My friend, people who think like you need to be running the world if we want a peaceful existence as opposed to the self destructive and wartorn existence we have.
things i did not expect to learn from this: - rotating a pentagon around a circle of fifths will produce a chromatic scale - the first half of the gamecube intro is the circle of fourths but pitch shifted
I realized from the decagon that two circles of fifths a tritone apart (and going in the same direction) is the same as two chromatic scales (circles of half steps) a tritone apart (and going in the same direction as each other), because a tritone plus a half step is a perfect fifth and/or because a tritone minus a half step is a perfect fourth.
Long story short: in music theory, the sequence of F - C - G - D - A - E - B (or its reverse) comes up a LOT. Each of those notes is an interval called a "perfect fifth" away from the next. So it's a sequence of fifths. Add in the five other notes common in Western music (the black notes on a piano) and you can make the sequence into a circle. It's handy for remembering things like which key has what sharps or flats, once you are used to it.
The 11-gon actually illustrates the principle behind cycloidal drives, a type of transmission. The inner gear (the polygon) having just one fewer teeth than the outer (the circle of fifths) gives it this unique rotational mode that acts as a 11:1 gear reduction. In this case, that means it will play every note 11 times before the polygon rotates once.
When used in this way, any regular polygon with A * B vertices (where A and B are positive integers) will behave the same as A copies of a regular polygon with B vertices. Because of this property, the really novel behavior will be on a the prime-numbered polygons. I wonder whether every sequence of intervals is possible?
@@lemming7188If you just mean in 12-EDO, the interval between any two adjacent (in time) chords must always be the same, due to a sort of time-independence symmetry (involves the geometric and interval symmetry of the circle as well), and, due to the symmetry of the polygons and the factors of 12 (1, 2, 3, 4, 6, and 12), the chords themselves must always be one of the following: (a) a single note, (b) two notes a tritone apart, (c) an augmented chord, (d), a fully diminished 7th chord, (e) a whole tone scale (as a chord), or (f) a chromatic scale (all 12 notes played at once) This is the same if you use the "circle of half-steps" instead of the circle of fifths, and is probably easier to understand for the "circle of half-steps". Anyway, this means the number of possible patterns so very limited I can list them: 1) The pentagon's pattern from the video 2) The heptagon's pattern (pentagon's pattern backwards) 3) The hendecagon's pattern backwards (same just using an arrow point out from the center in one direction) 4) The hendecagon's pattern 5) The decagon's pattern 6) The decagon's pattern backwards (should be the tetradecagon's pattern) 7) The triangle's pattern 8) The nonagon's pattern (the triangle's pattern backwards) 9) The octagon's pattern (the square's pattern backwards) 10) The square's pattern 11) The hexagon's pattern 12) the dodecagon's pattern (Note that the reason we only have backwards and forwards for each multi-note chord is because none of factors of 12 is relatively prime with anything less than it other than 1 and the factor minus 1.) Interesting how there are 12, just like there are 12 notes in the scale (in 12-EDO). I'm not sure if that's a general pattern though. By the way, to check if the similarity between the circle of fifths and circle of half-steps applies in other EDO's, you need to use intervals that are n steps in m-EDO where n and m are relatively prime.* *To explain further: "m-EDO" means "m Equal Divisions of the Octave" (or similar), and the smallest interval in such a system is a 2^(1/m) ratio or frequency or wavelenth. To get an interval cycle that passes through every note of m-EDO, you need an interval whose ratio is 2^(n/m) where the greatest common divisor of n and m is 1. In 12-EDO, n must be 1 (single half step), 5 (perfect fourth = 5 half steps), 7 (perfect fifth = 7 half steps), 11, (major seventh = 11 half-steps) or possibly other numbers like -1 (half-step in other direction) or 13 (minor ninth) that are octave-equivalent to those, so we just have the circle of fifths and the circle of half-steps, where-as other intervals cycle before getting to every note: whole step (2^(2/12)=2^(1/6)) generates 6-EDO, e.g. a whole tone scale minor third (2^(3/12)=2^(1/4)) generates 4-EDO, e.g. a fully diminished seventh chord major third (2^(4/12)=2^(1/3)) generates 3-EDO, e.g. an augmented chord tritone (2^(6/12)=2^(1/2)) generates 2-EDO, e.g. two notes a tritone apart in each octave minor sixth (2^(8/12)=2^(2/3)) generates 3-EDO major sixth (2^(9/12)=2^(3/4)) generates 4-EDO minor seventh (2^(10/12)=2^(5/6)) generates 6-EDO octave (2^(12/12)=2^(1/1)=2) generates 1-EDO one note in each octave major ninth (2^(14/12)=2^(7/6)) generates 6-EDO, etc. In other EDOs, you would have more cycles that go through every note, for example, in prime number EDOs like 31-EDO, every single interval generates such a cycle.
I like the attention to little details. The little wind up the polygons do in the opposite direction before turning regularly and the slow down at the end of the rotation. You didn't have to do that. It didn't help majorly with the visualization, but you did it anyways. Kudos.
It's insane to see the consequences of modular arithmetic in mod12 (the arithmetic of clocks, i.e. 6 + 7 = 1, 8+8 = 4, etc) in music so clearly. For example, 11 = -1 (as in one hour before 12:00, that is, one hour before 00:00). You can see that the effect of an 11 sided polygon is the same as a "1 sided polygon" (aka, a needle), but ticking backwards due to the minus sign. The same happens with 7 = -5, that's why a 7 and 5 sound the same but backwards. More generally, this happens with any two numbers a and b that add up to 12 (or a multiple of 12), like 3 and 9, because 9 = -3.
If there's gonna be a follow-up, it would be really cool to have the notes play in a few octaves, then do a gentle bandpass on the middle frequencies. You'd get a cool variant on that staircase illusion, and hitting C again wouldn't be as stark!
Indeed, "imperfect" polygons are way more useful musically-speaking than "perfect" polygons. The "everything's a little broken, and that's ok" thing applies here gracefully!
have you ever noticed that the triangle you're describing can be flipped to be the other? major and minor chords are just reflections of each other. blows my mind
9:26 I knew it was coming, but it still gave me chills. 13-gon: same as 11 14-gon: faster tritone-apart chromatic scale 15-gon: fast repeating augmented chords? 16-gon: fast repeating dim 7 chords? 17-gon: go away 18-gon: whole-tone chords, _really fast_ 19-gon: leave me alone
I expect all the prime-number-gons will do either chromatic scales or circles of fifths due to a couple of symmetries of the situation. Actually, all n-gons where n is relatively prime with 12 (so isn't divisible by 2 or 3) should have this property. The first non-prime one of these is 25, which should play the circle of fifths in the same direction it rotates since it's one more than 24, which is 2 times 12.
Its a great visual illustration of how tool incorporates 11's in scales and timing and polyrhythms for the exact same effect. its really pretty simple but it comes off as next level if you have the ear for it.
I am very curious to see what this would look and sound like for equal divisions of the octave other than 12 (the best ones might be 5, 7, 17, 19, and 22, because they are relatively small, and contain one and only one circle of fifths).
@@lasstunsspielen8279 Yes but polygons that have a number of sides that is equal to a divisor of 60 but not of 12 will make chords that aren't heard here
The nonagon going clockwise makes me think of some kind of cartoony Industrial Revolution-era factory scene, while going counterclockwise it just makes me think of a video game major boss intro.
Very cool. I just KNOW your videos will blow up soon. In any case, it'd be neat to see this again with non-regular polygons. Keep up the awesome content
8:43 Years ago, I used to draw stars of different #'s of vertices in different ways, so that I draw them accurately without drawing the vertices first. I wondered what a 12 pointed star would sound like on a piano, with each vertex being given a note on an octave. I played exactly this. The Hendecagon here is still part of my piano practice routine.
Music for your nightmares Haha. It all sounds like terrifying circus music because of all the chromaticism and tritones. The 11-sided shape was semi-reminiscent of tubular bells only more disturbing somehow 😎
This rendering of tone intervals as a polygon of rotation is very clever! Now let's consider the IRREGULAR polygons of n sides. Not only could this be a very easy way for students to visualize the triads and chord extensions, but perhaps also pick up a preliminary sense of how cadences work,
Honestly quite disappointing results, but that should have been expected because 12 is so divisible. Repeating this same exercise with chromatic scale instead of circle of fifth could be more interesting. Or using major scale, only 7 notes.
Until now, I used to think that shape and music were unrelated. After watching this video, however, I realized that such things can be interconnected. I found it particularly fascinating how the number of angles in a shape corresponds to the difference in the number of notes played simultaneously. While I've had some interest in shapes, I've never really been into music. After watching this video, I feel like my understanding of music has improved compared to before. 10706
Nice video! Starting from the music end would be interesting - what's the irregular polygon that plays a major scale for example? (is there one?) - is there a shape that plays a 2 5 1 chord sequence, or an arpeggio/short melody etc.?
i shouldve entirely been prepared to have king gizzard enter my brain the moment a nonagon was mentioned but here we are. nonagon infinity opens the door
Cool video!! You're showing some very neat aspects of modular arithmetic, how co-primality can be used to make encodings, and how that fails (makes a chord vs a single note) when there's common divisors. How encryption and number theory overlap with music is just awesome (but also makes sense if you compare the maths). Thanks for sharing!!
This is interesting, but I would actually love to hear this where we hear the exact notes that are played where points touch the circle and not only when the exact contact points of the notes of the circle of fifths is touched. Using the example of the pentagon. If the top point is touching C, the next point is touching a slightly sharpened D, next point is touching a much more sharpened E. next point is touching a slightly sharpened Db and final point is touching a more sharpened Eb. Anyone else get what I mean by that? And I'd also like to hear a steady transition of the motion travelling around the circle, like a sustained chord that is rising in pitch with exactly the intervals that the different polygons denote. Each of the 12 segments of the pie can be broken into 30 microtones/pitches. So for example, C to G (and each of the other segments) actually has 30 subdivisions between the 2 notes. Where the points of the polygons touch at these points is what I'd really love to hear.
I love that the pentagon basically just reverts the circle of fifths (y'know, 5 sides). Completely and utterly logical and intuitive in hindsight, but I doubt many would've guessed that on their own!
Hendecagon reminded me of the forest temple from Legend of Zelda OoT. Clockwise and counter clockwise it reminded me of one of the 4 temple in Majora’s Mask.
This got real interesting when the notes were played sequentially. I expected a pentatonic chord for 5, but god chromatics. I find this approach both smart and creative. Just what music theory needs, after centuries with a system full of exceptions. Good work! You could animate the interval classes 1 thru 6 into a lydian scale using the formula n * (-1)^(n+m), n in 0...6, m being 0 or 1 for major and minor resp, the latter being tonal mode: 0,11,2,9,4,7,6,5, sorted and relative to 0: -5, -3, -1, 0, 2, 4, 6. Swap the m and you have the locrian (most minor) scale mode. Notice that negative offsets are odd and the positive even. So an Archimedean spiral would draw these scales, y's are n and x 's are pc, making x a function of y, that way matching the linear pitch axis horizontally, like on the piano keyboard. So I don't believe in 4096 sets, but in the Major scale, the only one containing all 6 interval classes, or 7 including the root. Nice, eh?
For example, in 19 equal pitch divisions of the octave, the circle of perfect fifths can be described in steps of the tuning system as 0, 11, 3, 14, 6, 17, 9, 1, 12, 4, 15, 7, 18, 10, 2, 13, 5, 16, 8. It can be described in letters as F, C, G, D, A, E, B, F#, C#, G#, D#, A#, E# or Fb, B# or Cb, Gb, Db, Ab, Eb, Bb.
This is brilliant -- combining geometry and music and finding very interesting tonal patterns they create. I think there's a lot more to be investigated regarding this.
These are all written in Java using a graphical library called Processing (processing.org), and the built-in Java MIDI library for writing out MIDI, which then gets realized as audio with DAW plug-ins.
I'd be curious about an extension of this: Rotating a poygon on an arbitrary plane slicing a cone It would be an ellipse that touches, but draw rays from the center of the polygon, play notes when they cross one of the cone's vertical lines The height of the cone could represent ... something
That was fun. The later ones were mostly more interesting than the early ones. I' like to hear the 13-gon and the 17-gon being prime, which means none of the notes are played simultaneously - pure melody and fast. I would also like to hear what the polygons would sound like if instead of the circle of fifths ordering the straight chromatic scale ordering was used.
The nonagon nexus key is the basis of what youve stumbled upon. It encodes the solfeggio frequencies. A series of roation of the nonagon where the degrees are added up reveals this pattern. Going deeper through the math procrss of the nonagon nexus key, one can even derive the fine structure constant.
My hearing is gone. I cant hear anything coming through the tiny speakers of a cell phone, so correct me if im wrong and if you care to. Or perhaps tell me if i was right. The rotating octagoñ and the series of diminished chords whose roots ascend or descend in half stèps can be heard at the end of a song by the rock band "Queen". The song is "you take my breath away" and the effect is achieved with an echo/delay. The delay is roughly one second or 60 beats per minute. Each beat is divided into 3 segments or triplets. First a guitar playing descending half steps (3 per second) so that as the guitar descends 3 half steps an echo of the same guitar repeats those notes as it continues moving down. Evenually a vocal singing the same series of descending chromatic tones using the words "take my breath" over and over . Check it out. Check the entire song out. Or dont, i dont care.
What's interesting is every one of those sounds I've heard on a 1970s horror show or 1970 Syfy show. That is so interesting. I'm curious what would happen if you had unusual shapes such as a triangle that had two long sides and one short side.
all of this is fairly straightforward. the pent and heptagon seem odd at first but the circle of fifths is just that... FIVE, and its complement in base 12 is of course, seven... what i see interesting is that a minor chord is a mirror reflection of its major... CEG faces the opposite way to CEbG... FAC vs FAbC... etc etc...
Would be interested to hear each with square, saw, sine, and pulse waves sped up beyond the point where pitch discernment is perceptible (spinning the shapes as fast as possible along the Co5s). The geometric quality of the oscillation would probably be pretty pleasing when sped up and applied to wavetable synthesis.
i think the reason why it does this is that as the polygon rotates, notes are played in star patterns. the triangle creates the {12/3} star, which is actually three squares. It played all four squares at the same time. the square makes the {12/4} star, which is four triangles. {12/6} is six lines. the pentagon creates the {12/5} star, which is an actual star, where one line hits all 12 points and then loops back on itself. the {12/7} star, if it existed, would be equivalent to the {12/5} star, which is why it does the same thing as the pentagon. given the nature of the circle of fifths, if you find the notes five away from a given note, it will give the notes next to it on the chromatic scale. since going five points down the road is basically what the {12/5} star is, it makes a chromatic scale. similarly, constructing a {12/8} star will give you {12/4}, and so on.
Imagine having a wall of hand-cranked versions of this in a children's museum.
And the museum guard must be replaced every two days due to a nervous breakdown.
Imagine if it was a board with pegs and string where people could draw out a shape with the string and have it rotate
That sir is a brilliant idea.
That is avery good idea indeed!
My friend, people who think like you need to be running the world if we want a peaceful existence as opposed to the self destructive and wartorn existence we have.
things i did not expect to learn from this:
- rotating a pentagon around a circle of fifths will produce a chromatic scale
- the first half of the gamecube intro is the circle of fourths but pitch shifted
I guess they're called fifths for a reason
I realized from the decagon that two circles of fifths a tritone apart (and going in the same direction) is the same as two chromatic scales (circles of half steps) a tritone apart (and going in the same direction as each other), because a tritone plus a half step is a perfect fifth and/or because a tritone minus a half step is a perfect fourth.
It's not, it's just the same instrument, not the same notes at all
Playing fourths like that is called plagal harmony
@@blackmage1276quartal harmony usually.
I am not a musician. I have never understood “Circle of Fifths.” This has now raised my level of incomprehension by a power.
😂
Power greater or smaller than one?
Long story short: in music theory, the sequence of F - C - G - D - A - E - B (or its reverse) comes up a LOT. Each of those notes is an interval called a "perfect fifth" away from the next. So it's a sequence of fifths.
Add in the five other notes common in Western music (the black notes on a piano) and you can make the sequence into a circle.
It's handy for remembering things like which key has what sharps or flats, once you are used to it.
It's a tool that simplifies scales. You have to know what a scale is first. Go learn that.
Circle of fifths is just a fancy way of organizing every 5th note. It's a useful tool for musicians.
The 11-gon actually illustrates the principle behind cycloidal drives, a type of transmission. The inner gear (the polygon) having just one fewer teeth than the outer (the circle of fifths) gives it this unique rotational mode that acts as a 11:1 gear reduction. In this case, that means it will play every note 11 times before the polygon rotates once.
Interesting 🤔 I hear the Nintendo Game Cube start jingle
H E N D E C A G O N
The 11 polygon is actualy a fire ringtone
GameCube startup sound haha
Maybe an alarm, but not a ringtone
Same with the decagon
I find it funny, that it have 11 sides, but plays in 6/4
😂
Hendecagon: Oh wow, that's complex and interesting.
Dodecagon: What the fuck.
Hendecagon is the eighties computer jingle.
Dodecagon is what a concussion sounds like. every time.
The Hendecagon isn't complex, it's just playing the circle of fifths
@@nesquickyt I understand.
@@nesquickytThat is arguably complex.
When used in this way, any regular polygon with A * B vertices (where A and B are positive integers) will behave the same as A copies of a regular polygon with B vertices. Because of this property, the really novel behavior will be on a the prime-numbered polygons.
I wonder whether every sequence of intervals is possible?
Does this mean that theoretically any interval cycle could be represented by a Polygon with a vertex count that is Prime?
If true, could be a super interesting tool for classification. Would get extremely impractical though lol
@@lemming7188If you just mean in 12-EDO, the interval between any two adjacent (in time) chords must always be the same, due to a sort of time-independence symmetry (involves the geometric and interval symmetry of the circle as well), and, due to the symmetry of the polygons and the factors of 12 (1, 2, 3, 4, 6, and 12), the chords themselves must always be one of the following:
(a) a single note, (b) two notes a tritone apart, (c) an augmented chord, (d), a fully diminished 7th chord, (e) a whole tone scale (as a chord), or (f) a chromatic scale (all 12 notes played at once)
This is the same if you use the "circle of half-steps" instead of the circle of fifths, and is probably easier to understand for the "circle of half-steps".
Anyway, this means the number of possible patterns so very limited I can list them:
1) The pentagon's pattern from the video
2) The heptagon's pattern (pentagon's pattern backwards)
3) The hendecagon's pattern backwards (same just using an arrow point out from the center in one direction)
4) The hendecagon's pattern
5) The decagon's pattern
6) The decagon's pattern backwards (should be the tetradecagon's pattern)
7) The triangle's pattern
8) The nonagon's pattern (the triangle's pattern backwards)
9) The octagon's pattern (the square's pattern backwards)
10) The square's pattern
11) The hexagon's pattern
12) the dodecagon's pattern
(Note that the reason we only have backwards and forwards for each multi-note chord is because none of factors of 12 is relatively prime with anything less than it other than 1 and the factor minus 1.)
Interesting how there are 12, just like there are 12 notes in the scale (in 12-EDO). I'm not sure if that's a general pattern though. By the way, to check if the similarity between the circle of fifths and circle of half-steps applies in other EDO's, you need to use intervals that are n steps in m-EDO where n and m are relatively prime.*
*To explain further: "m-EDO" means "m Equal Divisions of the Octave" (or similar), and the smallest interval in such a system is a 2^(1/m) ratio or frequency or wavelenth. To get an interval cycle that passes through every note of m-EDO, you need an interval whose ratio is 2^(n/m) where the greatest common divisor of n and m is 1. In 12-EDO, n must be 1 (single half step), 5 (perfect fourth = 5 half steps), 7 (perfect fifth = 7 half steps), 11, (major seventh = 11 half-steps) or possibly other numbers like -1 (half-step in other direction) or 13 (minor ninth) that are octave-equivalent to those, so we just have the circle of fifths and the circle of half-steps, where-as other intervals cycle before getting to every note:
whole step (2^(2/12)=2^(1/6)) generates 6-EDO, e.g. a whole tone scale
minor third (2^(3/12)=2^(1/4)) generates 4-EDO, e.g. a fully diminished seventh chord
major third (2^(4/12)=2^(1/3)) generates 3-EDO, e.g. an augmented chord
tritone (2^(6/12)=2^(1/2)) generates 2-EDO, e.g. two notes a tritone apart in each octave
minor sixth (2^(8/12)=2^(2/3)) generates 3-EDO
major sixth (2^(9/12)=2^(3/4)) generates 4-EDO
minor seventh (2^(10/12)=2^(5/6)) generates 6-EDO
octave (2^(12/12)=2^(1/1)=2) generates 1-EDO one note in each octave
major ninth (2^(14/12)=2^(7/6)) generates 6-EDO,
etc.
In other EDOs, you would have more cycles that go through every note, for example, in prime number EDOs like 31-EDO, every single interval generates such a cycle.
That's why I think it'll be interesting to check out more primal numbered polygons, since 11 did factor a new sequence
@@lemming7188 somebody better do a paper on this
Hendecagon sounds like the Game Cube startup screen
That's why this so nostalgic but i don't know where the tune come from 😂
It also sounds like one of the sounds used in Brain Training for the Nintendo DS.
it sounds like something from the original paper mario's soundtrack but i can't remember where
@@Farvadude Sounds like the endless staircase from Mario 64
@@MT-pe8bh you're right that's it
You gonna F around and open a portal to another dimension you keep this up!
en.wikipedia.org/wiki/The_Music_of_Erich_Zann
It’s the nonagon, don’t you know? Nonagon Infinity opens the door.
hehehe, F
I like the attention to little details. The little wind up the polygons do in the opposite direction before turning regularly and the slow down at the end of the rotation. You didn't have to do that. It didn't help majorly with the visualization, but you did it anyways. Kudos.
Yeah, that was very nice!
Agree! I enjoyed that, too!
I challenge you to make a shape that looks like africa that plays Africa by Toto as it rotates.
that’s impossible
I challenge you to come up with a less zoomer idea
@@d3tuned378I challenge you to make a shape that looks like Africa that plays Africa by Toto as it rotates.
@@akneeg6782 that's the same idea
Mandelbrot plays Rosana.
It's insane to see the consequences of modular arithmetic in mod12 (the arithmetic of clocks, i.e. 6 + 7 = 1, 8+8 = 4, etc) in music so clearly. For example, 11 = -1 (as in one hour before 12:00, that is, one hour before 00:00). You can see that the effect of an 11 sided polygon is the same as a "1 sided polygon" (aka, a needle), but ticking backwards due to the minus sign. The same happens with 7 = -5, that's why a 7 and 5 sound the same but backwards. More generally, this happens with any two numbers a and b that add up to 12 (or a multiple of 12), like 3 and 9, because 9 = -3.
fascinating connections!
Which is also why 6 in either direction sounds exactly the same
Took the words right out my mouth 💯
@@Th_RealDirtyDan Wow, true! Did not even realize!
If there's gonna be a follow-up, it would be really cool to have the notes play in a few octaves, then do a gentle bandpass on the middle frequencies. You'd get a cool variant on that staircase illusion, and hitting C again wouldn't be as stark!
Shepard tones th-cam.com/video/PwFUwXxfZss/w-d-xo.html
decagon
Try a 120-45-15 degree triangle. You will get all the major or minor chords, depending on how you orient the triangle.
Indeed, "imperfect" polygons are way more useful musically-speaking than "perfect" polygons. The "everything's a little broken, and that's ok" thing applies here gracefully!
have you ever noticed that the triangle you're describing can be flipped to be the other? major and minor chords are just reflections of each other. blows my mind
@@louisaruth Yeah, its true that its isomorphic. Thats the main point of equal temperament. (Except for e flat and non perfect fifths)
@@lunyxappocalypse7071 really seems like something that should be discussed more often
9:26 I knew it was coming, but it still gave me chills.
13-gon: same as 11
14-gon: faster tritone-apart chromatic scale
15-gon: fast repeating augmented chords?
16-gon: fast repeating dim 7 chords?
17-gon: go away
18-gon: whole-tone chords, _really fast_
19-gon: leave me alone
I expect all the prime-number-gons will do either chromatic scales or circles of fifths due to a couple of symmetries of the situation. Actually, all n-gons where n is relatively prime with 12 (so isn't divisible by 2 or 3) should have this property. The first non-prime one of these is 25, which should play the circle of fifths in the same direction it rotates since it's one more than 24, which is 2 times 12.
Pentadecagon should be 3 simultaneous chromatic scales, each a major third apart.
@@jimmygarza8896 Technically that's the same as "fast repeating augmented chords," but I should have stated that they move in a chromatic loop.
I want to hear the 17-gon.
20 - Rick Rolled
The 11 sided one is such a cool rhythm. Like bossa nova played on a telephone
Subscribed BTW 😊
Sound like Gamecube intro:D
the rhythm isn't that interesting lol, it's just the notes
Press your luck gameshow
Its a great visual illustration of how tool incorporates 11's in scales and timing and polyrhythms for the exact same effect. its really pretty simple but it comes off as next level if you have the ear for it.
I am very curious to see what this would look and sound like for equal divisions of the octave other than 12 (the best ones might be 5, 7, 17, 19, and 22, because they are relatively small, and contain one and only one circle of fifths).
Also I'd be interested to see 60, just because the large number of divisors it has would make for lots of chord combinations
@@robo3007 True!
60 would sound the same as 12 but 5 times quicker
@@lasstunsspielen8279 Yes but polygons that have a number of sides that is equal to a divisor of 60 but not of 12 will make chords that aren't heard here
you forgot 31!!!
The nonagon going clockwise makes me think of some kind of cartoony Industrial Revolution-era factory scene, while going counterclockwise it just makes me think of a video game major boss intro.
photoshop flowey
the counter-clockwise one is actually really similar to a song called hyper zone 1 from kirby's dream land 3
Game Cube loading screen
the clockwise one sounds a lot like Nuclear Fusion from Touhou as well
Counterclockwise is just the first four notes of Hyper Zone 1 from Kirby’s Dreamland 3 (Final boss phase 1 theme)
Very cool. I just KNOW your videos will blow up soon. In any case, it'd be neat to see this again with non-regular polygons. Keep up the awesome content
Do it again with the 23TET circle of fifths. 23 being a prime number will surely create interesting microtonal patterns.
no regular polygon will play a chord, you'll go over the circle in all different intervals
@@SZebS did you watch the video? the polygons' vertices don't need to line up with notes
@@ataraxianAscendant did you read my comment? Polygons only play chords of more than one vertex is touching a note at once
@@SZebSi dont think sirfloll explicitly mentioned chords
@@sillyk2549 he didn't, i'm just saying what will happen because 23 is prime
8:43 Years ago, I used to draw stars of different #'s of vertices in different ways, so that I draw them accurately without drawing the vertices first. I wondered what a 12 pointed star would sound like on a piano, with each vertex being given a note on an octave. I played exactly this. The Hendecagon here is still part of my piano practice routine.
Triangle: Creepy. Mystery.
Square: Confusion. "Whodunnit?"
Pentagon: Going up. Going down.
Hexagon: Mysterious Grandfather clock. Watching the clock. Anticipation.
Heptagon: Running down stairs. Running up stairs.
Octagon: Being chased by the killer. Tumbling downhill..with the killer.
Nonagon: Mysterious Windmill. (both sides)
Decagon: ascending crystal stairs. Falling through glass.
Hendecagon: Cubes rolling.
Dodecagon: Stabby Stabby!
Is the hendecagon one just a reference to the GameCube intro (which it sounds like)
What a great idea for a video, Algo. I like the voice narrated ones. The pentagon and hendecagon are good candidates for shorts.
Music for your nightmares Haha. It all sounds like terrifying circus music because of all the chromaticism and tritones. The 11-sided shape was semi-reminiscent of tubular bells only more disturbing somehow 😎
I would love to hear this spread over more octaves
And right angle triangles would be interesting too
I hope you make more of these
hendecagon sounds like an old nintendo sound effect
game cube starting up 😂
reminded me of old school Sesame Street from the 70s
I had no idea what the pentagon would sound like but as soon as I heard the chromatic it makes perfect sense.
Oh right, because it's circle of FIFTHS
❤
This rendering of tone intervals as a polygon of rotation is very clever! Now let's consider the IRREGULAR polygons of n sides.
Not only could this be a very easy way for students to visualize the triads and chord extensions, but perhaps also pick up a preliminary sense of how cadences work,
Honestly quite disappointing results, but that should have been expected because 12 is so divisible. Repeating this same exercise with chromatic scale instead of circle of fifth could be more interesting. Or using major scale, only 7 notes.
Ooh why not TET-19 with the circle of sixths!
The chromatic scale will give you the same stuff but in a different order.
I’d love to hear this series using different scales instead of the circle of fifths.. fascinating video!
Until now, I used to think that shape and music were unrelated. After watching this video, however, I realized that such things can be interconnected. I found it particularly fascinating how the number of angles in a shape corresponds to the difference in the number of notes played simultaneously. While I've had some interest in shapes, I've never really been into music. After watching this video, I feel like my understanding of music has improved compared to before. 10706
Nice video! Starting from the music end would be interesting - what's the irregular polygon that plays a major scale for example? (is there one?) - is there a shape that plays a 2 5 1 chord sequence, or an arpeggio/short melody etc.?
Nice video. Interesting intersection between math (geometry, groups and modular arithmetic) and music.
This is not just an intersection imo, music is just as much applied maths like physics and informatics are
2:55 why did that sound like galaxy colapse (song) 4:14 sounded that same
The 11-gon had me saying "no whammy no whammy big bucks big bucks!" 🤣
Hendecagon is my new favorite shape. I'll take tritones and chromatics all day. Thanks for making this wonderfully interesting video!
i shouldve entirely been prepared to have king gizzard enter my brain the moment a nonagon was mentioned but here we are. nonagon infinity opens the door
Cool video!! You're showing some very neat aspects of modular arithmetic, how co-primality can be used to make encodings, and how that fails (makes a chord vs a single note) when there's common divisors. How encryption and number theory overlap with music is just awesome (but also makes sense if you compare the maths).
Thanks for sharing!!
Love it. I have had similair ideas combining it with the colour wheel of light.
This is mesmerizing. My favorite video ever. Thanks for creating this video!
8:30. Ah so that's how King Crimson writes their music.
It really Thelas my hun Ginjeet.
This is interesting, but I would actually love to hear this where we hear the exact notes that are played where points touch the circle and not only when the exact contact points of the notes of the circle of fifths is touched.
Using the example of the pentagon. If the top point is touching C, the next point is touching a slightly sharpened D, next point is touching a much more sharpened E. next point is touching a slightly sharpened Db and final point is touching a more sharpened Eb.
Anyone else get what I mean by that?
And I'd also like to hear a steady transition of the motion travelling around the circle, like a sustained chord that is rising in pitch with exactly the intervals that the different polygons denote.
Each of the 12 segments of the pie can be broken into 30 microtones/pitches. So for example, C to G (and each of the other segments) actually has 30 subdivisions between the 2 notes. Where the points of the polygons touch at these points is what I'd really love to hear.
8:22 peckidna from MSM third track on magical nexus be like:
I love that the pentagon basically just reverts the circle of fifths (y'know, 5 sides). Completely and utterly logical and intuitive in hindsight, but I doubt many would've guessed that on their own!
The dodecagon creates a beautiful shifting rainbow on the keyboard!
Hendecagon:
Progressive Metal. Thanks for posting.
So little kids next to a piano are just Dodecegons. Got it.
Hendecagon reminded me of the forest temple from Legend of Zelda OoT. Clockwise and counter clockwise it reminded me of one of the 4 temple in Majora’s Mask.
Dodecagon = horror movie music.
This got real interesting when the notes were played sequentially. I expected a pentatonic chord for 5, but god chromatics. I find this approach both smart and creative. Just what music theory needs, after centuries with a system full of exceptions. Good work! You could animate the interval classes 1 thru 6 into a lydian scale using the formula n * (-1)^(n+m), n in 0...6, m being 0 or 1 for major and minor resp, the latter being tonal mode: 0,11,2,9,4,7,6,5, sorted and relative to 0: -5, -3, -1, 0, 2, 4, 6. Swap the m and you have the locrian (most minor) scale mode. Notice that negative offsets are odd and the positive even. So an Archimedean spiral would draw these scales, y's are n and x 's are pc, making x a function of y, that way matching the linear pitch axis horizontally, like on the piano keyboard. So I don't believe in 4096 sets, but in the Major scale, the only one containing all 6 interval classes, or 7 including the root. Nice, eh?
I love how the 11-gon is literally just tarkus
HENDECAGON
I was a music theory major in college and I find this more than extremely fascinating
Very interesting!
I do wonder how it would sound in equivalents of the circle of fifths in other tuning systems (if there exist any)
They exist.
For example, in 19 equal pitch divisions of the octave, the circle of perfect fifths can be described in steps of the tuning system as 0, 11, 3, 14, 6, 17, 9, 1, 12, 4, 15, 7, 18, 10, 2, 13, 5, 16, 8. It can be described in letters as F, C, G, D, A, E, B, F#, C#, G#, D#, A#, E# or Fb, B# or Cb, Gb, Db, Ab, Eb, Bb.
sry bold questions maybe ..but why is the circle build like that...?
f.e. why the black & white keys near counter splited or opposite.
would love to see an extended version based on 31-tet or other tuning systems
Second this, also for 19-, 24- and 53-TET
This is brilliant -- combining geometry and music and finding very interesting tonal patterns they create. I think there's a lot more to be investigated regarding this.
Nonagon infinity mentioned 🗣️🗣️
This is highly interesting and very well done, thank you for putting it in such an understandable way!
How are you making these animations?
These are all written in Java using a graphical library called Processing (processing.org), and the built-in Java MIDI library for writing out MIDI, which then gets realized as audio with DAW plug-ins.
A randomized bounce bouncey ball could make an ineresting chord progression. Kind of like a wind chime.
8:45 OMG!!! NINTENDO GAME CUBE!?
I didn’t know that Pythagoras and Phillip Glass had a love child that made videos.
Very resourceful!!
1:04 Super Mario Sunshine >:0
8:47 Starting on C, it’s really grooving if you subdivide 3+2+3+2+2
Keith Emerson Agrees: th-cam.com/video/AGGpBXd7ToA/w-d-xo.html
0:57 - a villain sneaking closer to you
6:19 _______ infinity opens the door
Dodecagon got some stank
3:50 this is perfect for the I swallowed shampoo, probably gonna die, it smelled like fruit, that was a lie, meme.
Literally just fourier series
... not really? Or, I’m not seeing it
not related... we're looking at mod 12 arithmetic
@@StefaanHimpe yea youre right
no, just because you ar me watching a linear series of 1x? that's very ambiguous
huh?
I'd be curious about an extension of this:
Rotating a poygon on an arbitrary plane slicing a cone
It would be an ellipse that touches, but draw rays from the center of the polygon, play notes when they cross one of the cone's vertical lines
The height of the cone could represent ... something
poygon
Legend has it that this is how the crash bandicoot soundtrack was written
That was fun. The later ones were mostly more interesting than the early ones. I' like to hear the 13-gon and the 17-gon being prime, which means none of the notes are played simultaneously - pure melody and fast. I would also like to hear what the polygons would sound like if instead of the circle of fifths ordering the straight chromatic scale ordering was used.
The nonagon nexus key is the basis of what youve stumbled upon. It encodes the solfeggio frequencies. A series of roation of the nonagon where the degrees are added up reveals this pattern. Going deeper through the math procrss of the nonagon nexus key, one can even derive the fine structure constant.
I think it would be fascinating to have two separate but different polygons play at the same time
Gamecube, it's Marvin. Your cousin, Marvin Cube. You know that new bootup sound you're looking for? Well, listen to this! 8:20
So this is the math that was being communicated in Close encounters of the Third kind!
The hendecagon is some Emerson Lake and Palmer status.
I really wanted to see the circle featured
6:10 Really opened the door for me
NONAGON INFINITY OPENS THE DOOR
My hearing is gone. I cant hear anything coming through the tiny speakers of a cell phone, so correct me if im wrong and if you care to. Or perhaps tell me if i was right.
The rotating octagoñ and the series of diminished chords whose roots ascend or descend in half stèps can be heard at the end of a song by the rock band "Queen".
The song is "you take my breath away" and the effect is achieved with an echo/delay. The delay is roughly one second or 60 beats per minute. Each beat is divided into 3 segments or triplets. First a guitar playing descending half steps (3 per second) so that as the guitar descends 3 half steps an echo of the same guitar repeats those notes as it continues moving down. Evenually a vocal singing the same series of descending chromatic tones using the words "take my breath" over and over .
Check it out. Check the entire song out. Or dont, i dont care.
8:49 GameCube?
What's interesting is every one of those sounds I've heard on a 1970s horror show or 1970 Syfy show. That is so interesting.
I'm curious what would happen if you had unusual shapes such as a triangle that had two long sides and one short side.
Man is literally changing the way we understand music
Just when I learned to draw a circle you now add all these others to learn !
all of this is fairly straightforward. the pent and heptagon seem odd at first but the circle of fifths is just that... FIVE, and its complement in base 12 is of course, seven...
what i see interesting is that a minor chord is a mirror reflection of its major... CEG faces the opposite way to CEbG... FAC vs FAbC... etc etc...
You can mix polygons to play shifting chord sequences
i love the decagon! sounds very trippy
I realized the Hexagon play the notes in both rotations!
8:47 This is the exact riff used in Those Who Chant by Walter Bishop Jr!
Musician: hey polygon, what notes do you play ?
Dodecagon: All of them.
Knowing how the pentagon and hendecagon alone behaved, what would a 55-gon sound like?
9:34 That’s crack up 😂 it’s like I’ve had enough
What a cool demonstration. Thank you for this.
Aw man, I wanted to hear all of them at the same time at the end
what happens when you map a clave pattern and rotate it?
Would be interested to hear each with square, saw, sine, and pulse waves sped up beyond the point where pitch discernment is perceptible (spinning the shapes as fast as possible along the Co5s). The geometric quality of the oscillation would probably be pretty pleasing when sped up and applied to wavetable synthesis.
That's something I've been imagining since I was a kid. now I'm wandering how useful it cold be
Trippin' hard on Hendecagon, like trancedelic asmr to my adhd, thx! 🤩 Dodecagon is just Jason Voorhees suddenly standing behind you.
i think the reason why it does this is that as the polygon rotates, notes are played in star patterns. the triangle creates the {12/3} star, which is actually three squares. It played all four squares at the same time. the square makes the {12/4} star, which is four triangles. {12/6} is six lines.
the pentagon creates the {12/5} star, which is an actual star, where one line hits all 12 points and then loops back on itself. the {12/7} star, if it existed, would be equivalent to the {12/5} star, which is why it does the same thing as the pentagon. given the nature of the circle of fifths, if you find the notes five away from a given note, it will give the notes next to it on the chromatic scale. since going five points down the road is basically what the {12/5} star is, it makes a chromatic scale.
similarly, constructing a {12/8} star will give you {12/4}, and so on.
I recommend using Shepard tones to ensure that none of the notes is higher than the rest