Some additional thoughts: 1) It's worth noting that just because two notes are close together, that doesn't mean they're a comma. For instance, Partch's Genesis scale contains notes as close as 15 cents apart, but they're considered different notes and thus their difference isn't a comma. Or, I guess it could be if you were tempering that difference out in a different scale, but in the context of the Genesis Scale it doesn't count as one. 2) I glossed over this, but depending on which definition you like, it may not be accurate to describe Pythagorean tuning as a tempered scale. It tempers certain ratios, but it's built out of strictly just-intonation bits. It's just some of those bits get very complex. Some sources do call it a temperament, others don't. I'm largely agnostic on the issue, if I'm being honest.
Also: Extended Pythagorean, as described in much mediaeval Middle Eastern music, actually produces a third only 1.954 cents away from a pure 5/4 by stacking eight perfect 4/3 fourths and reducing by octaves, which in Western music would be spelled as a diminished fourth. That tiny, tiny comma is known as the schisma and is functionally tempered out in many just intonation performances and disappears outright in tunings like 41- and 53-EDO.
Thanks for the video! A question though. If I recall my history correctly, the pythagorean tuning produces 13 different notes, and the comma is the result of discarding either the Gb or the F# which are 24 cents away from each other, and then people discarded one or the other to produce a 12 tone series. I may well have misunderstood something, there, but if not, why not just use 13 notes and call it good? Might be hard to implement on fretted instruments but few care about that, and you could make a keyboard with 13 keys to an octave easily enough. Why didn't they do it? Triskaidekaphobia, or is there more to it?
@Laughing Daffodils: It goes back to the circle of fifths. The circle is only closed if we treat Gb and F# as enharmonically equivalent. If we don't, then the "circle" of fifths is really a spiral, and you're back to the problem discussed at the outset of this video: you can keep going up perfect fifths and never reach an octave.
I think Partch would disagree. I unfortunately do not have my copy of Genesis of a Music with me right now, but the interval 81/80, which occurs between several sequential notes in Partch's monophonic fabric, is pretty much universally called a comma both in Partch's writings and in the microtonal community in general. I think it's possible to conceptualize that something might be two different conceptual pitch classes and also a comma apart.
A bit of a tangent, but I watched a presentation showing how we use a comma to make the major scale and pentatonic scale and even in resolving polyrhythms. This is probably a concept you're already familiar with but maybe it would make for an interesting topic for a future video.
When I was in high school, and just learned that perfect fifths are 3:2, I sat down with my calculator, typed in 440, multiplied it by 1.5 twelve times, divided by 2 seven times, and got roughly 446. I assumed I messed up somewhere along the way, and put it out of my mind until years later, when I learned about intonation, temperaments, and commas, from Wikipedia. Vindication! (Interestingly enough, this comment contains 12 commas. 😉)
In 53-TET: - The perfect fifth are only 0.07 cents flat. - The major third is only 1.4 cents flat. - The minor third is only 1.3 cents sharp. - The harmonic seventh is only 4.8 cents sharp. 53-TET does not temper out the syntonic comma. It gets mapped to one step (22.6 cents).
Most pianos have 3 strings for every note to make it sound louder, so an old piano could have some of those strings off tuned from each other, thus getting a comma appart. Maybe..
Martin Roth's version of Barber's Adagio for Strings actually uses detuned notes like this. I love the effect. th-cam.com/video/6hKxAE-O6Uc/w-d-xo.html
You should listen to Christopher Bailey's "Concerto for shitty piano" christopherbailey.bandcamp.com/track/composition-for-s-piano-drum-samples-concr-te-sounds-and-processing-2013
It's also what gives Accordion it's distinctive buzz! There are multiple sets of reeds SLIGHTLY tuned off from each other. But when its played using that piano sound, it just feels like teaching piano. No student's piano is ever tuned!
Dang, what a nice introduction to commas! It's a fascinating, and insanely nerdy world. I personally am a fan of the various diesis..es. Related note...what is the plural of diesis?
More seriously my original plan for this video involved lots of diesis talk, but I decided that should probably be its own video. Which I probably won't make for a while 'cause I have some other tuning stuff I wanna do first and also I try not to do _too_ much tuning stuff in a row, but I didn't want to just cram them in when just talking about the big dog commas got me over 9 minutes.
So what you're really saying, then, is that 12-tone equal temperament tries to hide the syntonic comma by making it fade into the background of equal intervals. In other words, it's the original comma chameleon.
Having flashbacks to middle school orchestra, when I got bent out of shape once because SOMEONE behind me had their violin just slightly out of tune and I was not yet socially aware enough to not correct them
I've been playing guitar since I was 5, and have tuned by ear since I was 8... I can say, it pisses me off when someone's guitar is slightly out of tune... It makes me cringe, and sometimes I wanna rip the guitar away from the person instead of correcting them....
This reminds me of the chocolate bar illusion 12 P5s is almost the same as 7 octaves. That's neat because a perfect fifth is 7 half steps up and an octave is 12 half steps up. It's like an inversion
Really-excellent summary! Another thing worth mentioning: Quasi-coincidentally, Quarter-Comma Meantone tuning is audibly identical with 31 equal steps per octave (AKA “31TET” or “31EDO”). As you correctly pointed out, if you stack up only 12 quarter-comma-flat fifths, you end up with a horrible-sounding wolf fifth. The circle of fifths fails to close by a horrendous-sounding 41 cents. However, if you stack up 31 of them, the circle of quarter-comma-flat fifths fails to close by only 6 cents! 6 cents is only barely audible in-real-world music anyway, but spreading that already-tiny error across 31 fifths (18 octaves), the difference between a QC Meantone Perfect 5th and a 31TET P5, is less than 1/5 of a cent, which is for-sure completely inaudible. Even if you stack up *_12_* QC Meantone P5s against 12 31TET P5s, the difference is still only a little over 2 cents, which, in normal-speed music, is still pretty much inaudible.
This is great! I'm so happy that microtonal music theory is surging in popular TH-cam channels. You explained the idea of a vanishing comma better than most people I know! I'm looking forward to any other microtonal exploration you undertake on this channel.
*_Warning: wild speculation._* I've been messing around with note pitches lately and it gave me an idea. "What would be the single most profound way to get a fundamental frequency", I thought. Well, what is a frequency but cycles per second? And what's a second? It's defined as "the duration of 9,192,631,770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium-133 atom". Fancy science talk out of the way, we've got a note with the frequency of 9,192,631,770 Hertz. Let's call it *S.* That's way out of the hearing range of anything that lives on Earth, so I divided it by 2^25, i.e. lowered it by 25 octaves, which gave me the frequency of 273.961775600910186767578125 Hz (yes, that's a lot of digits you don't really need, but I'll keep them just for purity's sake). That falls within the range of the first-line octave and is very close to C#4 (the most accurate pitch for it I was able to find is 277.1826 Hz). S4 is higher than C#4 by 20.234496 cents. For comparison, the syntonic comma is 21.51 cents and the diaschisma is 19.55. I'm not quite sure where the idea goes beyond this point. Maybe it becomes the new A=432. I hope not. So far, there is one thing I will claim: _as the discoverer of this universally significant constant, I shall name the interval of 20.234496 cents... _*_The Temporal Comma._*
The problem with all of this is that a second is still an arbitrary unit of time, despite its being precisely defined. Similarly, designating any given frequency (regardless of of how many digits it has after the decimal point) as A, B, C, or any other pitch is also arbitrary. While absolute definitions for some things might be elusive, one can get a lot of mileage out of relative relationships.
It is a flaw, yes, but it is also a human construct, based on the way _humans_ perceive time. And those humans are the main audience around here, so I think my logic still applies.
@@jbh001 a second is an arbitrary unit of time, but the time of radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium-133 atom is not as much. Yet, arguably the most truly fundamental time in physics (at least, in its modern understanding) is the Planck time, as it depends only on the three fundamental physical constants - the Planck unit, the speed of light and the gravitational constant - with no numerical constants to multiply. However, it is 5.39 × 10^−44 seconds, so you will have to bring it down by quite a bit of octaves down to bring it to the human range of hearing - I'd say about a hundred and twenty?
@@thomaswinwood That's really neat. You can also do 4 * planck_freq / 2^(1/fine_structure_constant) to get 415.35 Hz. If you call that Ab then you get A440.05842Hz. Take that, A432-ers!
I'm definitely going to save this one for later, when people ask me questions about piano tuning. But on that _note_ , I would like to add that even octaves aren't safe. Not on the piano, at least. We tend to tune them ever so slightly wider than they ought to be, its size depending on the room or hall the instrument's in. A grand piano tuned with perfect octaves sounds off in big halls, for instance. What I'm taught in school is this: building up towards the higher registers, we aim for a twelfth to be perfect. Given that the fifth is sliiiightly smaller than perfect, the octave naturally compensates for this by the same amount. Thanks for sharing this, perhaps future students will find it useful too :)
The ever dreaded wolf note is the bane of violin luthiers. It took me 3 months, several minute adjustments and trying 4 different types of strings to quiet the wolf on my last violin. My cello on the other hand, is quite tame.
5:17 "it takes a while, but 53 perfect 5ths is actually almost identical to 31 octaves." 5:34 "these two notes wind up only 3.6 cents away from each other, which is pretty spot-on."
@@peteroselador6132 Nicer major and minor thirds/sixths/sevenths and alternate versions of these that don't exist in 12 equal Pretty close to quarter comma meantone It makes chords sound really good and has more melodic variety th-cam.com/video/Q2bmXl3zgdE/w-d-xo.html th-cam.com/video/L_cx4MylmXk/w-d-xo.html
One of the fundamental flaws in most tuning theory is the notion that dissonance is intrinsically bad. Dissonance is in fact extremely useful artistically. In my mind, the better temperaments are those that give us a whole palette of different sounds to choose from. Some keys are consonant and calm, others are more dissonant and edgy. Some will have nice fifths, others nice thirds. Equal temperament seems the worst choice of all, because every key is identical. And what's the point of having twelve keys if they all sound the same?
i actually LOVE the ever so slight dissonance you play at the beggining. i found out that a distorted guitar with that tiny bit of dissonance in maybe one string sounds so powerful
This is the scale I like best from a maths perspective: 1:125:2410:9 B:H 6:55:44:325:1836:253:28:55:3 3B:2H 9:548:25 B:H is the ratio of 2 to the square root of 3. This is also the ratio between the base of an equilateral triangle to its height. B:H is the geometric mean of 125:108 and 144:125 and is really amazingly close to both. The ratio between the two is 15625:15552. The scale is based on a progression from 10:9 to 9:5 where is step is the ratio 6:5. There are 9 major triads (3:4:5) and 9 minor triads (12:15:20). Once you see the relationship between these triads you will see that this crysraline structure is a thing of beauty in its own right.
Idea: Don’t use tuning systems. Tune every note to the root of the chord being played, for _each_ chord you’re going to play, changing the exact notes as you play.
That's basically adaptive just intonation. A capella groups such as barbershop quartets intuitively approximate it, and there's software to do it on MIDI virtual instruments. This approach tends to result in comma pumps, which are why even an expert group of singers can end up singing the tonic some cents sharp or flat from the pitch they originally sang it at, which makes it sound like the key slightly changed. If there's a comma pump you want to avoid, you can temper out the offending comma or just make it an interval you use.
Thank you so much, this is the best explanation I've seen of why we can't have perfect tuning systems. Other people explain that stacked octaves and stacked fifths don't equal each other, but nobody I've seen ever brought physics into it and proved it using mathematical equations.
You can fudge it well with enough notes, but it depends on how many distinct intervals and harmonics you want to represent and how well. I'm personally quite fond of 41-EDO but notation and keyboards for such a large gamut can be frustrating, not to mention instruments like guitars...
I think the fact that there are no perfect scales is one of the best things about tuning theory. Creating a tonal system (or even just choosing) can become a creative thing on it's own. Of course we don't really do that in the west, as we are kind of stuck with 12edo, but hopefully that will change. Look I'm not saying that 12 is bad, I'm just saying that you only ever order strawberry ice cream but maybe you could try some blueberry ice cream, you might like it. And try out 17 and 22, they are damn cool divisions of the octave.
This is hands-down one of the BEST explanations I've ever seen of the problems of tuning. I remember when I was in high school and tuning my guitar, I realized I could just tune it to chords I was playing. I'd play a G chord, then adjust the 'b' string and I noticed it sounded really good. Then I'd play an E chord and it would sound awful! I thought it was a problem with me, or with my guitar, and it used to drive me mad. It wasn't until years later that I understood what was happening.
So, just as an aside about Pythagorean tuning (length warning. Also Quack Quack): I love it, or offshoots of it. See, as with any interval you stack, you'll eventually hit a 'Moment of Symmetry' as described by Erv Wilson, an internally consistent gamut made up of only two step sizes. With the fifth, it's at 12 notes, then 17, then 29, then 41, then 53. Two cool things; 1. If you model where the intervals lie on something like a fretboard the note distribution looks almost exactly like the '7 diatonic notes and 5 accidentals' you find on a piano. 2. You can mess about with the parameters to get a stupid amount of variety that's internally consistent. Say you have an MOS at 17 notes, composed of 5 (roughly) Pythagorean commas and 12 semitones at about 90c. If you dont change the order of the notes, but there respective sizes (like making all the commas bigger and the semitones smaller) you get different tonal possibilities, but the fifths will still stay quite pure (less the 3 to 5 cents out), and you'll only ever have one wolf. MOS at 29? Note orientation still looks like piano keys. Very internally consistent. Can mess with it to get a tonne of variety. Only one wolf. Exactly the same for 41. Piano key arrangement. Consistent. Lots of potential variety. Only one wolf. I'd give 53 more than a passing mention if I didn't find it too big to be useful.
One of my (unauditioned and hence having a rather low average skill level) choirs for a while had a conductor who was mad keen on just intonation and recommended a book called "How Equal Temperament Ruined Harmony (and Why You Should Care)". It was slightly ironic that (IIRC - I must re-read it carefully) the book recommended mean tone rather than just intonation, pointing out that quarter-comma MT gives a scale with only one sort of whole tone, whereas the conductor was wont to rave about (among other topics) the difference between a "major" (9:8) tone and a "minor" (10:9) one. I seriously doubted that the individual singers tuned to each other closer than these differences anyway.
It just occurred to me that you could make an equal temperament scale where the perfect fifth is always just and the octave is slightly sharp. And now I want to know what that sounds like.
in the era of digital music I have a question -> is sticking to one tuning really necessary throughout a piece? Would it be possible to retune the instruments infinitely as a song is played to get perfect ratios? Start on any given note play the just intonation notes that follow and have the key fluctuate or even migrate over the course of the song? It would be interesting to have a song with a key change partway through to something near the original key without it ever sounding like the song had moved away from that key. Or would the whole song sound slightly out of tune the whole time?
I wish I never learnt all this stuff, I do not have perfect pitch but now whenever I play my guitar and songs/whatever in different keys I have the anxiety that it's out of tune and probably some note is and I just subconciously feel discomfort rather than knowing what is not in tune.
I feel like the easiest way to understand the pythagorean comma would be to first show that 4/3 is the inverse of 3/2, and then show the coincident harmonic to tune them, then go through the cycle of 5ths/4ths within an octave to show how when you arrive at the octave, you are now 23ish cents sharp from the pure octave. My grandpa showed me this in person at an actual piano by literally going through and tuning it that way when I was in high school and it blew my mind. Totally made music and tuning and the harmonic series and all of that suddenly make sense all in one swoop. Like, oh shit, this is where it all comes from 🤯
Very interesting video! I love hearing about harmonic ratios in different tuning systems. I've been thinking about trying to define an interval space made of vectors where the "fundamental" vectors are just intervals like P8, P5, M3, and H7 (harmonic 7th), and trying to visualize the Major and minor triads as triangles in that vector space. And I think 7-chords should be tetrahedrons (triangular pyramids) in that space. Now that I've seen this video I think I might be able to put into words the problems I've been running into. One of them is the IIm7 chord. If you base it on the "2" as (P5 + P5 - P8), then there's a comma on the chord-7th, which would be the root of the scale if it weren't for that comma. On the other hand if you base it on the "2" as (P8 - P5 - P5 + M3), then there's still a comma there, but it sounds less jarring for some reason if I play the chord-7th as the root of the scale. However, that means that a IIm7 - V - I progression doesn't sound as good because there's not a perfect fifth between the "2" and the "5" anymore. That's another comma I guess.
random recommendation: instead of deconstructing a song, how about comparing the melodic themes across albums? I'm thinking of Baroness and their Red, Blue, Green, Yellow and Purple albums... what can their tonal structures tell us about the intent behind the colour associations?
Hey nice this topic is finally covered Now if only you can also explain the difference between comma, diesis, schisma, kleisma, diaschisma and all those weird names I see all over in Wikipedia. Is there even a standard for naming what interval as which? Like, commas can be as small as 3 cents (Mercator's comma, 3^53:2^84) and as big as 65 cents (tridecimal comma, 27:26)
There's not a definite cutoff for comma size, though the larger your commas are, the more absurd your temperaments will be. On xenharmonic wiki pages, I've seen arbitrary comma cutoff tentatively proposed at ~100 cents, though of course there are some temperaments that temper out semitones (such as father, which causes 16/15 to vanish). I've never seen any interest in temperaments where the commas are larger than 16/15.
I just wanna say that this is like my shit. I am so excited about this video and I will share it so that precisely none of my friends can enjoy it with me
there's a "hermode tuning system" for electronic music, that keeps notes roughly where they belong in equal temper, but adjusts chords on the fly into just intonation. any observations would be appreciated
I appreciated the clearly explained mathematics about Pythagorean octaves and fifths never meeting again after 0. Always even or always odd. Is this the Pythagorean comma dilemma?
So, weird question that I think fits in with this video and the one you did a few weeks back, "How A London Orchestra Broke International Law", in which you talked about what the official correct frequency for a note is (and how it changed over the years)… A friend, who knows a lot more about music theory and professional music than I do, mentioned that something in the key of C♯ is not the same as D♭ -- that the notes aren't quite the same. Of course, he realizes and agrees that they're the same notes on a piano, but he says one (I assume the ♯) should be "brighter" than the other when sung or played on instruments that allow more variation. Can you explain this better for me? (I understand frequencies and a fair amount of physics; I just don't have much music theory background.) Does this mean they should, ideally, be different frequencies? Or does it only have to do with how a vocalist or instrumentalist would sing or play a note? And if they're different frequencies, then how does that translate to real life? Again, it obviously isn't going to make any difference on a piano. I'm also fairly sure most people now use digital tuners for their instruments… do any of those actually give different results if you set them to C♯ instead of D♭? Would even the most professional symphony orchestra tune differently based on that? And, the one place I could picture them actually tuning slightly differently in real life: do professional musicians tune differently between "matching" ♯/♭ keys for individual songs when recording a studio performance?
If the instruments aren't tuned in equal temperament, the exact tunings will be different for each scale because the "starting point" (root) that you're measuring from will be different. For example, let's look at the Pythagorean tuning mentioned because the Wolf Fifth sounds gnarly and distinctive. Because that interval would land on different scale degrees depending on which key you're in, each different key would have a different sour scale degree. This is a lot like Phrygian has a distinctive stank on it's second note, because it's a minor second from the root. But instead of the scale being offset a certain way, the actual tuning of the instrument is offset a certain way. For a final example, imagine a guitar where the A string is detuned, everything else normal. In a song in the key of G, the 2nd would sound off. In a song in the key of A, the root would sound off. It would be the same string (analogous to the interval in the tuning you're asking about) being "wrong", but because the key is different, where that "wrong" note falls in the key is changed.
In 12 equal the fifths form a circle (i.e. the circle of fifths). If you don't temper (Pythagorean) or temper more (meantone) they form a spiral with different notes for sharp and flats. (Cb)-Gb-Db-Ab-Eb-Bb-F-C-G-D-A-E-B-F#-C#-G#-D#A#-(E#). In your example C# would be used when in the key of D, Db would be used when in the key of Ab, i.e. same as standard key signatures. For Pythagorean pitches in ascending order C > Db > C# > D For meantone pitches in ascending order C> C# > Db > D A piano would most likely be tuned to 12 equal, but fretless instruments and voices could temper notes slightly depending on context.
With the advent of synthesizers and electronic instruments, I don't think commas are really a villain of tuning theory anymore. Any industry-standard DAW has more than enough computing power for someone to tune each note of a piece of music to whatever frequency ratio they want (essentially, we're not limited to 12 notes per octave anymore. We're limited to how many different frequency ratios per octave that a DAW can store). The comma shifts that occur in just intonation don't matter because we don't have to use physical instruments to make music anymore.
There is even a software called bitKlavier which can change note tuning at different rates during performance based on various gestures or settings. It’s kind of hit and miss though for the final output. A concept like that can easily ruin the illusion of thinking you’re hearing a ‘piano’.
In this day and age, is kind of an artificial problem. You can basically just use dynamic tuning, and for the majority of music it's gonna sound fine. Although polytonality may suffer, I seriously thing the musicians working today are slowly begining to turn the wheels on the abandonment of standardized tuning, and much like we have alreaddy started developing tones and timbres that can define a sound, artists may develop tunings that work as part of thwir esthetic. pop might be slow to get there, but I'm surprised I haven't herd more complex intervals in metal (and it's subgenres), and I *have* herd quite a bit in some contemporary jazz, and even just modern jazz. To be overly confident in the matter so it feels cooler if... sorry *when* I am right: "moving foreward the musical landscape will gradually abandon a=440hrz, 12-tone equal temprament, and will begin to experiment with other tunings. I predict pop's entry will be someone finding a specific weird tuning, possibly just intonation, and when it succeeds, finally record labels will permit the experimentation and a trend will start. The other genres will likely have earlier entries, I mean they already have lots of instances, but the normalisation is yet to come, althoug it *will* definetly come, mark my words! anyways, 12-tet is not gonna die, but it's gonna be less assumed, and at first there will be a bunch of instruments that are phased out as digital production reigns for a bit, but then people will start learning how to use their instruments in a more tone-dynamic way. Fretless strings, and other already pan-tonal instruments (I came up with that word now, so probably misused it) are gonna be the first ones to reintroduce non-12-tet accoustic playing in the mainstream, shortly followed by digitally altered electric instruments, people modifying their instruments, and new versions being made that are either adopted for specific tunings or can be adjusted* *(cw:gloom)all this hinges on the fact we don't all perish due to nuclear war, climate disaster, revolution backlash, and that we don't wind up in a society without any room for natural musical development due to totaliterianism or whateve, and I'm not comfortable predicting that either way confidently...
Hey, crazy question here. And there's an answer I have, but I want to get your take, and the reasoning behind it. What is the difference between C# and Db? Or F# and Gb? Why are we using different names for what I would assume is the same thing? Also, why are there no half steps between the E and F, for instance?
Very well explained, but one thing I am missing at the start is that for instance most musical instruments are tuned equally in temperament and that for instance the difference between a Cb and B can't be heard, as they are exactly the same tones on a piano! So instead of talking about intervals, it would be very good to first explain the non real interval variant on a piano, where a perfect fifth interval between notes is NOT the same change in frequency for all notes on the keybed as opposed to exact interval which is theory and in practice is used in voices only, where no instruments are playing, or where instruments are playing that are all tuned by just intonation. Basically if all instruments and voices are using the same tuning, there is no problem, but if one is using a different tuning, then there is problem which will be perceived as out of tune and then the question is:out of tune in what context?
So is this why it's important to continually be 'tuning' when playing in a brass band? That depending on what key and chord you're playing in the actual note you want to play may need to be tweaked to make it sound 'in tune'?
you can also ignore the comma completely and just compose while minding the "comma drift". there are lots of examples on youtube, i can provide some if someone requests.
Does anyone know how he is playing microtonal intervals? I haven't found any composition software like muse score which would allow you to compose microtonally.
Musescore will allow you to play any microtonal interval; when you select a note, on the right hand Inspector dialogue under the section Note, there is a Tuning section which you can set in positive or negative cents, e.g. -50 for a quarter note down or 200 for a whole step up from what's written
If you want to compose microtonal music extensively in notation, Dorico has amazing support allowing you to define the tuning of the natural notes as well as an arbitrary number of accidentals with whatever symbols you want. It is fantastic, but not cheap.
Great work, but you forgot to mention the most important part; This only matters for a handful of instruments, because most instruments, including the human voice let you tune each note independently.
One of the problems with Just intonation (as opposed to 12-EDO), is it all depends on what your root is to be able to tune a chord Justly. So depending on the voicing and the intended effect, is a C7 chord starting on E still a C7 chord but in first inversion, or is it an E minor, diminished 5th, add 6 chord? To figure out the correct Just tuning requires so many judgment calls on what the composer intended as to render it impractical to attempt for anything but simple compositions. I shudder at the thought of trying to figure out Just Intonation for a Wagnarian opera or Richard Strauss's orchestral works. I'd be curious if using 72-EDO tuning would make it worth the attempt.
There are many different quirks, all of which can be described under contemporary regular temperament theory. The difference between C and B# will matter in most meantone temperaments, and is known as an enharmonic relationship. In just intonation, all different versions of C are all 81/80 apart from each other, as are all different versions of B#. Some examples: In 12edo, C = B# In 19edo, C = BX In 31edo, C = B (sharp-and-a-half) Reading Terpstra's work on this (mostly about 31edo) is pretty helpful here. www.huygens-fokker.org/docs/terp31.html
Well, "objectively" is a tough thing to nail down, but I think it's less pleasant to listen to. The ET tritone is pretty well-approximated by a 32:45 ratio, whereas the pythagorean wolf fifth is (3^11):(2^18), which is kinda off-the-charts dissonant. It's too far from 2:3 to be approximated back to a perfect 5th, so it's just sort of stranded. I'm not really sure what the best just approximation for the Meantone wolf is, but I wouldn't be surprised if it was worse than the tritone too. It certainly sounds less pleasant, but my ears are obviously biased.
I wonder what it'd sound like if you used a different tuning sensibility for each chord? Like if there was some way to classify and perfectly order all the tuning methods (or all of them within a sensible subset, maybe), and then you apply each of them seperately to each possible chord, also put into an order. That's bound to have a unique feel to it, right?
All glory to the modern equal temperament, based on the beautiful half-tone relation of 2^1/12!!! In the equal temperament the interval of tritone has a relation of √2, isn't it a miracle of a higher nature?
I don't know if you do request but honestly it's a song relatively new and it's made me cry inside every time that I listen to it do you think you would be able to do Understanding Little Rover by the Stupendium
Hey Man! Would you be interested in analysing Interstellar Overdrive? People say its improvised heavily and has no structure whatsoever but personally I think there is a complex structure and hard work behind it. Its almost like a math problem waiting to be solved. Anyway thank you for all these great videos, they are very helpful. keep up the good work!
Can you take a look at some higher chord extensions? Like #15 or 17, etc? For example, I've found that an EbΔ7#15 resolves nicely to Dm chords with or without extensions. Or a F7#9(natural17) to CΔ chords
Wouldn't #15 just be the same as a b9? as long as the extension is in the top voice, octave placement seems irrelevant. 17 would just be a 3rd in the top voice.
@@graysonwilson-cacciapalle7989 I write it as a #15 when you already play a natural 9. When you see it as a polychord, for example DΔ7/CΔ7 it becomes clear why the #15 is spellef C# and not Db. Similarily a natural 17 is useful in combination with the #9. Since it creates a false major seventh chord when you voice the 5h in the upper octave. E. g. Bb7#9ㅒ17 could be Bb, D, Ab, C#, F, C Octave placement is important, at least for ensembles spanning more than 5 octaves because the overtones reinforce eachother. You wouldn't really use this chord for anything that doesn't span a wide register anyways since scoring it in a close position would sound too dissonant. Inversions would have to be very well thought out. (I couldn't really come up with any) since it would create a lot of jarring intervals.
So, you kind of handwaved this, but... is a cent 1/100 of an equal-tempered* halfstep? Or is it, as I suspect, context-dependent: 1/100 of a halfstep *in the temperament in which you're working*? If there is more than one such contextually-active temperament, does one take over? Or must you tag each cent-measurement? This is all 2001-level stuff, isn't it? :-) [ * or did I mean to say "just-intoned" there? ]
I am an aduocate of not tempering muſicke at all. Notes a ſyntonic comma apart ſometimes I conſider a differnt verſion of the ſame note, in other contexts I conſider it a completely different note. I can neuer make it definitiue. Context matters. Thoſe ſynotonic pairs ſerue different harmonic functions melodically and choꝛdally. 5 limit JI with occaſional higher limits as colour and effects is the onely way muſick ſhould be tuned.
As someone always fascinated by certain mathematical features of music, this video and some of the comments have at least shown me clearly where the limit of that mathematical fascination lies. Having said that, I suspect that what you're talking about explains why I can never satisfactorily tune a guitar by ear. Is that right? Or should I just buy more expensive guitars?
yes, that is exactly correct. When you tune strings of any instrument to be perfectly in tune with neighbouring strings, the result will inevitably drift in both directions.
If y'all liked this video you should probably check out the channel Early Music Sources, and their videos on temperament in the Renaissance! (spoiler: they didn't necessarily use 1/4-comma meantone!!)
Nicely done! I hope you don't mind my piggybacking off of your video and saying that, if anyone wants to check out an arguably wayyyy too dense, awkwardly narrated overview of tuning theory like is talked about in this video, I've got a tuning theory series on my channel, along with several general theory videos that also incorporate tuning theory. Cheers!
A Mathematical proof in a TH-cam video - nice! Better still I could follow it....well most of it. Good video - I do wonder how musicians in the era of Just Intonation coped. Equal temperament is a fudge / compromise but it makes life so easy. ( politicians take note )
so, for an interval that is the ratio of integers A and B, what is the function f(A, B) that defines (or orders) the relative pleasantness of different intervals? what makes a simple ratio simple?
QUESTION:What if one of my pieces doesn't come ou the way I want it to. Do I start over or do I go to another piece?Thanks in advance for any and all advice.
I sometimes wonder about whether we really have to use tempered scales, especially in electronic music. If a melody is just a sequence of intervals, you should be able to play each interval perfectly without having to worry about how many cents you are off any given reference scale. On the other hand maybe that gets you into trouble because returning to the fundamental might not be possible in a nice, simple interval. At the very least electronic musicians should be able to play every chord perfectly; in many melodic tracks you can hear the different frequencies moving in and out of phase and I wonder a lot about how much crisper chords could sound if they used perfect intervals instead. I feel like it's a path that's not well-trodden, even though everyone is going crazy about sound design at the moment.
You don't!!! You can play in any tunings you want! Other equal divisions of the octave! Crazy just intonation using primes up to 37! Non-octave monstrosities! Also - try "Alt-Tuner" it's a fun software for playing chords in-tune with each other as they move... www.tallkite.com/alt-tuner.html
@@Schindlabua Oh yeah, I'm everywhere Sevish and I co-host the microtonal podcast "Now&Xen." You should check it out! :) itunes.apple.com/us/podcast/now-xen/id1447552191?mt=2
Hey would you please analyse the piano intro from "firth of fifth"? I think it is one of the best piano sections in the story of progressive music, it deserves some attention, doesn't it?
Some additional thoughts:
1) It's worth noting that just because two notes are close together, that doesn't mean they're a comma. For instance, Partch's Genesis scale contains notes as close as 15 cents apart, but they're considered different notes and thus their difference isn't a comma. Or, I guess it could be if you were tempering that difference out in a different scale, but in the context of the Genesis Scale it doesn't count as one.
2) I glossed over this, but depending on which definition you like, it may not be accurate to describe Pythagorean tuning as a tempered scale. It tempers certain ratios, but it's built out of strictly just-intonation bits. It's just some of those bits get very complex. Some sources do call it a temperament, others don't. I'm largely agnostic on the issue, if I'm being honest.
Also: Extended Pythagorean, as described in much mediaeval Middle Eastern music, actually produces a third only 1.954 cents away from a pure 5/4 by stacking eight perfect 4/3 fourths and reducing by octaves, which in Western music would be spelled as a diminished fourth. That tiny, tiny comma is known as the schisma and is functionally tempered out in many just intonation performances and disappears outright in tunings like 41- and 53-EDO.
Thanks for the video! A question though. If I recall my history correctly, the pythagorean tuning produces 13 different notes, and the comma is the result of discarding either the Gb or the F# which are 24 cents away from each other, and then people discarded one or the other to produce a 12 tone series. I may well have misunderstood something, there, but if not, why not just use 13 notes and call it good? Might be hard to implement on fretted instruments but few care about that, and you could make a keyboard with 13 keys to an octave easily enough. Why didn't they do it? Triskaidekaphobia, or is there more to it?
@Laughing Daffodils:
It goes back to the circle of fifths. The circle is only closed if we treat Gb and F# as enharmonically equivalent. If we don't, then the "circle" of fifths is really a spiral, and you're back to the problem discussed at the outset of this video: you can keep going up perfect fifths and never reach an octave.
I think Partch would disagree. I unfortunately do not have my copy of Genesis of a Music with me right now, but the interval 81/80, which occurs between several sequential notes in Partch's monophonic fabric, is pretty much universally called a comma both in Partch's writings and in the microtonal community in general. I think it's possible to conceptualize that something might be two different conceptual pitch classes and also a comma apart.
A bit of a tangent, but I watched a presentation showing how we use a comma to make the major scale and pentatonic scale and even in resolving polyrhythms. This is probably a concept you're already familiar with but maybe it would make for an interesting topic for a future video.
Watching this video decreased the length of my watch later list from 81 to 80 videos.
That lowered your watch later list by a syntonic ，
Haha, I get it. 😂
@@xuly3129it lowered by 21 1/2 cents
Rookie numbers
Mine just increased From 1935 to 1936
When I was in high school, and just learned that perfect fifths are 3:2, I sat down with my calculator, typed in 440, multiplied it by 1.5 twelve times, divided by 2 seven times, and got roughly 446. I assumed I messed up somewhere along the way, and put it out of my mind until years later, when I learned about intonation, temperaments, and commas, from Wikipedia. Vindication!
(Interestingly enough, this comment contains 12 commas. 😉)
I took 110hz, multiplied by 1.5 53 times, divided by 2 31 times, and got... 110.2hz.
@@ValkyRiver rip
Yes, you actually get 446,0030364991...
@@ValkyRiver53edo user
In 53-TET:
- The perfect fifth are only 0.07 cents flat.
- The major third is only 1.4 cents flat.
- The minor third is only 1.3 cents sharp.
- The harmonic seventh is only 4.8 cents sharp.
53-TET does not temper out the syntonic comma. It gets mapped to one step (22.6 cents).
Nice
I kind of like the way it sounded when you played the two C#s together. It gives me an old piano vibe.
Most pianos have 3 strings for every note to make it sound louder, so an old piano could have some of those strings off tuned from each other, thus getting a comma appart. Maybe..
in pianos this dissonance feels very dreamy and psychodelic to me. in distorted guitars it sounds so much more powerful than if they were not there
Martin Roth's version of Barber's Adagio for Strings actually uses detuned notes like this. I love the effect.
th-cam.com/video/6hKxAE-O6Uc/w-d-xo.html
You should listen to Christopher Bailey's "Concerto for shitty piano"
christopherbailey.bandcamp.com/track/composition-for-s-piano-drum-samples-concr-te-sounds-and-processing-2013
It's also what gives Accordion it's distinctive buzz! There are multiple sets of reeds SLIGHTLY tuned off from each other. But when its played using that piano sound, it just feels like teaching piano. No student's piano is ever tuned!
Dang, what a nice introduction to commas! It's a fascinating, and insanely nerdy world. I personally am a fan of the various diesis..es.
Related note...what is the plural of diesis?
Dieses?
Dieses'? Diese? ‡'s?
Pretty sure it's "dice".
@@12tone true, true
More seriously my original plan for this video involved lots of diesis talk, but I decided that should probably be its own video. Which I probably won't make for a while 'cause I have some other tuning stuff I wanna do first and also I try not to do _too_ much tuning stuff in a row, but I didn't want to just cram them in when just talking about the big dog commas got me over 9 minutes.
So what you're really saying, then, is that 12-tone equal temperament tries to hide the syntonic comma by making it fade into the background of equal intervals. In other words, it's the original comma chameleon.
You come and go, you come and go...
Jeffry Jones I enjoyed this comment 🤪
Comma comma comma comma comma commaeleon
Having flashbacks to middle school orchestra, when I got bent out of shape once because SOMEONE behind me had their violin just slightly out of tune and I was not yet socially aware enough to not correct them
@@ze_rubenator They play everything too high.
I've been playing guitar since I was 5, and have tuned by ear since I was 8... I can say, it pisses me off when someone's guitar is slightly out of tune... It makes me cringe, and sometimes I wanna rip the guitar away from the person instead of correcting them....
I can't think of a more appropriate place to correct someone than in a classroom
@@samhodge7460 I wasn't the teacher, so it was a problem
@@MaraK_dialmformara I correct the teacher sometimes...
This reminds me of the chocolate bar illusion
12 P5s is almost the same as 7 octaves. That's neat because a perfect fifth is 7 half steps up and an octave is 12 half steps up. It's like an inversion
Really-excellent summary!
Another thing worth mentioning: Quasi-coincidentally, Quarter-Comma Meantone tuning is audibly identical with 31 equal steps per octave (AKA “31TET” or “31EDO”).
As you correctly pointed out, if you stack up only 12 quarter-comma-flat fifths, you end up with a horrible-sounding wolf fifth. The circle of fifths fails to close by a horrendous-sounding 41 cents.
However, if you stack up 31 of them, the circle of quarter-comma-flat fifths fails to close by only 6 cents! 6 cents is only barely audible in-real-world music anyway, but spreading that already-tiny error across 31 fifths (18 octaves), the difference between a QC Meantone Perfect 5th and a 31TET P5, is less than 1/5 of a cent, which is for-sure completely inaudible.
Even if you stack up *_12_* QC Meantone P5s against 12 31TET P5s, the difference is still only a little over 2 cents, which, in normal-speed music, is still pretty much inaudible.
This is great! I'm so happy that microtonal music theory is surging in popular TH-cam channels. You explained the idea of a vanishing comma better than most people I know! I'm looking forward to any other microtonal exploration you undertake on this channel.
*_Warning: wild speculation._*
I've been messing around with note pitches lately and it gave me an idea. "What would be the single most profound way to get a fundamental frequency", I thought. Well, what is a frequency but cycles per second? And what's a second? It's defined as "the duration of 9,192,631,770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium-133 atom".
Fancy science talk out of the way, we've got a note with the frequency of 9,192,631,770 Hertz. Let's call it *S.* That's way out of the hearing range of anything that lives on Earth, so I divided it by 2^25, i.e. lowered it by 25 octaves, which gave me the frequency of 273.961775600910186767578125 Hz (yes, that's a lot of digits you don't really need, but I'll keep them just for purity's sake).
That falls within the range of the first-line octave and is very close to C#4 (the most accurate pitch for it I was able to find is 277.1826 Hz). S4 is higher than C#4 by 20.234496 cents. For comparison, the syntonic comma is 21.51 cents and the diaschisma is 19.55.
I'm not quite sure where the idea goes beyond this point. Maybe it becomes the new A=432. I hope not. So far, there is one thing I will claim: _as the discoverer of this universally significant constant, I shall name the interval of 20.234496 cents... _*_The Temporal Comma._*
The problem with all of this is that a second is still an arbitrary unit of time, despite its being precisely defined. Similarly, designating any given frequency (regardless of of how many digits it has after the decimal point) as A, B, C, or any other pitch is also arbitrary. While absolute definitions for some things might be elusive, one can get a lot of mileage out of relative relationships.
It is a flaw, yes, but it is also a human construct, based on the way _humans_ perceive time. And those humans are the main audience around here, so I think my logic still applies.
@@jbh001 a second is an arbitrary unit of time, but the time of radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium-133 atom is not as much. Yet, arguably the most truly fundamental time in physics (at least, in its modern understanding) is the Planck time, as it depends only on the three fundamental physical constants - the Planck unit, the speed of light and the gravitational constant - with no numerical constants to multiply. However, it is 5.39 × 10^−44 seconds, so you will have to bring it down by quite a bit of octaves down to bring it to the human range of hearing - I'd say about a hundred and twenty?
@@thomaswinwood That's really neat. You can also do 4 * planck_freq / 2^(1/fine_structure_constant) to get 415.35 Hz. If you call that Ab then you get A440.05842Hz. Take that, A432-ers!
I'm definitely going to save this one for later, when people ask me questions about piano tuning. But on that _note_ , I would like to add that even octaves aren't safe. Not on the piano, at least. We tend to tune them ever so slightly wider than they ought to be, its size depending on the room or hall the instrument's in. A grand piano tuned with perfect octaves sounds off in big halls, for instance.
What I'm taught in school is this: building up towards the higher registers, we aim for a twelfth to be perfect. Given that the fifth is sliiiightly smaller than perfect, the octave naturally compensates for this by the same amount.
Thanks for sharing this, perhaps future students will find it useful too :)
The ever dreaded wolf note is the bane of violin luthiers. It took me 3 months, several minute adjustments and trying 4 different types of strings to quiet the wolf on my last violin. My cello on the other hand, is quite tame.
Is this the nerdiest episode yet? I love it!
5:17 "it takes a while, but 53 perfect 5ths is actually almost identical to 31 octaves."
5:34 "these two notes wind up only 3.6 cents away from each other, which is pretty spot-on."
Wait is this why 31tet is often lauded by microtonal composers? Smoother commas?
@@peteroselador6132 31-TET is meantone, whereas 53-TET is not. That means that chord progressions like I-vi-ii-V-I won’t comma pump.
@@peteroselador6132
Nicer major and minor thirds/sixths/sevenths
and alternate versions of these that don't exist in 12 equal
Pretty close to quarter comma meantone
It makes chords sound really good and has more melodic variety
th-cam.com/video/Q2bmXl3zgdE/w-d-xo.html
th-cam.com/video/L_cx4MylmXk/w-d-xo.html
One of the fundamental flaws in most tuning theory is the notion that dissonance is intrinsically bad. Dissonance is in fact extremely useful artistically. In my mind, the better temperaments are those that give us a whole palette of different sounds to choose from. Some keys are consonant and calm, others are more dissonant and edgy. Some will have nice fifths, others nice thirds. Equal temperament seems the worst choice of all, because every key is identical. And what's the point of having twelve keys if they all sound the same?
So you can change keys dramatically within one piece of music.
i actually LOVE the ever so slight dissonance you play at the beggining. i found out that a distorted guitar with that tiny bit of dissonance in maybe one string sounds so powerful
if you declare that every interval except the octave is a comma, you end up with those "but every note is C" videos
4/3 isn't the octave and 3/2 isn't the octave
and if they're both commas that we temper, that means 2/1 is also tempered
This is the scale I like best from a maths perspective:
1:1 25:24 10:9 B:H 6:5 5:4 4:3 25:18 36:25 3:2 8:5 5:3 3B:2H 9:5 48:25
B:H is the ratio of 2 to the square root of 3. This is also the ratio between the base of an equilateral triangle to its height.
B:H is the geometric mean of 125:108 and 144:125 and is really amazingly close to both. The ratio between the two is 15625:15552.
The scale is based on a progression from 10:9 to 9:5 where is step is the ratio 6:5. There are 9 major triads (3:4:5) and 9 minor triads (12:15:20).
Once you see the relationship between these triads you will see that this crysraline structure is a thing of beauty in its own right.
Idea: Don’t use tuning systems. Tune every note to the root of the chord being played, for _each_ chord you’re going to play, changing the exact notes as you play.
That's basically adaptive just intonation. A capella groups such as barbershop quartets intuitively approximate it, and there's software to do it on MIDI virtual instruments. This approach tends to result in comma pumps, which are why even an expert group of singers can end up singing the tonic some cents sharp or flat from the pitch they originally sang it at, which makes it sound like the key slightly changed. If there's a comma pump you want to avoid, you can temper out the offending comma or just make it an interval you use.
@@johnnycochicken done by Miles Davis AFAIK.
Also Collier does not evade comma pumps, but devises them to the benefit of his music.
“The power is yours!” Makes the perfect fifth equal to the ovtave
Thank you so much, this is the best explanation I've seen of why we can't have perfect tuning systems. Other people explain that stacked octaves and stacked fifths don't equal each other, but nobody I've seen ever brought physics into it and proved it using mathematical equations.
David MacMillan MinutePhysics has an excellent video on why you can't tune a piano.
@@cmarley314 Thanks, I'll have to check it out.
You can fudge it well with enough notes, but it depends on how many distinct intervals and harmonics you want to represent and how well. I'm personally quite fond of 41-EDO but notation and keyboards for such a large gamut can be frustrating, not to mention instruments like guitars...
I think the fact that there are no perfect scales is one of the best things about tuning theory. Creating a tonal system (or even just choosing) can become a creative thing on it's own. Of course we don't really do that in the west, as we are kind of stuck with 12edo, but hopefully that will change.
Look I'm not saying that 12 is bad, I'm just saying that you only ever order strawberry ice cream but maybe you could try some blueberry ice cream, you might like it.
And try out 17 and 22, they are damn cool divisions of the octave.
fuck yeah! You're not stuck with 12edo if you have a computer and ears! :D
12-TET is just 1/11 meantone
Off by an atom of Kirnberger (0.015 cents)
also, compare 19-TET And 1/3 meantone
This is hands-down one of the BEST explanations I've ever seen of the problems of tuning. I remember when I was in high school and tuning my guitar, I realized I could just tune it to chords I was playing. I'd play a G chord, then adjust the 'b' string and I noticed it sounded really good. Then I'd play an E chord and it would sound awful! I thought it was a problem with me, or with my guitar, and it used to drive me mad.
It wasn't until years later that I understood what was happening.
I found your channel via Origin of Everything and I'm so glad I did! Holy moly. I love music math. Thank you for presenting this.
So, just as an aside about Pythagorean tuning (length warning. Also Quack Quack):
I love it, or offshoots of it. See, as with any interval you stack, you'll eventually hit a 'Moment of Symmetry' as described by Erv Wilson, an internally consistent gamut made up of only two step sizes.
With the fifth, it's at 12 notes, then 17, then 29, then 41, then 53. Two cool things;
1. If you model where the intervals lie on something like a fretboard the note distribution looks almost exactly like the '7 diatonic notes and 5 accidentals' you find on a piano.
2. You can mess about with the parameters to get a stupid amount of variety that's internally consistent.
Say you have an MOS at 17 notes, composed of 5 (roughly) Pythagorean commas and 12 semitones at about 90c. If you dont change the order of the notes, but there respective sizes (like making all the commas bigger and the semitones smaller) you get different tonal possibilities, but the fifths will still stay quite pure (less the 3 to 5 cents out), and you'll only ever have one wolf.
MOS at 29? Note orientation still looks like piano keys. Very internally consistent. Can mess with it to get a tonne of variety. Only one wolf.
Exactly the same for 41. Piano key arrangement. Consistent. Lots of potential variety. Only one wolf.
I'd give 53 more than a passing mention if I didn't find it too big to be useful.
"The Greek mathematician Pythagoras of triangle fame" might be the best thing I've heard today lol
the tempered major third is 64% of a syntonic comma higher than just major third (about 2/3 of the way between 5 limit and pythagorean major third)
One of my (unauditioned and hence having a rather low average skill level) choirs for a while had a conductor who was mad keen on just intonation and recommended a book called "How Equal Temperament Ruined Harmony (and Why You Should Care)". It was slightly ironic that (IIRC - I must re-read it carefully) the book recommended mean tone rather than just intonation, pointing out that quarter-comma MT gives a scale with only one sort of whole tone, whereas the conductor was wont to rave about (among other topics) the difference between a "major" (9:8) tone and a "minor" (10:9) one. I seriously doubted that the individual singers tuned to each other closer than these differences anyway.
1200*log(81/80)/log(2) is the comma in cents... there are tons of other comma's that I haven't studied yet but yup!
It just occurred to me that you could make an equal temperament scale where the perfect fifth is always just and the octave is slightly sharp. And now I want to know what that sounds like.
there are some well-temperaments like that that sound pretty Pythagorean and pleasant (stretched octaves are preferred, after all...)
in the era of digital music I have a question -> is sticking to one tuning really necessary throughout a piece?
Would it be possible to retune the instruments infinitely as a song is played to get perfect ratios? Start on any given note play the just intonation notes that follow and have the key fluctuate or even migrate over the course of the song?
It would be interesting to have a song with a key change partway through to something near the original key without it ever sounding like the song had moved away from that key.
Or would the whole song sound slightly out of tune the whole time?
I wish I never learnt all this stuff, I do not have perfect pitch but now whenever I play my guitar and songs/whatever in different keys I have the anxiety that it's out of tune and probably some note is and I just subconciously feel discomfort rather than knowing what is not in tune.
I feel like the easiest way to understand the pythagorean comma would be to first show that 4/3 is the inverse of 3/2, and then show the coincident harmonic to tune them, then go through the cycle of 5ths/4ths within an octave to show how when you arrive at the octave, you are now 23ish cents sharp from the pure octave.
My grandpa showed me this in person at an actual piano by literally going through and tuning it that way when I was in high school and it blew my mind. Totally made music and tuning and the harmonic series and all of that suddenly make sense all in one swoop. Like, oh shit, this is where it all comes from 🤯
Very interesting video!
I love hearing about harmonic ratios in different tuning systems. I've been thinking about trying to define an interval space made of vectors where the "fundamental" vectors are just intervals like P8, P5, M3, and H7 (harmonic 7th), and trying to visualize the Major and minor triads as triangles in that vector space. And I think 7-chords should be tetrahedrons (triangular pyramids) in that space.
Now that I've seen this video I think I might be able to put into words the problems I've been running into.
One of them is the IIm7 chord. If you base it on the "2" as (P5 + P5 - P8), then there's a comma on the chord-7th, which would be the root of the scale if it weren't for that comma. On the other hand if you base it on the "2" as (P8 - P5 - P5 + M3), then there's still a comma there, but it sounds less jarring for some reason if I play the chord-7th as the root of the scale. However, that means that a IIm7 - V - I progression doesn't sound as good because there's not a perfect fifth between the "2" and the "5" anymore. That's another comma I guess.
The ascension at 4:12 is basically the entire Sound of Perseverance album in a nutshell
random recommendation: instead of deconstructing a song, how about comparing the melodic themes across albums? I'm thinking of Baroness and their Red, Blue, Green, Yellow and Purple albums... what can their tonal structures tell us about the intent behind the colour associations?
Hey nice this topic is finally covered
Now if only you can also explain the difference between comma, diesis, schisma, kleisma, diaschisma and all those weird names I see all over in Wikipedia.
Is there even a standard for naming what interval as which? Like, commas can be as small as 3 cents (Mercator's comma, 3^53:2^84) and as big as 65 cents (tridecimal comma, 27:26)
There's not a definite cutoff for comma size, though the larger your commas are, the more absurd your temperaments will be. On xenharmonic wiki pages, I've seen arbitrary comma cutoff tentatively proposed at ~100 cents, though of course there are some temperaments that temper out semitones (such as father, which causes 16/15 to vanish). I've never seen any interest in temperaments where the commas are larger than 16/15.
Stephen Weigel 7-tet tempers out 2187/2048
I am going to using tuning to highlight the wolf fifth, Because I think it sounds neat.
5:50 Well from my point of view, the Jedi are evil!
Its made pretty obvious in the prequels.
new video wen
I love the episodes where you marry music with lots of math!
I just wanna say that this is like my shit. I am so excited about this video and I will share it so that precisely none of my friends can enjoy it with me
there's a "hermode tuning system" for electronic music, that keeps notes roughly where they belong in equal temper, but adjusts chords on the fly into just intonation. any observations would be appreciated
I appreciated the clearly explained mathematics about Pythagorean octaves and fifths never meeting again after 0. Always even or always odd. Is this the Pythagorean comma dilemma?
So, weird question that I think fits in with this video and the one you did a few weeks back, "How A London Orchestra Broke International Law", in which you talked about what the official correct frequency for a note is (and how it changed over the years)…
A friend, who knows a lot more about music theory and professional music than I do, mentioned that something in the key of C♯ is not the same as D♭ -- that the notes aren't quite the same.
Of course, he realizes and agrees that they're the same notes on a piano, but he says one (I assume the ♯) should be "brighter" than the other when sung or played on instruments that allow more variation.
Can you explain this better for me? (I understand frequencies and a fair amount of physics; I just don't have much music theory background.)
Does this mean they should, ideally, be different frequencies? Or does it only have to do with how a vocalist or instrumentalist would sing or play a note?
And if they're different frequencies, then how does that translate to real life? Again, it obviously isn't going to make any difference on a piano. I'm also fairly sure most people now use digital tuners for their instruments… do any of those actually give different results if you set them to C♯ instead of D♭? Would even the most professional symphony orchestra tune differently based on that? And, the one place I could picture them actually tuning slightly differently in real life: do professional musicians tune differently between "matching" ♯/♭ keys for individual songs when recording a studio performance?
If the instruments aren't tuned in equal temperament, the exact tunings will be different for each scale because the "starting point" (root) that you're measuring from will be different.
For example, let's look at the Pythagorean tuning mentioned because the Wolf Fifth sounds gnarly and distinctive. Because that interval would land on different scale degrees depending on which key you're in, each different key would have a different sour scale degree.
This is a lot like Phrygian has a distinctive stank on it's second note, because it's a minor second from the root. But instead of the scale being offset a certain way, the actual tuning of the instrument is offset a certain way.
For a final example, imagine a guitar where the A string is detuned, everything else normal. In a song in the key of G, the 2nd would sound off. In a song in the key of A, the root would sound off. It would be the same string (analogous to the interval in the tuning you're asking about) being "wrong", but because the key is different, where that "wrong" note falls in the key is changed.
In 12 equal the fifths form a circle (i.e. the circle of fifths). If you don't temper (Pythagorean) or temper more (meantone) they form a spiral with different notes for sharp and flats. (Cb)-Gb-Db-Ab-Eb-Bb-F-C-G-D-A-E-B-F#-C#-G#-D#A#-(E#). In your example C# would be used when in the key of D, Db would be used when in the key of Ab, i.e. same as standard key signatures.
For Pythagorean pitches in ascending order C > Db > C# > D
For meantone pitches in ascending order C> C# > Db > D
A piano would most likely be tuned to 12 equal, but fretless instruments and voices could temper notes slightly depending on context.
Finally, an explanation of this I can actually understand! Sorry Adam Neely, I just didn't get your really old video on this type of thing.
Ah perfect 5ths sound so... Well, perfect
Great stuff! Have you thought about covering the Werkmeister tunings?
Damn, I once spent like 6 hours learning all this on Wikipedia, and you just explained it in 10 minutes
immediately perked up when you played that JI 5th
I love this tuning stuff.
Fascinating. I have a lyre tuned in pythagorean just tuning. sounds richer than equal temperament.
I should add that it's in a pentatonic scale, as well.
"The maths gets a bit tricky"
Me, a mathematician: uhu. Sure.
With the advent of synthesizers and electronic instruments, I don't think commas are really a villain of tuning theory anymore. Any industry-standard DAW has more than enough computing power for someone to tune each note of a piece of music to whatever frequency ratio they want (essentially, we're not limited to 12 notes per octave anymore. We're limited to how many different frequency ratios per octave that a DAW can store). The comma shifts that occur in just intonation don't matter because we don't have to use physical instruments to make music anymore.
@Joshua Sanchez Modulation in JI was always possible; it just wasn't systematized.
There is even a software called bitKlavier which can change note tuning at different rates during performance based on various gestures or settings. It’s kind of hit and miss though for the final output. A concept like that can easily ruin the illusion of thinking you’re hearing a ‘piano’.
I wonder what Andrew Huang could do with this, if it were presented to him as a challenge?
In this day and age, is kind of an artificial problem. You can basically just use dynamic tuning, and for the majority of music it's gonna sound fine. Although polytonality may suffer, I seriously thing the musicians working today are slowly begining to turn the wheels on the abandonment of standardized tuning, and much like we have alreaddy started developing tones and timbres that can define a sound, artists may develop tunings that work as part of thwir esthetic. pop might be slow to get there, but I'm surprised I haven't herd more complex intervals in metal (and it's subgenres), and I *have* herd quite a bit in some contemporary jazz, and even just modern jazz.
To be overly confident in the matter so it feels cooler if... sorry *when* I am right: "moving foreward the musical landscape will gradually abandon a=440hrz, 12-tone equal temprament, and will begin to experiment with other tunings. I predict pop's entry will be someone finding a specific weird tuning, possibly just intonation, and when it succeeds, finally record labels will permit the experimentation and a trend will start. The other genres will likely have earlier entries, I mean they already have lots of instances, but the normalisation is yet to come, althoug it *will* definetly come, mark my words! anyways, 12-tet is not gonna die, but it's gonna be less assumed, and at first there will be a bunch of instruments that are phased out as digital production reigns for a bit, but then people will start learning how to use their instruments in a more tone-dynamic way. Fretless strings, and other already pan-tonal instruments (I came up with that word now, so probably misused it) are gonna be the first ones to reintroduce non-12-tet accoustic playing in the mainstream, shortly followed by digitally altered electric instruments, people modifying their instruments, and new versions being made that are either adopted for specific tunings or can be adjusted*
*(cw:gloom)all this hinges on the fact we don't all perish due to nuclear war, climate disaster, revolution backlash, and that we don't wind up in a society without any room for natural musical development due to totaliterianism or whateve, and I'm not comfortable predicting that either way confidently...
Hey, crazy question here. And there's an answer I have, but I want to get your take, and the reasoning behind it. What is the difference between C# and Db? Or F# and Gb? Why are we using different names for what I would assume is the same thing? Also, why are there no half steps between the E and F, for instance?
Very well explained, but one thing I am missing at the start is that for instance most musical instruments are tuned equally in temperament and that for instance the difference between a Cb and B can't be heard, as they are exactly the same tones on a piano! So instead of talking about intervals, it would be very good to first explain the non real interval variant on a piano, where a perfect fifth interval between notes is NOT the same change in frequency for all notes on the keybed as opposed to exact interval which is theory and in practice is used in voices only, where no instruments are playing, or where instruments are playing that are all tuned by just intonation.
Basically if all instruments and voices are using the same tuning, there is no problem, but if one is using a different tuning, then there is problem which will be perceived as out of tune and then the question is:out of tune in what context?
Love the way this goes fast!
That midi piano just sounds out of tune to me whichever note is played. I'm always a bit distracted by that for some reason.
I came to this channel for music theory and stayed for the mathematical proofs.
So is this why it's important to continually be 'tuning' when playing in a brass band? That depending on what key and chord you're playing in the actual note you want to play may need to be tweaked to make it sound 'in tune'?
Yes, and the same also applies to both orchestras and choirs singing a-cappella as well as brass ensembles.
You used music to trick me into thinking about math!
you can also ignore the comma completely and just compose while minding the "comma drift".
there are lots of examples on youtube, i can provide some if someone requests.
Does anyone know how he is playing microtonal intervals? I haven't found any composition software like muse score which would allow you to compose microtonally.
I'm using Reason and automating the tuning wheel to detune by the appropriate proportion of a half step.
Musescore will allow you to play any microtonal interval; when you select a note, on the right hand Inspector dialogue under the section Note, there is a Tuning section which you can set in positive or negative cents, e.g. -50 for a quarter note down or 200 for a whole step up from what's written
Thanks so much!
No problem!
If you want to compose microtonal music extensively in notation, Dorico has amazing support allowing you to define the tuning of the natural notes as well as an arbitrary number of accidentals with whatever symbols you want. It is fantastic, but not cheap.
I love the way you draw wolves
Great work, but you forgot to mention the most important part; This only matters for a handful of instruments, because most instruments, including the human voice let you tune each note independently.
One of the problems with Just intonation (as opposed to 12-EDO), is it all depends on what your root is to be able to tune a chord Justly. So depending on the voicing and the intended effect, is a C7 chord starting on E still a C7 chord but in first inversion, or is it an E minor, diminished 5th, add 6 chord? To figure out the correct Just tuning requires so many judgment calls on what the composer intended as to render it impractical to attempt for anything but simple compositions. I shudder at the thought of trying to figure out Just Intonation for a Wagnarian opera or Richard Strauss's orchestral works. I'd be curious if using 72-EDO tuning would make it worth the attempt.
Is this why sometimes the difference between C and B# winds up mattering, or is that a different quirk of equal temperament?
There are many different quirks, all of which can be described under contemporary regular temperament theory. The difference between C and B# will matter in most meantone temperaments, and is known as an enharmonic relationship. In just intonation, all different versions of C are all 81/80 apart from each other, as are all different versions of B#. Some examples:
In 12edo, C = B#
In 19edo, C = BX
In 31edo, C = B (sharp-and-a-half)
Reading Terpstra's work on this (mostly about 31edo) is pretty helpful here.
www.huygens-fokker.org/docs/terp31.html
so if some tunings have a wolf fifth, does that mean there’s also a tuning out there with a wolf fourth? Only I guess it’d be sharp instead of flat.
Any tuning with a wolf fifth and a perfect octave has a wolf fourth
Or a wolf octave?
Quick question: Do you think the wolf fifth is objectively more dissonant than the equal tempered diminished fifth?
Well, "objectively" is a tough thing to nail down, but I think it's less pleasant to listen to. The ET tritone is pretty well-approximated by a 32:45 ratio, whereas the pythagorean wolf fifth is (3^11):(2^18), which is kinda off-the-charts dissonant. It's too far from 2:3 to be approximated back to a perfect 5th, so it's just sort of stranded. I'm not really sure what the best just approximation for the Meantone wolf is, but I wouldn't be surprised if it was worse than the tritone too. It certainly sounds less pleasant, but my ears are obviously biased.
@@12tone What about the 7/5 tritone? Is it as consonant as the minor third (6/5) and the harmonic seventh (7/4)?
wow, amazing overview!
8:57 Go Planet!
What if you like the wolf fifth
Cool, so you can simply use JI on EVERYTHING!
I tune all my fifths to wolf fifths
I wonder what it'd sound like if you used a different tuning sensibility for each chord? Like if there was some way to classify and perfectly order all the tuning methods (or all of them within a sensible subset, maybe), and then you apply each of them seperately to each possible chord, also put into an order. That's bound to have a unique feel to it, right?
The Pythagorean comma is just a consequence of the fundamental theorem of arithmetic, but how many people know what both of those are?
I do... It's actually quite funny that the pythagorean comma might very well be the motivation to develop FTA in the first place :-D
All glory to the modern equal temperament, based on the beautiful half-tone relation of 2^1/12!!! In the equal temperament the interval of tritone has a relation of √2, isn't it a miracle of a higher nature?
I don't know if you do request but honestly it's a song relatively new and it's made me cry inside every time that I listen to it do you think you would be able to do
Understanding Little Rover by the Stupendium
Hey Man! Would you be interested in analysing Interstellar Overdrive? People say its improvised heavily and has no structure whatsoever but personally I think there is a complex structure and hard work behind it. Its almost like a math problem waiting to be solved. Anyway thank you for all these great videos, they are very helpful. keep up the good work!
Yes. I understand.
Can you take a look at some higher chord extensions? Like #15 or 17, etc?
For example, I've found that an EbΔ7#15 resolves nicely to Dm chords with or without extensions.
Or a F7#9(natural17) to CΔ chords
Wouldn't #15 just be the same as a b9? as long as the extension is in the top voice, octave placement seems irrelevant. 17 would just be a 3rd in the top voice.
@@graysonwilson-cacciapalle7989 I write it as a #15 when you already play a natural 9. When you see it as a polychord, for example DΔ7/CΔ7 it becomes clear why the #15 is spellef C#
and not Db.
Similarily a natural 17 is useful in combination with the #9. Since it creates a false major seventh chord when you voice the 5h in the upper octave. E. g. Bb7#9ㅒ17 could be Bb, D, Ab, C#, F, C
Octave placement is important, at least for ensembles spanning more than 5 octaves because the overtones reinforce eachother.
You wouldn't really use this chord for anything that doesn't span a wide register anyways since scoring it in a close position would sound too dissonant.
Inversions would have to be very well thought out. (I couldn't really come up with any) since it would create a lot of jarring intervals.
So, you kind of handwaved this, but... is a cent 1/100 of an equal-tempered* halfstep?
Or is it, as I suspect, context-dependent: 1/100 of a halfstep *in the temperament in which you're working*?
If there is more than one such contextually-active temperament, does one take over? Or must you tag each cent-measurement?
This is all 2001-level stuff, isn't it? :-)
[ * or did I mean to say "just-intoned" there? ]
The former is 'cents' and the latter is 'relative cents'
I am an aduocate of not tempering muſicke at all. Notes a ſyntonic comma apart ſometimes I conſider a differnt verſion of the ſame note, in other contexts I conſider it a completely different note. I can neuer make it definitiue. Context matters. Thoſe ſynotonic pairs ſerue different harmonic functions melodically and choꝛdally. 5 limit JI with occaſional higher limits as colour and effects is the onely way muſick ſhould be tuned.
If you played a chord of all C notes, with octave spacing of 3:2 (I think it's C1, C4, and C6), would it affect harmony similar to a G?
"Can you make three tonics sound like a fifth, or at least function as a fifth?"- in broader terms.
As someone always fascinated by certain mathematical features of music, this video and some of the comments have at least shown me clearly where the limit of that mathematical fascination lies.
Having said that, I suspect that what you're talking about explains why I can never satisfactorily tune a guitar by ear. Is that right? Or should I just buy more expensive guitars?
yes, that is exactly correct. When you tune strings of any instrument to be perfectly in tune with neighbouring strings, the result will inevitably drift in both directions.
If y'all liked this video you should probably check out the channel Early Music Sources, and their videos on temperament in the Renaissance! (spoiler: they didn't necessarily use 1/4-comma meantone!!)
Analyse "Telemóveis" by Conan Osíris, it's a really interesting and unique song.
Nicely done! I hope you don't mind my piggybacking off of your video and saying that, if anyone wants to check out an arguably wayyyy too dense, awkwardly narrated overview of tuning theory like is talked about in this video, I've got a tuning theory series on my channel, along with several general theory videos that also incorporate tuning theory. Cheers!
Well, we humans perceive pitch logarithmically. So hear me out, guys-LINEAR EQUAL TEMPERAMENT
You mean the harmonic series?
The difference beween the two C#s was pretty obvious to my ear.
A Mathematical proof in a TH-cam video - nice! Better still I could follow it....well most of it. Good video - I do wonder how musicians in the era of Just Intonation coped. Equal temperament is a fudge / compromise but it makes life so easy. ( politicians take note )
so, for an interval that is the ratio of integers A and B, what is the function f(A, B) that defines (or orders) the relative pleasantness of different intervals? what makes a simple ratio simple?
Wow great job! Loved it
QUESTION:What if one of my pieces doesn't come ou the way I want it to. Do I start over or do I go to another piece?Thanks in advance for any and all advice.
I couldn't differentiate those C# notes :<
I'm confused
I sometimes wonder about whether we really have to use tempered scales, especially in electronic music. If a melody is just a sequence of intervals, you should be able to play each interval perfectly without having to worry about how many cents you are off any given reference scale.
On the other hand maybe that gets you into trouble because returning to the fundamental might not be possible in a nice, simple interval.
At the very least electronic musicians should be able to play every chord perfectly; in many melodic tracks you can hear the different frequencies moving in and out of phase and I wonder a lot about how much crisper chords could sound if they used perfect intervals instead. I feel like it's a path that's not well-trodden, even though everyone is going crazy about sound design at the moment.
You don't!!! You can play in any tunings you want! Other equal divisions of the octave! Crazy just intonation using primes up to 37! Non-octave monstrosities! Also - try "Alt-Tuner" it's a fun software for playing chords in-tune with each other as they move... www.tallkite.com/alt-tuner.html
@@stephenweigel Awesome! That's exactly the kind of thing I was looking for., I'll give it a whirl! :D
Schindlabua Awesome :D Googling the word “xenharmonic” opens up many cans of worms, too!
@@stephenweigel I might have seen you comment on Sevish's music before actually. :P
@@Schindlabua Oh yeah, I'm everywhere
Sevish and I co-host the microtonal podcast "Now&Xen." You should check it out! :)
itunes.apple.com/us/podcast/now-xen/id1447552191?mt=2
My thinky parts hurt.
6:27 butterfly
Hey would you please analyse the piano intro from "firth of fifth"? I think it is one of the best piano sections in the story of progressive music, it deserves some attention, doesn't it?