Pythagorean Comma

[@FuroTaki]

“ Wolf”gang Amadeus Mozart person will probably cry too…

[@ralyks._rgw]

its almost 1am, and im sitting here taking notes because i wanted to finally understand what just intonation and commas really are and you did a great job teaching it! So much so now I cant fall asleep because I want to learn more

[@dfhwze]

sound is one of the most fascinating aspects of life

[@excitinguniverseofmusictheory]

It's wiggly air that tickles my basilar membrane. Visceral enjoyment!

[@tfprimeguy101]

Beep hoop

[@codoof1]

This video is made so well to the point it makes the bleak intonation of the computer generated narration obsolete

[@excitinguniverseofmusictheory]

Thanks for the kind words!

[@gandalfthegrey2171]

I agree, first time I could get through a video that had C.G narration, imo the video would be better narrated by a real person but it's still a fantastically educative, interesting and accurate video. There are many other music theory and maths videos out there which are not as interesting or are misleading and untrue.

[@excitinguniverseofmusictheory]

The fake narrator means I don't have to set up a microphone, and y'all don't have to endure listening to my weird Canadian accent

[@gandalfthegrey2171]

Aw I love the Canadian accent, way better than American. But fair enough dude :)

[@flipperpitstudio]

Absolutely my thought. 😁👍

[@SebastienZunino]

It’s without any doubt my future favorite channel !!! Thank you so much to share all this knowledge !!! Greetings from French who live in Spain !!! 🎸🇫🇷☯️🌎

[@excitinguniverseofmusictheory]

Thanks Sebastien! I'll try to make one of these every once in a while. The challenge is trying not to repeat subjects that have been covered well by all the other KZbinrs.

[@alexgabriel5423]

You in a way missed the boat...because Ottoman Classical Music and Byzantine music use the Pythagorean comma as a fundamental unit to build a whole tone>>9 Pythagorean commas make up one whole tone...then 1 tone + 1 comma give the first or small sharp, 1 tone + 4 commas give the second or large sharp. A first small flat is a tone - 1; a second or large flat is a tone -4 commas. Thus there are 4 accidentals between Do and Re, and a sharp is never the same as a flat in these musical traditions. The Ottoman tanbur shows the system...it s still seen and heard in Turkey. Byzantine music of the Greek Church can be played on it too although the Church does not admit instruments....

[@angelmendez-rivera351]

To be more accurate, these forms of music use a justly intonated version of 53-EDO, where 31 of the steps are Pythagorean commas, and 22 of them are syntonic commas, but in most contexts, the two are interchangeable, since their difference is inaudible.

[@nickrobinson2023]

So well explained. It's a very complex matter if you start to look at all the maths and ratios. It can be overwhelming and off-putting for the average person like me. So thanks for that. Thumbs up and I will share. I will check your Patreon channel to see what it's all about. Thanks and keep up the good work.

[@pentapandamusic]

This helped me a lot, maybe you should feature this video and "What is a mode?" in your channel page so that they get more exposure. Right now in your channel page, all people see is a bunch of scales.

[@excitinguniverseofmusictheory]

yeah, that's a good point. I think I have the scale videos organized into playlists but they still obscure the few pedagogical videos in there

[@pentapandamusic]

@@excitinguniverseofmusictheory thanks again for the teaching! Great content

[@lorenzojw04]

Wow this video was actually insanely interesting lol, glad I learned this

[@MusicWithDan]

Werckmeister III uses the Pythagorean sixth for notes D-flat and B-flat. D-flat coincidentally relates to C by a ratio of 256/243.

[@chrispa23]

This voice sounds like the upstart crow. love it.

[@geoffroymb]

Nice comparison with the earth’s rotation!

[@firasb-ck1dj]

Thank you very much for this beautiful explanation

[@Anonymous-th6mq]

I now understand the flaws of Pythagorean tuning

[@excitinguniverseofmusictheory]

It's subtle, isn't it? I'm glad this video demonstrated it well for you. It's not an easy concept to explain with just words.

[@angelmendez-rivera351]

This is not a flaw of Pythagorean tuning, so much as it is a flaw with the diatonic scale and chromatic scale trying to force the intervals to work with only 12 divisions of the octave. There are many workarounds to this issue that do not involve giving up on Pythagorean tuning altogether.

[@angelmendez-rivera351]

2:42 - 2:50 No, it is called a perfect fifth. Also, this tuning need not apply to the fifth degree of a scale, so designating this interval as a fifth is completely arbitrary. It is only called a fifth in the context of the diatonic scale. 2:51 - 2:56 No, the ratio is 3/2, rather than 2/3. Fundamental intervals such as this one are defined as ascending in frequency.

[@excitinguniverseofmusictheory]

All salient points. What we call a "fifth" is only so (in one sense) if it marks a generic interval of 5 steps in a scale or pitch class set -- not just a diatonic scale. But I think you can appreciate that most people watching this video will understand that what we refer to as a 5th is an interval of 7 semitones in a 12-tone chromatic universe. And to your second point I agree that 3/2 is the better way to express that ratio, but in the narration you'll hear that it's referencing the second and third harmonics. Showing them on the screen as 2:3 is really not going to confuse anyone, especially since just seconds later it explains that ascending by a fifth is done by multiplying by 1.5, or 3/2. I appreciate someone with your attention to detail and obvious expertise in this subject watching my videos! Thank you and I hope you enjoy the other materials in this series. Cheers

[@angelmendez-rivera351]

@@excitinguniverseofmusictheory Thank you for your kind response, I appreciate it. *What we call a "fifth" is only so (in one sense) if it marks a generic interval of 5 steps in a scale or pitch class set -- not just a diatonic scale.* Technically, you could re-adapt the language in such a manner, but non-diatonic scales that are not related in some fashion to the chromatic scale with 12 steps are xenharmonic, and thus use a completely different type of intervallic vocabulary. Besides, if you want to use a mathematically sound naming scheme, we need to shift the numbering of steps, because what we call the "unison" actually should be called the zeroth degree of the scale. However, I understand this is beyond the scope of what you are saying, so I will refrain from making a deal out of that. *But I think you can appreciate that most people watching this video will understand that what we refer to as a 5th is an interval of 7 semitones in a 12-tone chromatic universe.* I understand as much. All I am trying to do is to clarify the proper context of your statements, and to make sure that people are aware that there do exist other contexts where the things as presented in the video are not necessarily true. I think this clarification is actually extremely important, because, without this clarification, people have historically, and in the future, will continue to, gotten under the impression that this is how _all_ music, universally and in all eras, works and has always worked, and that is just not true. It is true of one specific musical tradition, which while dominant in our culture, is not dominant everywhere. Keep in mind KZbin is a global platform, and while the predominant use of the English language would normally indicate that the things in the video spoken about are only applicable to Westerners, English is a primary language all over the world. Most countries have English as a second language and a mandatory language in schools. So, when people come around and see this video without clarification, they are going to start treating this tradition of music as the only real tradition of music. We already have a lot of elitism coming from music theorists and musicologists. We need to avoid accidentally having people reinforce those negative biases on top of that by using mathematics. I say all this because I have experienced it in first-person, I am not presenting baseless speculation or nitpicking something that will have no effect on people's biases. *And to your second point I agree that 3/2 is the better way to express that ratio, but in the narration you'll hear that it's referencing the second and third harmonics.* I do not believe this addresses my contention. I explain further below. *Showing them on the screen as 2:3 is really not going to confuse anyone, especially since just seconds later it explains that ascending by a fifth is done by multiplying by 1.5, or 3/2.* Actually, I disagree. I do believe this will be confusing to many people, especially those closer to the beginner level who are only hearing about the mathematics for the first time ever. If you show the number 2/3 on screen, but then say the number 3/2 voiced, even if they conceptually point to the same idea, presentation-wise, that is a contradiction. To be consistent, you really should either only use 2/3, or only use 3/2, not both. But since the convention in this musical tradition you are presenting about is that of ascending scales, rather than descending scales, you really should use 3/2. This is a minor point, and I do not think it takes away from the video, but it is not a point I think I should retract. *I appreciate someone with your attention to detail and obvious expertise in this subject watching my videos!* Thank you, I appreciate your response.

[@excitinguniverseofmusictheory]

@@angelmendez-rivera351 if I ever redo this video, I'll take these points into consideration and rewrite the script for better accuracy. Cheers

[@saxachewon8062]

Thanks for the excellent explanation. I find it easier to understand numbers with decimals instead of fractions.

[@adambelarouchia4429]

Dude these vids are so amazing!

[@excitinguniverseofmusictheory]

hey thanks!

[@nickrobinson2023]

Adding a link to your Patreon channel would be helpful :)

[@shaofu424]

clear and concise good work be safe, be happy, work hard peace \m/

[@jadefoo455]

yes

[@VS3d0v]

Hey, what a tremendous work with all these scales videos. Pity there aren’t too many lessons though.

[@excitinguniverseofmusictheory]

I agree, the lesson-type videos take a disproportionately massive amount of time to produce. I’m impressed by KZbinrs like 3blue1brown who makes beautiful animated lessons of such high quality so often. If I had more time and a lot more skill that is what I would aspire to.

[@pentapandamusic]

Could introduce equal temperament at the end

[@dhaighy]

Dude you're a genius.

[@excitinguniverseofmusictheory]

oh STOP IT.

[@pazserenaeuna]

Wow amazing and beautiful

[@pavelmanEC]

It needs a second part.... For werckmaister, just intonation....

[@excitinguniverseofmusictheory]

I agree! The sequel should continue the story with temperament, meantone, 12TET and other tuning strategies.

[@MichaIsraelKovler]

seems with pythagorean tuning each note has a character of its own. And each chord/interval - as well. Wolves are beautiful animals - and so is that interval. It is totally NOT wrong, but different. But hey, since when different is bad? I have a guess but this is not the place for it. Try listing to the Wolfe interval with "unprecedented equal temperament ears"!

[@chrisrosenkreuz23]

Kind of poetic that you mentioned the leap year, since the descending Pythagorean comma is exactly -23,46 cents which is exactly the degrees of tilt that the Earth axis has, which in turn makes the seasons possible, which in turn make the leap year possible

[@angelmendez-rivera351]

The Pythagorean comma is not exactly -23.46 cents. The number of cents is not a rational number. Also, the axial tilt of the Earth is not constant, but it varies, according to the Milankovic cycles.

[@chrisrosenkreuz23]

@@angelmendez-rivera351 I was making a point about the different ways of measuring the harmony of the universe around us. Music is not different from geometry in the way most people think. The fact that there are slight differences in the numbers does not make that correlation go away.

[@angelmendez-rivera351]

@@chrisrosenkreuz23 What you talked about has nothing to do with geometry, nor the universe in general.

[@chrisrosenkreuz23]

@@angelmendez-rivera351 yes it has, the implication was obvious, but somehow you missed it. Better luck next time.

[@angelmendez-rivera351]

@@chrisrosenkreuz23 No. Insisting that a non-existing relationship is there does not make it exist. I have a degree in physics. I am not missing anything here. I am just not so naive to buy nonsense.

[@dragonfly873]

Thank you a lot, thanks

[@MH-ln6pv]

Ah, so this is what spambots get up to in their off days.

[@excitinguniverseofmusictheory]

The answer is “email spam and music theory” ... “Alex, what are two things that offer no benefit to mankind?” Yes for 400 go again

[@luoshengying]

2.29 a Good metaphor

[@alexpierce4380]

Yay band you uploaded

[@Anonymous-th6mq]

Alex Pierce yay band

[@alexpierce4380]

Anonymous Medina is god

[@mr.peterson9575]

How many octaves does it take until a fifth is a fifth again?

[@excitinguniverseofmusictheory]

"Until a fifth is a fifth again" is an odd way to phrase the question, since every one of the fifths is a fifth; but I think what you're asking is if we can keep going up and up and up and find a place where the fifths are equal to a whole number of octaves. In other words, does the circle of not-equal fifths ever meet back up with itself. The answer is no. Phrasing the question a different way, you're asking if there is some value of k or m where (3/2)^k is equal to 2^m. As an equation: (3/2)^k = 2^m for some integers k and m. Rewrite the left-hand side to make it easier to compare: (3/2)^k = 3^k/2^k So we have 3^k/2^k = 2^m Multiplying both side by 2^k: 3^k = 2^(k+m) This equation implies that 3^k must be a power of 2. However 3^k can never be a power of 2, because any power of 3 will have 3 as a prime factor, and any power of 2 will only have 2 as a prime factor. Since 3 and 2 are distinct prime numbers, it is impossible for 3^k to equal 2^(k+m) for any values of k or m. Therefore there is no way to keep adding justly tuned fifths and end up being equal to a whole number of octaves. The circle of fifths will keep going on and on getting more out of tune the more you keep adding fifths. This is evident by the fact that our equally tempered 12-TET tuning uses powers of the square root of 12, which is an irrational number which by definition can never become an integer by raising to a power.

[@bgqt]

Caseoh thug shake skibidi gyatt

[@jkmarksongs]

8:30 Waerewahh super sir

[@PeterJ42a]

At 6.22, one fifth down from B flat is E flat; not C flat.

[@excitinguniverseofmusictheory]

The narrator says “giving us E flat”, the consonant at the end of “us” is not well enunciated, and makes it sound like “giving a C flat”

[@worldofmakam]

Concise and to-the-point. You should delve into the discussions around these problems. If you can continue this series and provide an English subtitle I can translate them into Turkish.

[@ianring]

I agree! This video just barely scratches the surface of an incredibly deep topic. Good point about the subtitles, I’ll add those soon (for the hearing impaired as well as for translation)

[@MarwarMusicSchoolJodhpur]

Sir, what is frequeancy of Harmonium notes ?

[@excitinguniverseofmusictheory]

I have no special expertise in this. Google search brought up this: see the section on "tuning" www.keshav-music.com/harmonium-faqs

[@sorrychangedmyusername3594]

This is just advanced music on methematics

[@pavelmanEC]

Hi, a question. What do you use in order to create the animations?

[@excitinguniverseofmusictheory]

The videos are created using Apple Motion, which is basically the cheaper entry-level version of Final Cut Pro www.apple.com/ca/final-cut-pro/motion/

[@ruckers1624]

Thank you for sharing this video. It's very good - but that voice... annoyingly Hawkins like :D . A bit on the "mathy" side... would have liked more practical examples of what a pure 5th sounds like compared to a just/equal temperament 5th. Question though: no one ever explained tuning a harpsichord starting on the pitch d. That threw me off. All historical context for all historical tuning that I know of start on either C and/or more modernly A-440ish. All known historically preserved tuning forks are on a reference point C between a 5th below modern pitch and/or as much as a 5th above (Italian Organs from the Renaissance period). My personal tuning fork I use at home for my collection is C at 493.9hz. Just curious about your frame of reference, if you have time and don't mind responding. In every case thanks for the video and for sharing. Always willing to learn something new!

[@excitinguniverseofmusictheory]

ruckers1624 hi! The choice of using D as the origin tone was prescribed because of the harpsichord sample library I used to generate the audio. It has settings to change the temperament, and when I chose “Pythagorean”, the wolf was equidistant from D, not the C. Also convenient that I found a D fork on amazon to use as a graphic so I went with it. Not that it matters to the tuning; it’s more about how far you travel in the Line or Fifths in either direction than where you started from. I didn’t do any historic research to see if that was a normal place to start on the line. Thanks for pointing out that oddity! If I ever redo the video in higher resolution I will follow the more normal routine for tuning. Cheers

[@fgonzalez90]

Also, D lies right in the center of the natural pitches in the FCGDAEB progression (note the symmetry from D vs. the asymmetry from C), so it's a good starting point to create an "even" circle of fifths/fourths. Perhaps, that's why the dorian mode was so popular in medieval music.

[@ruckers1624]

Thank you to you both for the great replies it's very plausible and of course very interesting for me to rethink this from a more theoretical view rather than a practical View. I like the idea of D being a total sensor y Dorian mode might have lasted so long. Very clever way of thinking even if it's not backed by any historical data. Thank you again for sharing and for taking your time to reply.

[@excitinguniverseofmusictheory]

@@fgonzalez90 I visualize it like putting a belt around your waist that doesn't quite fit perfectly. If you start wrapping at your left hip, the buckle will be at your right hip and won't do up without a little tug to pull it together - that SQUEEZE is the Comma!. If you start wrapping from the back, the buckle will be in the front and you'll have to >TUG< and suck in your gut to do it up. Comma. The buckle is where the WOLF is, and you can tune your scale starting wherever you like so that the Wolf isn't in an inconvenient spot, like in the middle of your back... or between E and F on a harpsicord. I can empathize that old-timers tuning an instrument with Just fifths would want that comma to be somewhere that it's not going to be perceptible when playing diatonic music in common keys like C, or F, or D etc.

[@chisaomusician7752]

What picture of the planets and zodiac is that at the end?

[@excitinguniverseofmusictheory]

en.wikipedia.org/wiki/Petrus_Apianus#/media/File:Ptolemaicsystem-small.png

[@johnsontrimble7881]

2:51 I think you meant 3:2. 2:3 would be an undertone

[@marcusenevoldsen9336]

Does anyone know how to subtract a syntonic comma (81/80) from a just fifth. Please, i am begging you. I just can't figure it out.

[@excitinguniverseofmusictheory]

well it would be whatever the frequency times 3/2 (up a fifth), divided by 81/80 (down the comma). Or multiply by 80/81, the reciprocal. Same diff. unless I misunderstood the question?

[@marcusenevoldsen9336]

@@excitinguniverseofmusictheory Right, I found out

[@garyreidmusic]

This doesn't explain How or Where we add the compensation for the missing bit...

[@whyyeseyec]

The universe is not perfect - fifth or otherwise. Don't believe me? Ask pi.

[@angelmendez-rivera351]

@@whyyeseyec π has nothing to do with the universe not being perfect. There is nothing wrong with π.

[@jond532]

why cant the universe just be perfect

[@angelmendez-rivera351]

What do you mean?

[@jond532]

@@angelmendez-rivera351 The 12 perfect fifths don't loop around , there isn't 360 days in a year, pi isn't 3, The speed of light isn't 300million meters per second. A cat doesn't get 10 lives.

[@angelmendez-rivera351]

@@jond532 12 perfect fifths do not perfectly approximate 7 octaves, but 53 perfect fifths are indistinguishable from 31 octaves. The difference between the two intervals is called Mercator's comma, with ratio 3^53/2^84. This interval is so small, it is not audible, so it is as if 53 perfect fifths were exactly equal to 31 octaves. There are not 360 days in a year, but this is not important. 360 is not a special number, or at least, not more special than 365 or any other number. Also, how many days there are in a year varies over time. The speed of light is 299 792 456 meters per second, not 3E8 meters per second, but this is only because of our choice of unit of length, the meter, which is arbitrary. We totally could just use a unit of length u such that 299792457/300000000 m = 1 u. Then the speed of light is equal to 3E8 u. This, still, is completely arbitrary, and there is nothing that makes 30000000 more special than 299792456 as a number. In fact, if we use hexagesimal instead of decimal, the numbers will look completely different, even with the same units. In the most natural units, the speed of light is 1 Planck length per Planck time. There is nothing imperfect about this. IDK what you are talking about. Cats are superior creatures. They get however many lives they want to get. Us mere mortals are in no position to judge how many lives a cat should or should not have. How dare you.

[@angelmendez-rivera351]

@@jond532 Also, π is not 3, but aside from being an integer, 3 is not special either. I fail to see how π not being 3 makes the universe not perfect. Besides, in the grand scheme of things, π is not that important of a constant. e is a far more important mathematical constant, if you ask me. Even more important is discussing functions, rather than constants. The exponential functions are fundamentally important in mathematics. There are other mathematical constants that are even more important, I would argue, such as sqrt(-1) or ζ(3), or γ.

[@jond532]

@@angelmendez-rivera351 I disagree I think 3, 12 and 360 are special numbers. I don't think the units we use are arbitrary.

[@carultch]

Why is it called a comma?

[@excitinguniverseofmusictheory]

The word "comma" came via Latin from Greek κόμμα, from earlier *κοπ-μα = "the result or effect of cutting" (for etymology see wikt:κόμμα#Ancient_Greek) en.wikipedia.org/wiki/Comma_(music)

[@janzenalleman2952]

Cmon now 1 dislike??

[@excitinguniverseofmusictheory]

h8rs gonna h8 h8 h8 h8 h8

[@eric4709]

Video = 10/10 !! Narrative = 0 /10 Can we PLEASE have a proper human voice ??

[@excitinguniverseofmusictheory]

"Daniel" the artificial narrator is deeply offended

[@eric4709]

How can an artificial person be offended? Surely that proves the point !@@excitinguniverseofmusictheory

[@excitinguniverseofmusictheory]

The real reason is more practical... using the artificial voice means I can make these without doing any fussy audio recording

[@eric4709]

@@excitinguniverseofmusictheory That I can understand .... but .... why spoil such an excellent video with what is an uninspiring and monotonous sounding voice over ? The video is worth soooo much more Surely it is worth the effort - even if you have to employ someone to do the reading of text ?

[@MichaIsraelKovler]

@@excitinguniverseofmusictheory I'll buy you a coffee for a human voice. despite, I watched it ALL

[@DerLiesl]

A=432 physically hurts

[@DerLiesl]

I do love your channel, though

[@criburgir]

It doesn't matter what A sounds like, what matters is the intervals between the notes

[@vodorigagargoyle1640]

you're brain is out of tune

[@eugenedumitrescu2187]

Lame low-effort audio

[@excitinguniverseofmusictheory]

Good observation