minutes in an hour too, it's a number with a lot of factors. I'm not suggesting we pick these numbers purely at random.
Wait, I'm comment scrolling while listening to the audio - he doesn't get to this point eventually? I thought that was going to be the whole wrap-up for the video.
I struggled with this stuff in school and really hated it! I'm not into just memorizing stuff, i just wanted to play music and I had a good grasp of music as it's own language. More recently (many, many years later) I've gotten a good grasp of it by using it and teaching it to others. I always appreciate when someone presents this in a way that connects all the dots and i often relearn something that I'd forgotten.
Jazz: let’s make it a circle
Every other shape: *It really do be like that sometimes*
I personally prefer the Circle of Life. It moves us all.
What about neco arc. It moves me
yo u not dead?
I agree supreme leader
I just don’t think the “dodecagon of fifths” has the same ring
Circle of 4ths is feeling left out
It's the circle of fifths but backward
@@ldbonq Indeed, I thought it might have been a worth a mention in the video though!
When memorising the different scales (Geef De Aap Een Bord Fis & Finnen Beschouwen Estlanders Als Deskundige Geschiedenisschrijvers are Dutch mnemonics for the scales)
I realised that if you go up 1 whole note, from C to D for example, you will go 'up' two sharps or down two flats.
You make it all make so much sense. Thank you for breaking things down the way you do! I appreciate what you do.
I love this series! I learn so much from them! Keep them coming. Thanks
I’ve seen this TikTok / shorts soo many times, but I’m happy to see it in full length!!
Your videos are so so good man!!! Niceee
Thank you so much for explaining this, you are doing a great job for people who are learning! ❤️
Loving every bit of this video!
It has to do with prime factors and relative primes.
If we index each of the 12 intervals from 0 to 11 (i.e. 2^(i/12), i=0,1,...,11), the only intervals that span all 12 notes are those with an index m that is relative prime with 12 (m ⊥ 12), i.e. m and 12 do not share prime factors. In the 12 tone system, the only intervals that do this are the minor 2nd (m=1), the 4th (m=5), the 5th (m=7), and the major 7th (m=11)!
The cool thing is that if we divide the octave in 13 intervals (2^(i/13)), which has been proposed and people have put music out using it already, any interval will do this, since 13 is prime!
The set of the intervals that compose the circle of the m-th interval can be constructed like this: C = {2^( (m*i%12)/12 ) | i ∈ N, m ∈ Z}. Just the indexes: J = { m*i%12 | i ∈ N, m ∈ Z}
I think the fact adding 1 makes every other number coprime to 13 because 13 is prime is key to some factorization algorithm, but I don’t know the name of it and I don’t know why adding 1 would help with factorization.
I like your funny words magic man
Thank you for getting to the wave forms.
I appreciate the use of the piano visual it makes the videos much more engaging!!!!
Circle of octaves is the real mvp
I could listen to your explanations for hours. You make everything so interesting and easy to understand.
I didn't take music lessons, just an intro to music theory course (I play piano as a hobby). I think the circle of 5ths the best symbol for the structure of western music itself, and the piano is the instrument that is the ultimate in getting the mind and body connected and flowing in western 12 tone music. I keep a circle of fifths on the music stand, and for every warm up I play all major scales, counter clockwise (circle of 4ths technically). Then when I get to the G major scale, I move up a whole tone to A then run through the minor scales. All the while I make sure my fingers are in the right positions for each scale on both hands. After doing this pretty much every time I warm up, I've been able to learn songs much more intuitively this way since my hands know where to go instinctively. I think of it as a "bottom-up" approach to learning music as opposed to learning it from the "top down" through extensively studying theory and thinking about particular keys and scales etc. I'm also able to improvise much more fluidly, although pretty slowly since I've only been playing a couple years.
This was one of the most informative videos you've done Charles.. I've never heard anyone explain WHY 5ths and not anything else. Thanks so much!
Might be handy as a starting point to describe the major scale as two tetrachords separated by a whole tone. If the first tetrachord can start a major scale then the second one also must be able to start a major scale but in order to keep the same tetrachord pattern the 7th of that new scale must add a sharp each time. Same the other way with flats (but with the 4th note).
Been playing piano for like 7 years and trumpet for 4, and it feels so great to know what all this terminology is music means. Ex. Augmented, diminished, dominant chord, major 7 with #9’s, etc. Thanks to Charles, I’ve been learning so much more about musical terminology
And once you start using the circle, you also keep discovering ways to relate every other musical concept to it.
For example, I just related modes, and saw in the circle why eg. Phrygian would have 4 flats relative to the same namesake ionian, which is a conversation I had two days ago but couldn't visualize.
Tres cool, it just keeps on giving.
As an amateur musician who never got much formal music theory, I’d love to see a video with your take on even temperament, keep up the great content!
Great stuff Charles!
I would totally get a course that discussed all these intriguing naturally occurring perfections in music. Even the stuff that’s less practical. It’s incredible! 🤯
A good portion of my warmups focus on the circle of 5ths now. Scales, chords, arpeggios, just playing through the circle. It felt like cracking a musical code once I started thinking that way.
You ask yourself very interesting questions. Really appreciate your stuff
This is really good man, thanks for making it. I'm gonna share this with my guitar students, should make for some interesting Q&A. Bravo!
Great explanation! I frequently try to explain things like this to my friends and they just give me blank stares. It feels good to be part of a community that understands the language of music
Your production quality has improved dramatically, man!
Hi Charles. That's such a neat way of describing the circle of 5ths. I teach transposition using the circle... writing for trumpet in Bb... move 2 places clockwise etc.
Really loved that right around 3:15, we got a series of dramatic reveals from an old-time radio drama.
This is the best video on the circle of 5ths I've seen so far. Thanks.
3:22 woah that Semitone ascended diminish sounds very cool and classical
Wow, that explanation of sharps and flats in the circle of fifths helps a lot to figure out what key a written music sheet is, nice!
Just watched this for the first time and only just getting into music theory. This is so cool! The circle of 5ths to music, is like the periodic table to chemistry. The more you explore it, the more mind blowing it is! (Same with the way elements naturally fall into the periodic table and your mind gets blown when you analyse it).
Thank you for such an amazing explanation of a fabulous thing.
See I needed you 7 years ago when I first started my music program in college 😂 even before then with music theory. Keep educating the world with music Charles 💪🏿🔥
THE TRITONES AHHHHHHHHHHHHHHHHHHHHHHHH
Stop you're gonna summon a fucking DEMON
This absolutely blew my mind.
I imagine Jacob Collier has probably worked out another set of intervals that work in the same fashion, probably on the complex plane (the circle of sqrt(-5ths) perhaps?)
any tonal system with N notes behaves identically to this if N = j(j+1) for some integer j! so you can have a 20 note or 30 note or 72 note system with a functional diatonic scale and "circle of fifths" (but probably not fifths but another interval).
SpongeBob is the only one who has become one with the Circle of Fifths. For he can draw a perfect circle
Avery, STOP STALKING ME ON UA-cam
Thank you for spreading truth.
One of my favorite songs written by Bill Evans is Since We Met. Its intro (follow the bass) is literally around the ENTIRE circle of fifths backwards, starting with F. I am pretty sure Evans plays it three times in a row, and then a few more times. The circle of fifths is fascinating!
You are one of the few people who make me actually WANT to learn music theory. You show that this can be fun as hell!
My favourite trick that results from the nature and symmetry of music is that bug buzz word "negative harmony". The fact that you can take the intervals in a chords as they relate to the key centre (e.g. a G7 in relation to C) and invert them, and get a chord which has the same harmonic tendencies (G7 becomes Fm6 which resolves to C just as strongly as G7) is kind of mind blowing. Add to the fact that each of those two resolutions have a completely opposite feel (G7 -> C feels overwhelmingly positive and content, while Fm6 to C feels melancholy or sad or nostalgic), giving rise to the aforementioned buzzword...
It's just incredible how our ears can not only detect these mathematical relationships and symmetries in combinations of air vibrations at varying frequencies, but provide pleasure from them too
I was literally thinking about this yesterday.
What I love with music theory and tuning theory is that all the math nerds come in full force giving a perspective on it I never realized or understood before
This reminded me of when I was in a steel pan band and I took the time to look at how the drums were constructed.
The tenors (highest voice, melody) were a single drum with notes arranged by the circle of 5ths. The second drums (I'm forgetting the name) were two drums containing each whole tone scale. I played the triple cellos (also called triple guitars), which was 3 drums containing each diminished 7th chord. The bass drums were also arranged by the circle of 5ths. So looking at these drums was my entryway to the same revelations you're discussing here.
There's definitely something to the idea that maybe the piano shouldn't be the standard for music theory pedagogy - some people think a steel drum arranged in the circle of fifths helps associate the relationship of notes better than a linear keyboard
Probably its not the best teaching in yt, but man, you brought so much enjoyment into it
8:53 YOKED! Looking like you single handedly carried that grand into the room!
Jordan Rudess in his Keyboard Wizardry video teaches an exercise which runs through all the 6ths in succession (not just major 6ths) as arpeggiated chord progressions (but you can play them any way you want) which are pretty much just chord tone substitution changes, C-Am-F-Dm-Bb-Gm-Eb-Cm but a very interesting pattern occurs when you wind up traversing all 12 keys in the process and returning back where you started. With every cycle, e.g., C to Cm everything drops down a half step. From C, 7 notes later it becomes Cm, then B, then Bm, then Bb, then Bbm etc. Theres actually 24 chords in all and we'll travel through the Circle of Fourths (Circle of Fifths counter clockwise) in order including also all the Relative Minors. The fingering and chord inversions are even symmetrical. If you begin in root position on a Major triad, ascending the highest note a whole step will bring you to the next chords 1st inversion this time being a minor. Then ascending the Middle note a half step to 2nd inversion returning to a Major. Next ascending the lowest note a whole step brings us back to Root position, only this time on a Minor triad. Everything takes turns in alternating from Whole/half steps and major/minor triads. Its such a complete balance. You don't even need to know the notes or anything about music to play this elaborate progression correctly. Just begin with any chord inversion and alternate your fingers accordingly. Once you complete the cycle of all 24 triads, the process continues the same with C major in root position so you may want to practice this exercise also starting with the other two chord inversions which gives us three different ways/shades/voicings of playing the progression. Another interesting thing is its only practical in ascending form. At least in physicality as far as being played on a keyboard, not in theory. So its kinda opening me up more to when Jacob Collier says every note is the same. 5ths and 4ths are the most obvious, but now 6ths are also the same, and on the Circle of Fifths, when we count all the relative minors. And 6ths of course are inversions of Thirds just as 4ths are inversions of Fifths. So we actually can play the Circle of 6ths in reverse but this would be inverting them to 3rds, thus traversing the Circle of Fifths in clockwise motion, and mirroring its 6th counterparts these can only be played in Descending form. Its hard to explain why in words but easy to realize on the piano keyboard. You just run out of space in one direction and the fingering is a lot more cumbersome. Almost makes one think the Circle of fifths was actually designed backwards when you think about it. This is why so many prefer to call it the Circle of 4ths. I suppose we chose 5ths because 4ths are considered dissonant in classical music since they aren't included in the Overtone Series. But we also need to realize that a circle has no direction and clockwise is just a mental construct for the sake of convention. This is why Buddhism distinguishes between conventional truth and Absolute Truth, and others may even go as far as to say there are no Absolutes. Even modern science is starting to put forth that time is not linear but goes in both directions from the present. We even live out our days in repeating cycles, and even our very anatomy conforms to a circadian rhythm. Funny I just now realized the correlation between the 24 hours on a clock and the 24 major/minor Triads in Music. A wise Qabbalistic Rabbi once said all things are complex in their simplicity. Here we have a mere 12 notes, and really only usually use 7 at time in musical context, yet the possibilities are still endless. It reminds me of the number for Pi. Its less than 4 yet still seemingly infinite. While on this Philosophical tangent might as well end by including the Numbers 3, 7, & 12 which are found everywhere, not just music theory. We can see it in the Calendar, Earth Lay lines, even the Hebrew alphabet. And when white light passes through a Prism (triangle of 3 sides) it gets broken into the 7 Color spectrum which many mystics affirm relate to the seven musical notes staring with Do as Red. Colors also have specific frequencies. I was always curious to see if you can create a painting which can be transcribed into music and vice versa what it would look and sound like. But then we'd also have to take geometry into consideration since shapes also reveal frequencies like demonstrated in water and sand formations which would make things rather complex. Id always Felt that music is so much more than just another form of entertainment, hobby and craft. Even before discovering Beethovens renown quote "When I open my eyes I must sigh, for what I see is contrary to my religion, and I must despise the world which does not know that music is a higher revelation than all wisdom and philosophy." Tibetan Lama and author T. Lobsang Rampa always said his main mission in life was to devise an aura reading machine which can detect health issues and even warned that Kirlian Photography was actually made to defeat this purpose. Dark muddy colors in ones aura reveal problem areas whereas bright colorful shades indicate good health and vigor. The cure would be to shine a particular color on the troubled area by which reversing the results back into a vibrant color. This can also be achieved by transmitting a particular frequency onto the area since again color and sound are correlated which is why Player pianos (aka Pianola) originally played by color. Rampa even once said in his best seller "You Forever" that you can judge a person by their taste in music. Funny, I always also already believed that as well. Tesla even said "if you want to find the secretes of the universe think in terms of energy, frequency and vibration". We even see indications of this in the Bible where God created the universe using sound. And even the Walls of Jericho were brought down by weaponizing sonics. Im really curious if science has ever used the Nature of the Harmonic Series and the Circle of Fifths against any theories and experiments. And more curious to know why not? Sound is actually the most purest form we can apprehend. So why wouldn't we use it as a yard stick. Especially after realizing its exceptional mathematics and aesthetics. Buddhism has attested since 500BC or earlier that the physical universe (solidity) is an illusion and all things are made of Vibrations. It took modern science nearly 3 Millennia to catch up and admit an Atom is 99.9% Empty and that the quantum mechanics of super string theory is the glue holding this holographic universe together. More like playing the song we perceive as our reality. In the Book "The Quantum and the Lotus" a discourse between a Physicist turned Buddhist and Buddhist turned physicist both agree that the Universe is one giant Symphony when touching upon the topic of Quantum Physics and String theory
Im just curious why you limited the circle of 6ths to only Major 6ths when it paints the fuller picture by alternating between all the relative keys as well. Interesting that they would comprise a diminished chord though. I never thought of that. Something else to add. And I preconceived doing the same exclusively with minor 6ths would render an Augmented chord by the process of elimination. But flatted 7ths yielding the Whole Tone scale I did not expect. I'd suppose at this point that doing the same with all the possible intervals would render every scale. Diatonics, Harmonic minor, Double Harmonic Minor etc. Its amazing how many ways there are to look at it. Does it ever end lol, God forbid. The miracle is in the infinite. Best lesson on the Circle of 5ths by far, most are just regurgitated basics, booorring. Hats off to you and Bravo good sir. Subscribed and well earned. Also love your effortless fluid seamless piano playing. You make it look so easy lol
i loved this video so much
Loving the theory videos!
checking out other concepts emerging from the circle of 5ths on the channel would be cool! maybe some advanced stuff like negative harmony or somthing in that tier, to blow people's minds, including mine
2:07 well done charles you are correct that is a sound 💕
Its actually fascinating this relationship between stacking b7s & the whole tone scale.. and the relationship between stacking 6ths & diminished chords.. what an interesting thing to consider! there must be some jazz that explores this? (!)
Like anything in life, there's nothing fascinating about it once you understand it. Don't get me wrong, I find music fascinating, but not this, because it's so simple. Do you find 2 + 2 = 4 fascinating?
A minor seventh is an inverted major second. If you were to write down the circle of minor sevenths so that you get for example C-Bb-Ab-Gb-E-D clockwise, then you'd naturally get C-D-E-Gb-Ab-Bb if you were to go counter clockwise, because a minor seventh and a major second are inversions of each other.
It's the exact same thing with fifths and fourths: if you have the circle of fifths clockwise, then you also have the circle of fourths counter clockwise.
Charles only alluded to this inversion thing at the end with the minor second and major seventh. He probably should have explained inversions and then done their circles in order of usefulness - or versatility, or more like amount of notes, really. It would have gone octave/unison (1 note), tritone (2 notes), major third/minor sixth (3 notes), minor third/major sixth (4 notes), major second/minor seventh (6 notes), minor second/major seventh (12 notes), and perfect fourth/perfect fifth (12 notes, but with relatively consonant intervals rather than dissonant ones).
In fact I have 3 levels of categorizing intervals:
- Melodic: you get an infinite amount of possible intervals, which are what constitutes a melody.
- Voicing: every interval within an octave (11 of them), because intervals are more or less consonant than their inversion. For example, a fifth is more consonant than a fourth, and a major seventh is more consonant than a minor second, etc. This means that for example a minor ninth would be considered the same as a minor second.
- Harmonic: every interval up to the tritone (6 ot them), for purposes like this; circles, or the specific notes of a chord regardless of the voicing. Inversions of intervals are really the same, harmonically. You know how a C major chord is a C major chord no matter the inversion, like for example C-E-G in the same octave or G-E-C spanning 3 octaves? Well C major is a triad, and I'd call these 6 harmonic intervals dyads, as in chords of 2 notes.
Well the 6's and Diminshed world connection is interesting i thought because of Barry Harris 6th diminished stuff. And same for his brothers & sisters (whole tone)
So yeah idk, i guess 2+2=4 IS interesting if you like that sort of thing.
Thanks for the reply, will get thru it later x
@@charlottemarceau8062 Fair enough
@@althealligator1467 if I were to get philosophical, 2+2=4 would be fascinating because of the beauty of math and the human mind. But I'll spare you that. Lol
@@lovelyunknown You know, fair enough, I'm all about people perceiving the world differently, as well as not being able to know anything fully without being able to doubt it, so I'd consider being fascinated by anything possible. I just don't think almost anyone in the world puts 2+2=4 in question very often, me included, so I wouldn't confuse fascination or wonder with appreciation or reliability, you know?
In 24TET, the circle of neutral 3rd is sometimes used and it's equivalent to neutral 6th if you go in the opposite direction
So interesting !!!!
If you alternate between major and minor 3rds (or major and minor 6ths), just like with extended tertial major and minor harmonies, you can get most every pitch in the chromatic scale. BUT, you have to alternate between major and minor. Like so:
C - E - G - B - D - F# - A - C# - F - Ab - C (this one is repeated, yes...) - Eb - G (also repeated) - Bb. So, we only had two repeating pitches before covering all 12 chromatic pitches within the octave.
Another chord prog.:
C - Eb - G - Bb - D - F - A - C# - E - G# - B - D# (repeated) - F#. Only one repeated pitch here.
In both cases, I'm using an Aug broken chord, or two major 3rds in a row when I get to the 13th above the initial pitch (in the case of the 1st example) or the 11th (in the 2nd example). So its not always major - minor - major - minor, or minor - major - minor - major. But it sounds cool when you stack all the notes on top of each other, and it also sounds cool when you use these alternating thirds (or alternating sixths) in chord progressions. Otherwise, yeah, you get the full dim. 7th or the aug triad with any attempt to have a circle of ONLY major or minor 3rds or major or minor 6ths... And of course with 2nds and 7ths, it kinda goes the whole tone or simply chromatic scale routes, or if you alternate between major and minor 2nds, either the half-whole octatonic/dim. scale, or the whole-half octatonic/dim. scale.
So yeah, extended tertial broken chords/pitches and circle of 5ths for the win, but yanno, we can always mess with alternating 3rds and 4ths, etc. with more inventive chord prog.😁😁😁😁😁😁😁😁
A circle of all major 3rds or all minor 3rds may not be useful but when taking the c major scale and turning that into a circle of thirds i.e. CEGBDFA helped me immensely accelerate my ability to know he 3 intervals of any triad on the spot as well as just stacking thirds in general and expanding those triads to 7th chords and further extensions. You are one of my favorite musicians, love your content!
Great video, super fun, accessible and educational as always! Thank you for the amazing work!!
The only thing I couldn’t help thinking throughout is, when one looks at the history of the diaonic scale and how temperaments came to be... well our scale IS actually designed around the fifth isn’t it? (3/2 frequency ratio, +/- 2cents for twelve-tone equal temperament if I remember correctly)
I feel guilty that I get to watch this video for free. Thank you for taking time to go over this in your own fresh way. It's a core principle and you make it so plain and simple. I, a seasoned musician, found it to be helpful to pier my foundations in music theory.
Nice!
Could you please do video about Pat Martino's method in light of these circles. He focuses on diminished and augmented triads so I thought it might be quite interesting!
The chorus of Yes’s Awaken is just the circle of fifths and it makes it feel like it’s always pointing to home yet perpetually rising. And every measure you add more accidentals or change keys. I’m not sure which would be the best interpretation
I love that song partly for that reason that progression is so cool
2:06 musical genius at work ❤
4:45 Literally my brain at this point!
Hey Charles, great explanation. Idea: you should try to get your hands on an accordion and fool around with it. I got one during the pandemic (Roland FR-1xb), it was super interesting. The Stradella bass system is all based on the circle of fifths, as you describe in this video.
Bonus points if you can find a chromatic button accordion, soon piano keys will seem so clumsy in comparison.
the "circle of 6ths" is still really useful for some composition/analysis - in particular, Béla Bartók composed his music with an "axis" system, and there's a big rabbit hole to fall down reading theory on that!
Music Theory is tricky but once you learn it, it makes songwriting so much easier
Love your explanation. ❤ How about circle of major 2nds???
Hello, a recommended song or songs to do would be Rex Orange County’s new album or even some of his older songs. He’s got quite a lot with piano or just dope key and chord changes. I love your content
Hey Charles! You should do a video on Into the woods (Broadway with bernadette Peters)
Made an educational video essay about this a few months ago! Polychron productions made a short video entirely on the math behind this too. It's a modular system of 12. 5 and 7 are the prime numbers and the only non factors of 12. Prime numbers visit every note just one time. Factors create equal divisions.
I would love to se you break down something by Darren Korb. Great vid btw!
My freshman music theory prof actually had us memorize key signatures by going around the circle of 5ths. That was really useful for me, who already knew key signatures really well, to help me understand the circle of 5ths, and for my classmates who didn’t know key signatures to understand them better.
You can also have a circle of 4ths with all the same features as the circle of 5ths, but it is basically just a more condensed version that goes the other way around.
I've also found constructing a circle out of alternating major and minor thirds is useful. It lets you see the relationship between all possible major and minor triads, e.g. A minor sits between F major and C major, share a lot of the same keys (A minor and C major share all the same keys) and flow nicely into one another. Similarly, by combining adjacent triads you can get more complex chords that are a flavor of both.
I really like that the twelfth root of 2 exist. Just multiply or divide a note frequency in hz and you get the next or previous note respectively.
Two cool things I would add to this:
Why are minor seconds, fourths, fifths, and major 7ths special? And by special I mean that their circle hits all 12 notes in some octave. First represent those intervals as jumps of 1, 5, 7 and 11 semitones. Now let's think about what it means to not hit all the notes. It means that we come back to our starting note, e.g. C, too soon. But it can be a C from any octave, so it's any note that differs from where we start by *a multiple of 12 semitones* since an octave is a jump of 12 semitones. When we go around a circle of some interval, we're adding that many semitones each time, e.g. the circle of fifths involves adding 7 semitones over and over. If you try this, the first time you hit a multiple of 12 is 84: 7, 14, 21, 28, 35, 42, 49, 56, 63, 70, 77, 84. It took us 12 steps to get back to a multiple of 12 (the C 7 octaves above where we started), hence we hit all the 12 possible notes. Let's try with major 6ths, i.e. steps of 9 semitones: 9, 18, 27, 36. That only took us 4 steps to get back to another C (3 octaves above where we started). The reason why this loops sooner is that 9 and 12 share some factors, specifically their highest common factor is 3. Hence it takes 4 steps (i.e. 12÷3) to reach another multiple of 12 - another C. If you do this with fifths, you'll find that 7 (the semitones in the interval of a fifth) has no common factors with 12 except 1. Hence their highest common factor is 1, so it takes 12 steps (i.e. 12÷1) to reach another C. So the reason why minor 2nds, fourths, fifths, and major 7ths are special is because 1, 5, 7 and 11 have a highest common factor of 1 with 12 (mathematicians call this coprime).
The other thing involves temporarily forgetting that there are 12 semitones and rediscovering *why* that's the case. In its purest form, a note is a frequency of vibration. The first things we'd have discovered is if you double the frequency (e.g. halving the length of string you're plucking) and play both together it sounds nice. So ×2 sounds good, in some way. We now call these octaves, so when you take a C and apply ×2 to it's frequency you get a C and octave higher. The same works with ÷2.
The next thing you'd discover is that ×3 also sounds nice, and you might use ÷2 to bring it a bit closer to the original note, giving us ×1.5. Basically: simple ratios sound nice.
Now that we have 2 ratios to play with, let's see how they interact. Well first I ×2 my starting frequency over and over and listen to how they sound. Next, I ×1.5 it over and over and listen to it. I'm getting all sorts of different new notes. But something amazing happens when I do the 12th ×1.5. It sounds almost exactly like a note I've heard before, the note I got when I did my 7th ×2. If we declare that it is the same note, then if we keep doing ×1.5 we're just going to get notes we've heard before that are higher octaves. So, up to changing octave, we've found all the notes that you can find with ×1.5. If we use ÷2 to bring all the notes near each other, we have these 12 notes next to each other, which we call semitones. And we've just rediscovered western music theory! *The 12 semitones simply are the different notes you get when applying ×1.5 to the frequency*, and bringing down to the same octave. We call these ×1.5 intervals a "fifth", which makes sense as a name after you invent the major scale which we haven't got to yet. So the miracle is that applying 12 jumps of a fifth takes you back to the same note you started at, just 7 octaves higher. This is not obvious that this would work. It's a *mathematical miracle* that is the reason music works the way it does. The circle of fifths *is how the 12 semitones are invented*. That's how important it is.
However, there's a tiny problem. 12 jumps of exactly ×1.5 isn't actually quite equal to 7 jumps of ×2. It's so very close but not quite right. Hence, we actually don't quite use ×1.5. This is where different tuning systems come into it, and in equal temperament tuning we use approx ×1.498. This means that the fifth you hear on a piano doesn't sound quite as nice as that perfect ×1.5 ratio, but it's so close that we can't tell and we get our perfect circle of fifths.
Hey Charles. I absolutely love these music theory videos, idk if you take requests but would it be possible for you analysis the Tron Legacy soundtrack? Especially the songs Son of Flynn, The Overture, and Derezzed.
This has been the only Co5ths video I’ve ever seen that really made sense to me somehow. Thanks
Circle of minor second covers all also 😊
I’m the professional musician in my family and my younger brother is the music hobbyist, but I’ll never forget the day when while I was home for Christmas break from college, he and I were just fiddling at the piano, because why not?? He has perfect pitch and I have really good relative pitch, but he’ll always admit I’m the better music reader…but here’s what happened that day of just having fun at the piano.
While playing some chords he was trying to figure out from a song we both knew (don’t recall what song), he suddenly stops and starts plunking a standard V-I resolution. Nothing anyone with a halfway decent musical ear hasn’t heard before, right? Well…he slowly starts doing a few more of them in slow succession and slowly realizes something…then starts doing EVERY…SINGLE…ONE and realized he came back to where he started and when I tell you I lost my damn mind 🤯🤯🤯🤯🤯
I scream (in sheer shock of course) at him with “Do you know what the heck you just did??” He had NO IDEA that he just completed the circle of 5ths, something taught in advanced theory classes! He can read music and knows the very basics of theory, but nothing beyond what’s taught in a beginner music theory class…but he’s always been able to hear stuff that even I can’t hear easily!
He has the ear, but doesn’t even realize he knows this stuff, and I’ve said for YEARS he’d be a stellar theory and aural skills teacher of some kind if he wanted to go that direction, but he never really wanted to do music professionally and there’s nothing wrong with that obviously, but maaaaaaan it still baffles all of our family as to why! He has the skills, all he needs is the knowledge!!
I'm just learning this and I'm using:
Clockwise -
•Cool Guys Dead Ass Eat Butt (Flat G)
Counter Clockwise -
•Cringy Fedora (Flat BEAD) (Sharp F)
I've messed around with keys for years. And I was very much a natural musician in regards to drums since I was 10. I joined school band when I was 11 or 12, and while I could play everything and was voted best musician in the schools in 8th grade (the last year I finshed before quiting band), I very much didn't pay attention as much as I should have. So I never learned LOTS of stuff properly. Now that I'm passed the "just digging in and improvising wonderful stuff" phase, and with years (sadly) of on and off playing, I'm in the "deeply and passionately interested in the actual understanding of music" phase of learning. So I'm just now coming to actual music theory at 30, and although it's more difficult to retain new knowledge at this age compared to being 18 and younger, the excitement and passion does most the heavy lifting.
People like Charles and Adam Nealy are incredible and indescribably useful and motivating, also. So thanks to all the musical content creators for actually giving the free gift of a thorough understanding and knowledge of music.
All major scales have the same shape on bass and guitar (until you get to the B and E strings because they are tuned at different intervals from the other strings). Same with minor scales, modes. And respective chords. Ie. Major chords have the same shape. It's a super interesting trick that I probably leaned on a little too heavily when I first learned improvisation.
A lot of impressionist composers looked at the minor third intervals and the major third intervals. They started composing with them, particularly as they were symmetrically splitting the octave..