Timeline for Why can't notes be tuned according to a defined frequency?
Current License: CC BY-SA 4.0
20 events
| when toggle format | what | by | license | comment | |
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| Jun 19, 2020 at 17:14 | comment | added | user50691 | That is not part of the OP so what purpose does it serve? There is a lot of ambiguity in the question and some responses. Just tuning also tunes that 5th exactly to 3:2. | |
| Jun 19, 2020 at 15:15 | comment | added | Kevin | @ggcg: This is how things currently work, but it is not how they worked historically. See for example Pythagorean tuning (tunes fifth to exactly 3:2) or quarter-comma meantone (tunes major third to exactly 5:4). | |
| Jun 19, 2020 at 0:32 | comment | added | user50691 | The ratio of a 4th times the ratio of a 5th does in fact give you 2, an octave. It is clear that one is creating a subjective inconsistency for the sake of argument. At least in this context the answer is not clear and leads to more ambiguity. Perhaps the author could provide the "definition" of consistency and where the actual inconsistency arises. | |
| Jun 18, 2020 at 23:57 | comment | added | phoog | @ggcg the assumption of just intonation is that intervals that aren't rational aren't perfectly "in tune." That's the origin of the second definition. Obviously "perfectly in tune" is somewhat subjective, but it is certainly true that if you tune a keyboard using acoustically pure intervals it doesn't turn out very well, and that does indeed arise from the logical inconsistency between those points. The fact that "this is not how intervals are defined" is in fact required by the logical inconsistency, which is why discussing the inconsistency is an answer to this question. | |
| Jun 18, 2020 at 21:17 | comment | added | user50691 | Additionally the desire to enforce your mathematical constraint is not necessarily logical in the first place. You painted yourself in a corner with that statement | |
| Jun 18, 2020 at 21:16 | comment | added | user50691 | Kevin your statement is patently false. This is not how intervals are defined universally. This is why we have 12tet, 12th root of 2 | |
| Jun 18, 2020 at 20:05 | comment | added | Kevin | @ggcg: The definitions are: 1) An octave consists of (exactly) a 2:1 ratio. 2) Smaller intervals than an octave exist and are (always) rational numbers. 3) There are finitely many notes in an octave. 2 is prime, so if you have any ratio which is not 2:1 or a power thereof, you can't multiply that ratio by itself repeatedly to get to some power of 2:1. That means stacking a rational interval on top of itself repeatedly will never land exactly on its starting point (in a higher octave). | |
| Jun 17, 2020 at 21:11 | comment | added | phoog | @chepner The one I usually use is the I-IV-ii-V-I progression, or some variant of that. If you tune C major, F major, and G major in just intervals then your D-to-A fifth is the very sour-sounding 40:27 ratio. If you raise the A to fix that, you lose the relative minor, and if you raise the E to fix that, you've admitted defeat by abandoning just intonation on the tonic chord. | |
| Jun 17, 2020 at 20:01 | comment | added | chepner | @phoog Some intervals would correspond, but not all. I tried leaving the details out to avoid having to discuss the ones that do :) | |
| Jun 17, 2020 at 19:55 | comment | added | phoog | @chepner you've chosen a poor example, because the ratios (15:8):(3:2):(5:4) do indeed conform to the 15:12:10 ratios of a just minor chord. Or are you talking about the G# of an E major chord? | |
| Jun 17, 2020 at 16:59 | comment | added | Pete Kirkham | @BlueRaja-DannyPflughoeft by 'our ears prefer' you mean 'the minds of people brought up with western music'. There is little evidence it really is innate biology, as indigenous tribes do not share such preferences newscientist.com/article/… | |
| Jun 17, 2020 at 16:09 | comment | added | user50691 | What definitions are "logically inconsistent"? | |
| Jun 17, 2020 at 12:48 | comment | added | Kilian Foth | @Ashley Yes, that's exactly it. This is why stacking fifths on top of each other results in pitches that slowly become ever higher compared with equidistant ones. | |
| Jun 17, 2020 at 12:27 | comment | added | Ashley | "... twelve perfect fifths almost but not quite correspond to seven perfect octaves". Just checking I understand: is the arithmetic here 2^7 = 128 but (3/2)^12 = 129.746? | |
| Jun 17, 2020 at 0:08 | comment | added | user28245 | Slight nitpick: the third prong of the trilemma wasn't stated. You can have all perfect octaves and all perfect fifths be just... you just end up needing infinite notes in an octave to do it (Dbb, C, B#, Ax#, who says they're the same? They're not, we just pretend they are). Ergodic theorem says it's even dense. And yes 19/12 is one of the continued fraction convergents to log_2(3/2). As an accident, 2^(4/12) is an acceptable approximation to 5/4 as well. | |
| Jun 16, 2020 at 21:04 | comment | added | BlueRaja - Danny Pflughoeft | @LeloucheLamperouge: Biologically, our ears prefer combinations of frequencies with ratios containing small integers. Physically, this is because sound waves are additive and thus have a resultant frequency dependent on that ratio. Mathematically, 12 notes are chosen because different powers of 12th-root(2) give numbers very close to 5/4, 4/3, and 3/2, so we're able to create several nice combinations with just those 12 notes. Why that's true has to do with continued fractions, or something. | |
| Jun 16, 2020 at 18:26 | comment | added | chepner | Say you want to play something in C major using just intonation. You can compute all the frequencies you want using the interval, and (for example), you get a particular frequency for E. But with those frequencies, the intervals using E as the root won't quite be in tune: you can only tune your instrument to play in one key. Using equal temperament, all the intervals work out mostly correct regardless of what key you choose, but every interval except the octaves deviate to some degree away from the "idea" interval. | |
| Jun 16, 2020 at 16:43 | comment | added | Lelouche Lamperouge | Why did we decide C * 3/2= G (i.e. a fifth) instead of calling it , say D ( a 2nd) or E(a third) . | |
| Jun 16, 2020 at 15:46 | comment | added | Scott Wallace | Indeed, so it is. And further: we cannot have perfect major thirds (ratio 5:4) or perfect minor thirds (6:5) with any equal temperament. Math rules. | |
| Jun 16, 2020 at 9:43 | history | answered | Kilian Foth | CC BY-SA 4.0 |