Honestly math fags need to teach math like this.
>>17079831This is cuteDo one for cantor's diagonal argument
>>17079831ironically 1:sqrt2 is a very musically pleasing ratio and we use it all the time, exactly, in equal temperament
>>17079953Tell us more, Harmonious Anon.
>>17079955Well, the most harmonious intervals are traditionally considered to be the unison (1:1) and the octave (1:2)The perfect fifth (2:3) is traditionally considered to be the next most harmonious interval. Other pleasing intervals can be constructed from simple integer values: the perfect fourth (3:4), the major third (4:5), the minor third (6:5), et cetera et cetera.Pythagoras considered the perfect fifth to be of prime importance, and constructed his scale by stacking fifths upon fifths. Yet by doing this, we can never quite perfectly arrive at an octave: the difference between 12 fifths and 7 octaves is (3/2)^12:7(2/1)^7, which is 531441/524288 and sounds noticeably out of tune. Even the 5:4 major third has to be approximated by 81:64 which is quite sharp.In fact, no rational interval can be used as the basis through which to derive all other intervals, and trying to do so leads into infinite spirals of different note identities. This can actually sound quite cool, but is difficult to implement in real life with physical instruments.So, let us consider irrational intervals. What interval sits exactly halfway between the unison and the octave? It is precisely 1:sqrt2. if we raise sqrt2 by the same ratio again, we get sqrt2:(sqrt2^2) which is obviously equivalent to sqrt2:2.If we extend this principle to be able to split the octave into twelve, which had traditionally been the number of distinct note identities in western music, we end up with a semitone of 1:the twelfth root of two, which we can repeat twelve times to end up with a perfect octave. It turns out that by doing so we can very well approximate many 'just' (low-integer) ratios whilst avoiding the problems of infinite pitch spirals, wolf intervals or pythagorean commas. See picrel for a diagram representing this. If you want to read more about why this is the case, you may find this article to be of interest:https://en.xen.wiki/w/Riemann_zeta_function
>>17080002Thank you, Harmonious Anon. That was both interesting and informative.Any you have impressed upon me just how non-natural “notes” are. The atoms we choose from the myriad frequencies are a choice. A preference. But they are arbitrary and they may be others.Tonight’s rabbit hole just opened.
>>17080003>there may be othersYes! there is an infinite continuum of harmonic possibility. The twelve note system is simply a practical compromise and a cultural norm. Arabic, persian and turkish music routinely use many more than twelve notes per octave, for example. I reccomend listening to Yasamin Shahhosseini,who plays oud, if you have never heard any.
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>>17080005The thing about microtonal music that rarely get mentioned though, is that the microtones are usually still framed within sub-tones around the 12 tones of normal 12-tone music. Like basically just semitones and quarter-tones around basic 12-tone structure. in the history of music this goes back very very far, I forget the name of the Pythagorean philosopher who covered this but it does go back to like hundreds of years BC. From a scientific perspective, one could rationalize this according to the idea that the harmonic series of overtones does have a natural mathematical relationship with musical notes, plus the fact that human ears only have a particular sensitivity to discerning one pitch from another (is 440 Hz that much different sounding from 442 Hz, outside playing them together in a chord?) That said the Pythagorean system of music based on integer ratios is a very powerful lingering cultural heritage which seems to transcend all disparate cultures and may be tapping into some deep mathematical structure that unites minds with the universe
>>17080152>The thing about microtonal music that rarely get mentioned though, is that the microtones are usually still framed within sub-tones around the 12 tones of normal 12-tone musicBy whom?>From a scientific perspective, one could rationalize this according to the idea that the harmonic series of overtones does have a natural mathematical relationship with musical notesRight. What's the "natural mathematical relationship" for a semitone?> the Pythagorean system of music based on integer ratios is a very powerful lingering cultural heritage which seems to transcend all disparate culturesThis is clearly not the case given the huge variety of scales people actually use.
>>17080152>normal 12 tone musicright, from the perspective of someone to whom 12-tone music is 'norrmal'.But there is something special about using 12 equal tones as opposed to say, some other similar number, like 11 or 13, in order to approximate the natural pitch ratios that we find consonant. if we plot Z(2πx/ln(2)) around that region we can see this (picrel).In fact, 12 is not the only number that has this property, though. 19, 31 and 53 are also especially successful in this regard, but perhaps 12, by merit of just being a smaller number to deal with, is more easily applied and so has become relatively culturally dominant. However, 12-tone systems, both equal and unequal, are only really good at approxomating intervals up to the 5-limit, and intervals in the 7- or 11-limits generally have to extend the number of notes available.>outside playing them together in a chordAh, but is playing notes together not the very definition of harmony? When we hear two notes in combination, we hear not only the original two notes f1 and f2 but also the combiation tones f3=f1+f2 and f4=f1-f2. This is perhaps another reaso why pure integer ratios are so preferable, as they are much more likely to generate combination tones which are also consonant.
>ITT: people with no musical ear think music is about the handful of consonant tones you get from small integer ratios
>>17080212>handfulanon there are literally more possible ways to combine those types of intervals than any culturally prevalent system of music has ever managed to encapsulate.>people with no musical earoh, i am also a professional musician. did this thread trigger some sort of insecurity for you? maybe you should work on that.
>>17080230Your argument is irrelevant. Only 2:1 and 3:2 are indisputably consonant intervals. We could bicker about 5:4 and 6:5 (I don't think pure thirds necessarily sound any more harmonious than variations based on ugly ratios) but whatever, you can have them for free. Let's say following mathematical elegance gives you four naturally appealing intervals. That means 2/3rds of the twelve note scale are nothing special to the ear. They function to create various degrees of dissonance and there's no inherent psychoacoustic reason to use those specific intervals. Composers aren't even limited by instruments with fixed tunings or timbres anymore. They're just limited by conventions and traditional music theory.
>>17080298wow, you are really retarded
>>17080298if you're giving me fifths and thirds, then you're also giving me fourths and sixths because they are just inversions of the same. so that's already 8/12 notes.starting from C we haveC 1/1Eb 6/5E 5/4F 4/3G 3/2Ab 8/5A 5/3(this collection of notes is called the 5-odd limit)now, let's think about triads, which are the building blocks of harmony. From nothing but those 3 intervals and their inversions we can already construct C, Cm, Ab, Am, F, and Fm. The remaining notes C#, D, Bb and B can all be constructed as the fifth or third from one of the notes we already had. Ergo, literally every note in the 12-tone system is a consonance. some are just more consonant than others! They are not just random and the fact that you would even suggest that implies that you have never really thought about this with any depth.I should mention that depending on what sequence of fifths and thirds you use, you can arrive at slightly different values for each note, which can cause issues in certain circumstances. The great thing about 12-tet is that it eliminates this problem and allows each note to be close enough that it can moonlight several harmonic identities simultaneously.
>>17080438>if you're giving me fifths and thirds, then you're also giving me fourths and sixths because they are just inversionsDo fourths sound like fifths to an NPC like you?
>>17080438i forgot about F#/Gbbut it can be constructed a number of different ways:7/5 from C6/5 from Eb (6/5 twice from C)5/3 from A (5/3 twice from C)etc etc.1: sqrt2 is actually a nice way to encapsulate all of these, which is what equal tempered instruments use
>>17080447yes. a fifth up is the same as a fourth down and vice versa. the fact that inverted interval pairs are very closely related, mathematically and experientially, is not exactly controversial.If I am singing a melody with my friend, and I am supposed to be harmonising a fifth below them, but my voice is too high to reach those notes, I can instead harmonise a fourth above and imply the same fundamental. in fact our brain will literally fill in and hear the missing fundamental.
>>17080451>>Do fourths sound like fifths to an NPC like you?>yes.I'm not surprised. You're essentially a biological LLM regurgitating babby's first music theory lesson. You don't actually hear intervals so my point is totally lost on you.
>these people who are far smarter than me need to dumb down everything to my levelShut the fuck up.
>i am le heckin' smort for being able to follow babby's first music lessonIf you had attended babby's second music leson, you'd have learned that a 4th is an unstable interval in tonal harmony, as opposed to the very stable 5th. But never mind that. You don't need to know anything about music theory. You can plainly hear it wanting to resolve to a 3rd even if you don't know what a 3rd is. This board is crawling with actual qualialess subhumans.
>>17080453>>17080466C2 and G2 are the 2nd and 3rd harmonic of C1G2 and C3 are the 3rd and 4th harmonic of C1both can fulfill the role of implying a fundamental C1. the missing fundamental is a known psychoacoustic phenomenon and is much more ubiquitous than the 'rules' of common practice tonal harmony, which are only relevant in one particular musical tradition at one time period, and even within that context are broken all the time.And you're ignoring the fact that context matters; an F together with a C sounds different depending on wether the implied harmony is Csus4 (unstable) or simply F(stable). If you think that every time a fourth is used the implication is a sus4 chord then.. you're exactly as retarded as I thought you were.
>>17080475Retard, there is nothing you can say that will change the way a 4th sounds. Nothing. There is no discussion to be had about this.
>>17079831Hey OP, your maths thread sucked so much that music fags came over to spray gay all over it.
>>17080478so you think C Eb G C sounds more dissonant than C Eb G B because the first contains a fourth and the second changes that fourth to a major third?
>>17080480>>17080480I don't need to "think" anything. I can pick up a guitar, pluck the two lowest strings and hear the sound it produces. It's not harsh, but there is something incomplete and incongruent about it. Anyone who actually hears can do the same and hear for himself. What I think is that you don't even understand what you're regurgitating, because you dismiss the core of tonal harmony as "one particular musical tradition" (which happens to be the most prevalent musical context for the 12-note scale), then immediately make irrelevant talking points about how a 4th might sound in "musical context". Now here's something that's really gonna throw you into a loop: I was specifically and deliberately talking about the sounds of intervals on their own, without context, but it is precisely BECAUSE THEY SOUND THAT WAY ON THEIR OWN that basic tonal harmony follows the conventions that it does.
>>17080486So pick up your guitar and play an open E major chord. What are the two notes on top? B and E? oh, that's a fourth. So an open E major chord is a dissonance now? you're obviously a beginner lol just stop, you're embarrassing yourself.
>>17080488I'm talking to you about an interval. Why are you replying with an "argument" about chords? Are you not only deaf but also literally retarded?
>>17080466What the fuck are you rambling about?
>>17080505If you don't know then I'm not talking to you.
>>17080488You're misunderstanding my point completely. What I was saying is that if you have a fixed pitch and you build a scale on top of it, based only on intervals that would please Pythagoras mathematically AND sound inherently harmonious to a typical human, you only get a 5-note scale. 4:3 is a nice ratio, but how harmonious it sounds (on its own) is debatable. If tonal harmony (that's been dominating popular genres for centuries) is any indicator of common taste, a 4th doesn't stand on its own.I'm not disputing that you can take a plain triad, double the root and get a chord with a 4th that sounds perfectly good alongside the other pitches. I'm saying it doesn't justify extending the scale, not by the original construction criteria. Yeah, you can take the 3:4 ratio that emerges from the chord, apply it to the root pitch and get a new pitch, but that's a jump through pure abstraction, not something you intuitively converge on by messing with two strings. If you had neither the mathematical reasoning to go by, nor the chord as a reference, you wouldn't perceive a detuned 4th as such, the way you can hear it when an octave or a 5th is off.
>>17080002>What interval sits exactly halfway between the unison and the octave?the tritone which is the least harmonious interval in the scale
>>17080536my dear boy, you obviously have yet much to learn. I wish you the best of luck.
>>17080628well, I rather like it. But the tritone is undoubtedly very important in functional harmony, being the basis of the V7-I cadence for example. And the equal tempered tritone is exactly equal to 1:sqrt2.
>>17080653I'm 2-3 stdevs above you in IQ and probably a much better composer than you, but whatever helps you cope. I'm sure you felt real smart getting filtered and spouting irrelevant trivialities like they're supposed to be revelations.
>>17080536>I'm not disputing that you can take a plain triad, double the root and get a chord with a 4th that sounds perfectly good alongside the other pitches.at that point you're just agreeing with him that if you have a fifth you also have a fourth so how is he wrong?>that's a jump through pure abstraction, not something you intuitively converge on by messing with two strings.unproven but ok, i can entertain it. assuming it's true, so what? even if you don't arrive at it just by ear, you still arrive at it naturally when you start systematizing music
>>17079831>they should teach math like in Sesame Streetokay dear, it's time to go to bed now
>>17080763>at that point you're just agreeing with him that if you have a fifth you also have a fourth so how is he wrong?But the context was constructing a scale. You can get an "emergent" 4th between existing scale pitches, but you don't get an F from a C just by selecting inherently harmonious-sounding intervals with nice ratios over the C.>assuming it's true, so what? even if you don't arrive at it just by ear, you'll naturally arrive at it when you start systematizing musicSo you start systematizing, jump through more abstract hoops, arrive at 12-TET and use that for centuries. But then you start to get bored with it. Now what? People have been trying to generalize the reasoning behind 12-TET for a ages. You have systems like 19-TET, 31-TET etc. but the music based on them tends to sounds like crap, because most of the intervals just sound like detuned versions of 12-TET ones. Not even the 7ths in 31-TET actually sound that consonant, even though they're good approximations of 7:4 and should "theoretically" be pleasing to the ear.Empirically, it looks like using muh harmonics, muh just intonation, muh small fractions etc. to navigate beyond the familiar musical landscape leads to dead ends. So on one hand, you have the musically deaf trying to peddle bad-sounding Western microtonality as progress (if you don't like the emperor's new clothes, you just need to train your eyes better). On the other, you have brainlets who think 12 is a magical number. Both are obviously wrong. The problem is that all these N-TET systems are trying to generalize the wrong things because they're driven by pure abstraction. What does generalize well is the idea of stable and unstable intervals, tension and resolution.
