10 ms·

Why do we tune in 5ths?

81 points by ofou 5y ago

BenoitEssiambre 5y ago
On multiple occasions, I've stumbled upon really old high school physics textbooks and they usually had a chapter on the physics of musical frequencies. It seems to be a weirdly forgotten subject. The octave is a doubling/halving of frequencies, and so is very recognizable. The fifth is the next lowest whole number frequency ratio 3/2, 2/3. So liberal use of fifth interval makes notes often end up low whole number ratio frequencies of each other which sounds harmonious.
- BenoitEssiambre 5y ago
  Also, if I remember correctly, every note in the major scale is a 3/2,2/3 ratio from a note in the lower or higher octave following the standard order of key signatures as you jump across consecutive octaves, which, I assume, is how they arrived at that standard order.
  - klodolph 5y ago
    That’s called “Pythagorean tuning” and it’s possible but it’s not especially common. The system we use is a compromise derived from that system. If you use pure 3:2 ratios you get the so-called “wolf tones”, but we don’t get those any more. The current most common system is equal temperament, which is a type of meantone tuning, where you take the 3:2 ratio and alter it. Equal temperament alters it to 2^(7/12) ≈ 1.4983, and it has the nice property that you get twelve notes and it’s all perfectly symmetrical, it doesn't matter which note you start on. Before equal temperament, there were various other systems. One popular one was quarter-comma meantone, which took four stacked 3:2 ratios (81:16) and squashed them to equal two octaves and a 5:4 ratio (80:16), which results in a value of 5^(1/4) ≈ 1.4953. You see it in some old organs—which may have, say, separate keys for G# and Ab. Then there are various non-meantone tunings, various “just” intonations, etc. In just intonation, a chord like C E G will have a 4:5:6 ratio. You can’t do this with all major triads, though, so you end up with some major triads that sound correct and others that sound out of tune. Our modern system is a compromise where all major triads are equally out of tune, rather than having some better and some worse.
    - lucas_codes 5y ago
      Wolf tones are certain notes which don't sound very well which vary between instruments due to their construction. I think you mean the Pythagorean comma.
      - bonzini 5y ago
        You are talking of the same thing. The typical wolf tone in quarter comma meantone was G#-Eb, which sounded so bad that some instruments added a separate key for Ab.
      - klodolph 5y ago
        Actually, "wolf interval". Pythagorean comma is the difference between enharmonic notes in Pythagorean tuning, e.g., difference between Ab and G#. Not what I was talking about. With any 12-tone system, you might have eleven fifths which are 700-x cents wide, on average... and then a diminished sixth (rather than a fifth), which is necessarily 700+11x cents wide as a result. The "wolf interval" is the diminished sixth (commonly G# to Eb, or C# to Ab). In equal temperament, the diminished sixth is equal to a fifth. You might also call something other than a fifth a wolf interval. Basically, an interval which normally sounds good, except you chose an enharmonic version of it, which sounds bad. "The" wolf interval is the quarter-comma meantone diminished sixth, which is 2^7 / 5^(11/4) = 1.5312. The Pythagorean wolf interval is 2^7 / (3/2)^11 = 1.4798. It's equal to a perfect fifth, minus the Pythagorean comma.
        lucas_codes 5y ago
        I see. I was confused because string players talk a lot about our wolf notes/wolf tones (especially violists and cellists), but I hadn't come across the term wolf interval.
        klodolph 5y ago
        That's because these tuning systems have been out of favor since about 1700. You might encounter some organists who know about it, but even then, there are only a handful of organs tuned this way in North America. https://en.wikipedia.org/wiki/Meantone_organs_in_North_America https://en.wikipedia.org/wiki/Meantone_organs_in_North_Ameri... It was Werckmeister who popularized and advocated systems which don't have wolf intervals (so-called "well temperament", as in "Bach's well-tempered clavier". The composers who cared about wolf intervals generally predate Bach, to give you an idea. Nowadays (since the 1980s) I can press a button and put my synthesizer or electronic piano in whatever tuning I want, so I can play with quarter-comma meantone temperament without finding one of the organs on the list, and experience it myself.
- moehm 5y ago
  > The octave is a doubling/halving of frequencies Side note: Except when in praxis you have to accord for non-perfect masses, then you have to stretch the tuning. https://en.wikipedia.org/wiki/Stretched_tuning https://en.wikipedia.org/wiki/Stretched_tuning
- metamet 5y ago
  > It seems to be a weirdly forgotten subject. I was fortunate to have had the opportunity to take a Physics of Music course in college that counted as the physics requirement. It was really nice to have the mysteries what makes music appealing be an academic focus for the summer. We even had a project at the end where we built an instrument. I made a fairly rudimentary xylophone with some pipes and a hand pipe cutter.
- BadassFractal 5y ago
  The physics of music comes up a bunch in music production. When working with different waveforms, filter types and specific bass frequencies, you always have to make sure you know what your spectrum looks like, or you'll have phase issues, or you won't hit the desired levels for each instrument and so on. You end up staring at fundamentals and harmonics a lot in order to get the timbres and the mix you have in mind.
kazinator 5y ago
If the instrument is tuned in fifths then, indeed, a diatonic scale pattern that proceeds perpendicularly across the finger board at the same position has four notes on each string. However, these four notes are sometimes spaced differently: different patterns of tones and semitones. When an instruent is tuned in fourths, scales use three notes. There are ways to conform to many of the string-to-string pattern variations by leaving out fingers: a - b c' # index ring pinky e f - g # index middle pinky - c - d It seems that the fifths tuning is quite tyrannical in this regard. Now when it comes to the major scale, that scale is actually a stack of two identical tetrachords, which are a fifth apart. These tetrachords have identical fingerings: # tuning in fifths now: g - a - b c c - d - e f So now if you have your fingers spread out in the right way to play the C-F tetrachord, all you have to do is preserve that when going to the next string. I suspect violinists must be exploiting this sort of relationship; favoring fingerings that show tetrachord symmetries. The Dorian mode (D to D mode of C major) also has two identical, stacked tetrachords, so if we slide one position down that pattern, we are still good: g - a - b c - d c - d - e f - g [ Dorian box ] [ Ionian box ] But, having observed that, these fingering patterns are not easy even in isolation, regardless of whether there is a change in the pattern going to another string.
- contravariant 5y ago
  To some extent they should have some similar properties since a fourth is basically the inverse of a fifth (modulo an octave). In theory you should also be able to shift a tetrachord on an instrument tuned in fourths you'll just go the other way and be off by an octave.
  - klodolph 5y ago
    Being off by an octave is pretty important for melodic lines, which is usually how the violin is used… so while you can translate chords from tuning in fifths to fourths or vice versa, in my experience, you need to relearn to play if you're doing melodies.
  - kazinator 5y ago
    If you go the other way, the tetrachords share a note: C D E F, G A B C Swap these: G A B {C} D E F So, tuning by fourths, it lines up, indeed, when we take into account the shared note, e.g. these two L-shaped tetrachord patterns: D - E F D - E F C + A - B C = A - B{C} G G Or: E F E F C - D + B C = B{C}- D G - A G - A
fallingmeat 5y ago
That has to be one of the more civil examples of online discourse I’ve ever seen.
- eyelidlessness 5y ago
  To be fair, you’re the first commenter. But I applaud you setting a good (sorry) tone!
  - a_t48 5y ago
    They may have been talking about the comments in the posted link.
    - eyelidlessness 5y ago
      They were, I was trying to bring some of that good tone here.
jedimastert 5y ago
I've thought about this a fair bit myself, and I think it's a combination of a couple of the reasons stated. First, it's a reasonable range in which every other note can be played within the string before. But more importantly, I think it's just because it's incredibly easy to hear a fifth for tuning. Humans are fairly adept at hearing minute differences between two of the same frequency (i.e. a few hertz or cents off) when two notes are exactly the same, a fifth above share the most harmonics within that reasonable fingering range.
- dhosek 5y ago
  Although it's typical to tune orchestral strings using harmonics to match the pitches. For the cello, viola and violin, this is done by comparing the 2nd and 3rd harmonics (at the octave and twelfth). For the bass, tuned in fourths, one compares the 3rd and 4th harmonics (at the 12th and second octave).
dhosek 5y ago
A lot of speculation but no real answers. I find the argument about resonances between strings to be spurious since it really only applies to notes played on open strings which are a distinct minority of the notes played (not to mention that there are distinct advantages to avoiding open strings, such as the lack of the ability to apply vibrato on an open string). Ultimately, it's a matter of string length, I think. It's worth noting that in the cello, viola and violin, all four fingers are used independently in fingering notes, but on the double bass (unless you're Joel Quarrington), you finger 1, 2, and 3+4 together since the littlest finger doesn't quite have the strength to press the string down on its own and the ring finger doesn't have the reach of the little finger. But, like all the respondents in the discussion, I don't really know either.
- analog31 5y ago
  Also, bass is most commonly tuned in fourths unless you're Red Mitchell.
- kayson 5y ago
  Your point about resonances isn't really true. The other strings can resonate "sympathetically" even when you play a fingered note. For example, vibrato can be achieved on open strings by vibrating on the finger position that corresponds to the note an octave higher than the open string. This is mostly used for the open G string, since as the lowest note, it can't be played anywhere else on the instrument. But its also used on other strings for different colors. Sympathetic resonance is also why some keys sound better on a violin than others. Violinists tend to like the keys with sharps: G D A E
  - dhosek 5y ago
    But that's not connected to fifths tuning. It can happen with any tuning (one “trick” with a dual neck 6/12 string electric guitar is to tune the strings on the 12-string neck to each of the diatonic pitches so they resonate with whatever note is played on the 6-string neck, and then of course there are the sympathetic strings on Indian stringed instruments which have a similar function). Curiously, while orchestral strings tend to be more comfortable in the sharp keys, as a double bass player, I was equally at home (if not more so) in the flat keys that tend to be favored by winds.
fathyb 5y ago
I always hum the first two notes of the Dallas opening theme to remember what a fifth is.
- antegamisou 5y ago
  Make your life simpler and memorize SOAD's Toxicity intro.
eyelidlessness 5y ago
I pretty much only play guitar in nonstandard tunings. Not all of them are oriented around fifths, but they often are at least in part. Repetition of finger positions across strings is one motivation. While some of the linked comments suggest reachability, the standard guitar tuning affords a lot more for many common guitar styles. But nonstandard tunings open up a lot of other styles because you can use partial fifths on open strings more readily than with standard. Two things I think (particularly) guitarists miss: 1. If you’re not doing chord/scale/mode transitions standard tuning is optimized for, you’re putting a lot of cognitive load either on your playing or your practice. 2. Translating guitar tuning to fifths tuning is a lot easier than vice versa if you’re reading tabs or standard sheet music.
- tasty_freeze 5y ago
  I've often wondered what guitarists are thinking about when they use alternate tunings. I'm sure if you play in one tuning long enough you can conceptualize how all the notes related to chords and keys, etc. But if you frequently change tunings, when you learn a song does it just become a matter of memorizing finger patterns, or do can your brain actually figure out the mapping such that you are thinking in terms of chords and intervals?
  - mabub24 5y ago
    A lot of guitarists have no problem with playing open tunings because they can pretty seamlessly transcribe the notes along one string to the whole tuning, like how the 2nd fret of the low D string in open D is an E note so thus a finger across the whole fret is an "E chord". It's once you get into alternate tunings that are outside the "open tuning" norm that it becomes, at least for me, memorizing finger patterns and getting fine coordination locked in. The theory gets thrown out for "what sounds good" and "hmm that sounded about right".
  - eyelidlessness 5y ago
    For me, once I’m playing it’s almost entirely muscle memory. Writing or learning, the tuning doesn’t matter at all it’s just developing that muscle memory. The thing nonstandard tunings does for me is it gives me a base state to work from that lets me use patterns and techniques suited to it. Drop D is very good for bass-heavy fifth chord stuff. CGD--- is really good for fingerstyle techniques. Other weirder tunings like some used by Michael Hedges let you rethink whole chord relationships.
pessimizer 5y ago
A lot of people seem to be answering by explaining what 5ths are. That isn't really the question. The question IMO is why they tune to 5ths rather than something else like 4ths, as a guitar often is. 4ths are really just 5ths backwards, but I've always wondered that about violin, too. edit: Ah, the last answer is very clear and good.
elihu 5y ago
I think it's mostly just that 5ths give you a good balance between ergonomics and range, and 5ths are relatively easy to tune by ear. Something interesting about violin tuning: if you tune by ear, you get a 3:2 ratio between the frequency of adjacent strings. If we're in the key of G, we can call the G string 1/1 in just intonation notation. The next string is D, or 3/2. Then we have A at 9/4. That's an octave and a just major second. The interesting thing is that when you get to E, it's 27/8. It's an octave and a major sixth, sort of. If we ignore the octave, it becomes 27/16 which is close to the more commonly used just major 6th which is 5/3. If a violin is tuned to 12-tone equal temperament then the distinction is moot. But in a just tuning, that open E may well be a wrong note depending on what you're playing.
- kragen 5y ago
  Violins don't have frets.
  - mandmandam 5y ago
    They didn't mention frets... Tuning the open strings relative to each other, at least for non perfectly pitched people, results in tuning the strings justly rather than to equal temperament.
    - mrob 5y ago
      The 5th is the least tempered interval in 12 equal temperament (excluding the untempered octave and unison), with less than 2 cents difference from the just interval, and the violin only has 4 strings, so in practice it's probably close enough.
      - kragen 5y ago
        Yes, it would be hard to be less tempered than octave and unison :)
      - elihu 5y ago
        The errors stack, so you get about 6 cents error between the G and E string. That's enough to be a perceptible difference.
spoonjim 5y ago
All this music theory stuff is not the answer. The answer is because the typical male hand can play the fourth with the pinky finger. So tuning in fifths gives the instrument the most range.
srcreigh 5y ago
Just don't use a guitar tuner. Or a piano. Your open C won't be C, it will be 440 * pow(sqrt(2,12),7) which is ~0.75Hz flat. (Does this mean guitars and pianos are out of tune? Yes.)
jancsika 5y ago
What is the music history version of HN where you can get a drive-by answer from one of the (probably many) musicologists who have devoted their lives to studying the history of the violin?
- icelandicmoss 5y ago
  > What is the music history version of HN In the 11 hours since you commented, the answer appears to be... HN :)
Waterluvian 5y ago
A tangent: I’m learning piano and musical notation finally clicked for me. Musical notation is a high level language. It gives you a program that you can run on any instrument. And it gives you the ability to modify it with relative ease. When playing clarinet in high school I felt that notion is this archaic thing that we’re stuck with because of the morons in the past. But I’m realizing it’s this beautifully refined language, developed over centuries.
- PaulDavisThe1st 5y ago
  It might be refined and developed over centuries, but it's also the expression of overwhelmingly a single culture. Western musical notation is not useless for denoting the music of other cultures, but it's also not very good and is frequently quite bad. If you think of western musical notation as "the system", you're at risk of remaining ignorant of the beautifully refined languages, developed over centuries, in other cultures. It's also a highly abstract language that seems to frequently, somewhat like Javascript, lead its users to have very little familiarity with what is going on at lower levels.
  - kansface 5y ago
    As a classical guitarist, western notation is not particularly good at expressing the music our own culture. The high open E string can be and is regularly played on the other 5 strings too, potentially at the same time! Natural and artificial harmonics, which string to play on and which fingers to use, and all sorts of right hand notation all feel very much bolted on cluttering the actual score. The continual coming and going of voices due to technical limitations means that music has to be experimentally decoded with much backtracking.
- interfixus 5y ago
  They tried to teach me various instruments when I was a boy. I didn't mind the playing, but Lord Almighty how I hated the notation stuff. It simply made no sense whatever, and was - as you describe it - clearly invented by morons. Only when I ventured into tech-land in my teenage years did any of it finally click: So that's what they mean by 'octave' - a doubling of frequency. Why couldn't they just have said so? Harmonies, chords, what have you - it was all just frequencies and sine waves. Didn't musicians know anything? These many years later, I still don't quite get it, but do of course realise there must be some sort of point to it all. High level language? Yes, I get it. Bach wrote in Python, not C. Time to revisit.
boothby 5y ago
Fiddlers don't always tune in fifths. There's a couple of tunes I play in DDAD, my favorite for its three octaves of awesome resonance. Classical violinists tend to sneer at such tunings, outside of rather esoteric baroque players, I believe. It's a shame, because different tunings can really change the tone of the instrument in interesting ways. As the comments allude, fifths make it easy to play a large range of harmonious music. And, no doubt. A lot of fiddle tunes are written for cross-tuning, and they're tortuous to play at speed in GDAE.
- BoxOfRain 5y ago
  >It's a shame, because different tunings can really change the tone of the instrument in interesting ways. I find this with guitar, but guitarists on the whole aren't snobbish about alternate tunings. DADGAD is my favourite guitar tuning for similar reasons.
  - Zuider 5y ago
    DADEAE is another tuning that is good for resonant melodic playing without straining the neck. As you can see you have the D,A,E strings tuned in 5ths like a banjo or fiddle.
  - aidenn0 5y ago
    According to my dad, guitarists were more snobbish about tunings in the past. By the late 60s it was cool to experiment with tunings (at least for rock guitarists), but he says there definitely was a point where you were looked down on, particularly for open tunings
xscott 5y ago
I have an electric guitar with 5 strings tuned fifths (C G D A E), and a tenor guitar with 4 strings in fifths (currently D A E B). I've been mucking around with both for a few years. Conceptually, they're nice. You get a great set of movable bar shapes, giving you spread chords with the first, fifth, and easy access to a 9th (2nd), minor third, major third, or 4th (11th). So it's super easy to get the common chord "colors" (sus2, maj, min, sus4). Similarly, you can add the next string to get a maj 6th, min 7th, and maj 7th with a 4 string movable chord. However, it all sounds a little thin if you're playing in an acoustic style. Bar chords in standard guitar tuning sound better to my ear, so I'm not sure I believe any of the responses talking about better overtones. Maybe that's different with a violin and the music that's played, dunno. If it was overtones, why wouldn't the "double bass" violins use it? Even with the shorter length of the tenor guitar (23 inches), playing diatonic scales is unpleasant. My pinky finger feels week when it has to stretch that distance for 4 notes per string. And I really don't like the stretched out (three notes per string) pentatonics. Because of this, I think the answer to the link above really is about the scale size and ergonomics. I'll probably leave the tenor guitar alone (it's a novelty for me anyways), but I'm thinking I'll switch to CGDGBE or go back to Drop D on the electric guitar. All fifths just isn't that enjoyable for me.
- xscott 5y ago
  There was a King Crimson link on HN the other day. I should mention Robert Fripp's "New Standard Tuning" is all fifths up the point where the highest string would be way too thin or way too tight (CGDAEG). Even as is, that high G seems unhappy, but he makes it work :-) I think someone used to sell string sets for that, but I can't find it on Amazon atm.
ithinkso 5y ago
This article, and the comments, made me realize why some friends that I tried to teach coding just couldn't grasp it yet for me it seems kinda obvious. I don't understand music one bit, none, zero. I can read descriptions and even understand what different notes are, and the symbols for them but it's a very dry knowledge. Nothing I do make it 'click' for me and it feels so disjointed and foreign that I just got overwhelmed and I know it will never make any sense to me I can read a math paper, and even though I might not understand it at first, at least I know that if I just take a book that explains the terminology then I'll be able to grasp it. With music it's a complete opposite - nothing no matter what I do will make it understandable for me Edit: To give you and example, I have been studying the famous clip [0] about the rhythmic displacement. I have watched it thousands times, read tons of analysis etc. and I know exactly what was done and when. I just don't hear it being done - I know people aren't lying to me but I just cannot hear whatever happened at that famous point and what changed after [0] https://www.youtube.com/watch?v=UinRq_29jPk https://www.youtube.com/watch?v=UinRq_29jPk
- patrickthebold 5y ago
  I actually don't like this attitude. In every case it's a mostly a matter of interest and effort. If you have those you can learn something.
  - ithinkso 5y ago
    I agree, that's what I told my friends, to just keep learning because if I could why wouldn't you. The point of my post is that I have realized how unhelpful that comment is because if someone told me that about music today... all I see is an insurmountable mountain ahead even though I've read quite a bit about it already Mind you it's different with for example fluid dynamics or functional programming or even chemistry or biology, I have zero knowledge of it (nor interest) and all I know is that it's hard... but I know that if I spend years learning it I will succeed. I don't know how to explain this feeling I have with music - it's just foreign on another level
  - jessermeyer 5y ago
    It's because they treat music like a math problem -- something that only necessitates reading and writing. That's how you play math, after all, but that's not how music is played.
dr_dshiv 5y ago
Music/math nerds: "The Science of Harmony", from 2019, has some major breakthroughs. Enjoy. https://spj.sciencemag.org/journals/research/2019/2369041/ https://spj.sciencemag.org/journals/research/2019/2369041/
klelatti 5y ago
It may be of interest that even in classical music there are occasional exceptions to the normal GDAE violin tuning. In Mahler’s 4th symphony the violin solo is played on an instrument tuned to AEBF# which gives it a very distinct quality. https://www.violinist.com/blog/stormviolin/20173/21062/ https://www.violinist.com/blog/stormviolin/20173/21062/
bjowen 5y ago
There’s probably a PhD length answer that involves the transition from Renaissance to Baroque music, the plagues of the mid 17th century and the Little Ice Age. Violins have always been identified as being tuned in 5ths to the best of my knowledge, but the features mentioned above helped them out-compete other stringed instruments over a century when the fashion in music favoured those features. Some of them are - the construction favours better projection over the kind of mellow sound of a viol, the rounded bridge allows for better virtuosic playing rather than chords. Music from the late 16th to the time of Monteverdi had a tradition of florid embellishments (diminutions) which favour a more responsive instrument, like cornetto, recorder or violin. The plague of 1630 in Venice is thought to be a major factor in the decline of the cornetto, so the violin had an opportunity to fill that gap then. There’s also a change in the theory of harmony around this time from the Guidonian hand and hexachords to a more complete chromaticism. As someone’s already pointed out, fifths make for a very convenient arrangement of tones and semitones in order to play a scale. I suspect hexachords fit more naturally (!) across strings tuned to fourths - you would have ut-re—mi on the hard, soft and natural hexachords at your fingertips. As music theory moved away from this understanding of harmony, it would have also been more suited to the violin over the similar instruments tuned to fourths and thirds.
musicale 5y ago
We don't. :) https://hubguitar.com/music-theory/why-the-guitar-is-tuned-in-fourths https://hubguitar.com/music-theory/why-the-guitar-is-tuned-i...

81 points by ofou 5y ago

On multiple occasions, I've stumbled upon really old high school physics textbooks and they usually had a chapter on the physics of musical frequencies. It seems to be a weirdly forgotten subject. The octave is a doubling/halving of frequencies, and so is very recognizable. The fifth is the next lowest whole number frequency ratio 3/2, 2/3. So liberal use of fifth interval makes notes often end up low whole number ratio frequencies of each other which sounds harmonious.

BenoitEssiambre 5y ago
Also, if I remember correctly, every note in the major scale is a 3/2,2/3 ratio from a note in the lower or higher octave following the standard order of key signatures as you jump across consecutive octaves, which, I assume, is how they arrived at that standard order.
- klodolph 5y ago
  That’s called “Pythagorean tuning” and it’s possible but it’s not especially common. The system we use is a compromise derived from that system. If you use pure 3:2 ratios you get the so-called “wolf tones”, but we don’t get those any more. The current most common system is equal temperament, which is a type of meantone tuning, where you take the 3:2 ratio and alter it. Equal temperament alters it to 2^(7/12) ≈ 1.4983, and it has the nice property that you get twelve notes and it’s all perfectly symmetrical, it doesn't matter which note you start on. Before equal temperament, there were various other systems. One popular one was quarter-comma meantone, which took four stacked 3:2 ratios (81:16) and squashed them to equal two octaves and a 5:4 ratio (80:16), which results in a value of 5^(1/4) ≈ 1.4953. You see it in some old organs—which may have, say, separate keys for G# and Ab. Then there are various non-meantone tunings, various “just” intonations, etc. In just intonation, a chord like C E G will have a 4:5:6 ratio. You can’t do this with all major triads, though, so you end up with some major triads that sound correct and others that sound out of tune. Our modern system is a compromise where all major triads are equally out of tune, rather than having some better and some worse.
  - lucas_codes 5y ago
    Wolf tones are certain notes which don't sound very well which vary between instruments due to their construction. I think you mean the Pythagorean comma.
    - bonzini 5y ago
      You are talking of the same thing. The typical wolf tone in quarter comma meantone was G#-Eb, which sounded so bad that some instruments added a separate key for Ab.
    - klodolph 5y ago
      Actually, "wolf interval". Pythagorean comma is the difference between enharmonic notes in Pythagorean tuning, e.g., difference between Ab and G#. Not what I was talking about. With any 12-tone system, you might have eleven fifths which are 700-x cents wide, on average... and then a diminished sixth (rather than a fifth), which is necessarily 700+11x cents wide as a result. The "wolf interval" is the diminished sixth (commonly G# to Eb, or C# to Ab). In equal temperament, the diminished sixth is equal to a fifth. You might also call something other than a fifth a wolf interval. Basically, an interval which normally sounds good, except you chose an enharmonic version of it, which sounds bad. "The" wolf interval is the quarter-comma meantone diminished sixth, which is 2^7 / 5^(11/4) = 1.5312. The Pythagorean wolf interval is 2^7 / (3/2)^11 = 1.4798. It's equal to a perfect fifth, minus the Pythagorean comma.
      - lucas_codes 5y ago
        I see. I was confused because string players talk a lot about our wolf notes/wolf tones (especially violists and cellists), but I hadn't come across the term wolf interval.
        klodolph 5y ago
        That's because these tuning systems have been out of favor since about 1700. You might encounter some organists who know about it, but even then, there are only a handful of organs tuned this way in North America. https://en.wikipedia.org/wiki/Meantone_organs_in_North_America https://en.wikipedia.org/wiki/Meantone_organs_in_North_Ameri... It was Werckmeister who popularized and advocated systems which don't have wolf intervals (so-called "well temperament", as in "Bach's well-tempered clavier". The composers who cared about wolf intervals generally predate Bach, to give you an idea. Nowadays (since the 1980s) I can press a button and put my synthesizer or electronic piano in whatever tuning I want, so I can play with quarter-comma meantone temperament without finding one of the organs on the list, and experience it myself.
moehm 5y ago
> The octave is a doubling/halving of frequencies Side note: Except when in praxis you have to accord for non-perfect masses, then you have to stretch the tuning. https://en.wikipedia.org/wiki/Stretched_tuning https://en.wikipedia.org/wiki/Stretched_tuning
metamet 5y ago
> It seems to be a weirdly forgotten subject. I was fortunate to have had the opportunity to take a Physics of Music course in college that counted as the physics requirement. It was really nice to have the mysteries what makes music appealing be an academic focus for the summer. We even had a project at the end where we built an instrument. I made a fairly rudimentary xylophone with some pipes and a hand pipe cutter.
BadassFractal 5y ago
The physics of music comes up a bunch in music production. When working with different waveforms, filter types and specific bass frequencies, you always have to make sure you know what your spectrum looks like, or you'll have phase issues, or you won't hit the desired levels for each instrument and so on. You end up staring at fundamentals and harmonics a lot in order to get the timbres and the mix you have in mind.

kazinator 5y ago

If the instrument is tuned in fifths then, indeed, a diatonic scale pattern that proceeds perpendicularly across the finger board at the same position has four notes on each string. However, these four notes are sometimes spaced differently: different patterns of tones and semitones. When an instruent is tuned in fourths, scales use three notes. There are ways to conform to many of the string-to-string pattern variations by leaving out fingers: a - b c' # index ring pinky e f - g # index middle pinky - c - d It seems that the fifths tuning is quite tyrannical in this regard. Now when it comes to the major scale, that scale is actually a stack of two identical tetrachords, which are a fifth apart. These tetrachords have identical fingerings: # tuning in fifths now: g - a - b c c - d - e f So now if you have your fingers spread out in the right way to play the C-F tetrachord, all you have to do is preserve that when going to the next string. I suspect violinists must be exploiting this sort of relationship; favoring fingerings that show tetrachord symmetries. The Dorian mode (D to D mode of C major) also has two identical, stacked tetrachords, so if we slide one position down that pattern, we are still good: g - a - b c - d c - d - e f - g [ Dorian box ] [ Ionian box ] But, having observed that, these fingering patterns are not easy even in isolation, regardless of whether there is a change in the pattern going to another string.

contravariant 5y ago
To some extent they should have some similar properties since a fourth is basically the inverse of a fifth (modulo an octave). In theory you should also be able to shift a tetrachord on an instrument tuned in fourths you'll just go the other way and be off by an octave.
- klodolph 5y ago
  Being off by an octave is pretty important for melodic lines, which is usually how the violin is used… so while you can translate chords from tuning in fifths to fourths or vice versa, in my experience, you need to relearn to play if you're doing melodies.
- kazinator 5y ago
  If you go the other way, the tetrachords share a note: C D E F, G A B C Swap these: G A B {C} D E F So, tuning by fourths, it lines up, indeed, when we take into account the shared note, e.g. these two L-shaped tetrachord patterns: D - E F D - E F C + A - B C = A - B{C} G G Or: E F E F C - D + B C = B{C}- D G - A G - A

fallingmeat 5y ago

That has to be one of the more civil examples of online discourse I’ve ever seen.

eyelidlessness 5y ago
To be fair, you’re the first commenter. But I applaud you setting a good (sorry) tone!
- a_t48 5y ago
  They may have been talking about the comments in the posted link.
  - eyelidlessness 5y ago
    They were, I was trying to bring some of that good tone here.

jedimastert 5y ago

I've thought about this a fair bit myself, and I think it's a combination of a couple of the reasons stated. First, it's a reasonable range in which every other note can be played within the string before. But more importantly, I think it's just because it's incredibly easy to hear a fifth for tuning. Humans are fairly adept at hearing minute differences between two of the same frequency (i.e. a few hertz or cents off) when two notes are exactly the same, a fifth above share the most harmonics within that reasonable fingering range.

dhosek 5y ago
Although it's typical to tune orchestral strings using harmonics to match the pitches. For the cello, viola and violin, this is done by comparing the 2nd and 3rd harmonics (at the octave and twelfth). For the bass, tuned in fourths, one compares the 3rd and 4th harmonics (at the 12th and second octave).

dhosek 5y ago

A lot of speculation but no real answers. I find the argument about resonances between strings to be spurious since it really only applies to notes played on open strings which are a distinct minority of the notes played (not to mention that there are distinct advantages to avoiding open strings, such as the lack of the ability to apply vibrato on an open string). Ultimately, it's a matter of string length, I think. It's worth noting that in the cello, viola and violin, all four fingers are used independently in fingering notes, but on the double bass (unless you're Joel Quarrington), you finger 1, 2, and 3+4 together since the littlest finger doesn't quite have the strength to press the string down on its own and the ring finger doesn't have the reach of the little finger. But, like all the respondents in the discussion, I don't really know either.

analog31 5y ago
Also, bass is most commonly tuned in fourths unless you're Red Mitchell.
kayson 5y ago
Your point about resonances isn't really true. The other strings can resonate "sympathetically" even when you play a fingered note. For example, vibrato can be achieved on open strings by vibrating on the finger position that corresponds to the note an octave higher than the open string. This is mostly used for the open G string, since as the lowest note, it can't be played anywhere else on the instrument. But its also used on other strings for different colors. Sympathetic resonance is also why some keys sound better on a violin than others. Violinists tend to like the keys with sharps: G D A E
- dhosek 5y ago
  But that's not connected to fifths tuning. It can happen with any tuning (one “trick” with a dual neck 6/12 string electric guitar is to tune the strings on the 12-string neck to each of the diatonic pitches so they resonate with whatever note is played on the 6-string neck, and then of course there are the sympathetic strings on Indian stringed instruments which have a similar function). Curiously, while orchestral strings tend to be more comfortable in the sharp keys, as a double bass player, I was equally at home (if not more so) in the flat keys that tend to be favored by winds.

fathyb 5y ago

I always hum the first two notes of the Dallas opening theme to remember what a fifth is.

antegamisou 5y ago
Make your life simpler and memorize SOAD's Toxicity intro.

eyelidlessness 5y ago

I pretty much only play guitar in nonstandard tunings. Not all of them are oriented around fifths, but they often are at least in part. Repetition of finger positions across strings is one motivation. While some of the linked comments suggest reachability, the standard guitar tuning affords a lot more for many common guitar styles. But nonstandard tunings open up a lot of other styles because you can use partial fifths on open strings more readily than with standard. Two things I think (particularly) guitarists miss: 1. If you’re not doing chord/scale/mode transitions standard tuning is optimized for, you’re putting a lot of cognitive load either on your playing or your practice. 2. Translating guitar tuning to fifths tuning is a lot easier than vice versa if you’re reading tabs or standard sheet music.

tasty_freeze 5y ago
I've often wondered what guitarists are thinking about when they use alternate tunings. I'm sure if you play in one tuning long enough you can conceptualize how all the notes related to chords and keys, etc. But if you frequently change tunings, when you learn a song does it just become a matter of memorizing finger patterns, or do can your brain actually figure out the mapping such that you are thinking in terms of chords and intervals?
- mabub24 5y ago
  A lot of guitarists have no problem with playing open tunings because they can pretty seamlessly transcribe the notes along one string to the whole tuning, like how the 2nd fret of the low D string in open D is an E note so thus a finger across the whole fret is an "E chord". It's once you get into alternate tunings that are outside the "open tuning" norm that it becomes, at least for me, memorizing finger patterns and getting fine coordination locked in. The theory gets thrown out for "what sounds good" and "hmm that sounded about right".
- eyelidlessness 5y ago
  For me, once I’m playing it’s almost entirely muscle memory. Writing or learning, the tuning doesn’t matter at all it’s just developing that muscle memory. The thing nonstandard tunings does for me is it gives me a base state to work from that lets me use patterns and techniques suited to it. Drop D is very good for bass-heavy fifth chord stuff. CGD--- is really good for fingerstyle techniques. Other weirder tunings like some used by Michael Hedges let you rethink whole chord relationships.

pessimizer 5y ago

A lot of people seem to be answering by explaining what 5ths are. That isn't really the question. The question IMO is why they tune to 5ths rather than something else like 4ths, as a guitar often is. 4ths are really just 5ths backwards, but I've always wondered that about violin, too. edit: Ah, the last answer is very clear and good.

elihu 5y ago

I think it's mostly just that 5ths give you a good balance between ergonomics and range, and 5ths are relatively easy to tune by ear. Something interesting about violin tuning: if you tune by ear, you get a 3:2 ratio between the frequency of adjacent strings. If we're in the key of G, we can call the G string 1/1 in just intonation notation. The next string is D, or 3/2. Then we have A at 9/4. That's an octave and a just major second. The interesting thing is that when you get to E, it's 27/8. It's an octave and a major sixth, sort of. If we ignore the octave, it becomes 27/16 which is close to the more commonly used just major 6th which is 5/3. If a violin is tuned to 12-tone equal temperament then the distinction is moot. But in a just tuning, that open E may well be a wrong note depending on what you're playing.

kragen 5y ago
Violins don't have frets.
- mandmandam 5y ago
  They didn't mention frets... Tuning the open strings relative to each other, at least for non perfectly pitched people, results in tuning the strings justly rather than to equal temperament.
  - mrob 5y ago
    The 5th is the least tempered interval in 12 equal temperament (excluding the untempered octave and unison), with less than 2 cents difference from the just interval, and the violin only has 4 strings, so in practice it's probably close enough.
    - kragen 5y ago
      Yes, it would be hard to be less tempered than octave and unison :)
    - elihu 5y ago
      The errors stack, so you get about 6 cents error between the G and E string. That's enough to be a perceptible difference.

spoonjim 5y ago

All this music theory stuff is not the answer. The answer is because the typical male hand can play the fourth with the pinky finger. So tuning in fifths gives the instrument the most range.

srcreigh 5y ago

Just don't use a guitar tuner. Or a piano. Your open C won't be C, it will be 440 * pow(sqrt(2,12),7) which is ~0.75Hz flat. (Does this mean guitars and pianos are out of tune? Yes.)

jancsika 5y ago

What is the music history version of HN where you can get a drive-by answer from one of the (probably many) musicologists who have devoted their lives to studying the history of the violin?

icelandicmoss 5y ago
> What is the music history version of HN In the 11 hours since you commented, the answer appears to be... HN :)

Waterluvian 5y ago

A tangent: I’m learning piano and musical notation finally clicked for me. Musical notation is a high level language. It gives you a program that you can run on any instrument. And it gives you the ability to modify it with relative ease. When playing clarinet in high school I felt that notion is this archaic thing that we’re stuck with because of the morons in the past. But I’m realizing it’s this beautifully refined language, developed over centuries.

PaulDavisThe1st 5y ago
It might be refined and developed over centuries, but it's also the expression of overwhelmingly a single culture. Western musical notation is not useless for denoting the music of other cultures, but it's also not very good and is frequently quite bad. If you think of western musical notation as "the system", you're at risk of remaining ignorant of the beautifully refined languages, developed over centuries, in other cultures. It's also a highly abstract language that seems to frequently, somewhat like Javascript, lead its users to have very little familiarity with what is going on at lower levels.
- kansface 5y ago
  As a classical guitarist, western notation is not particularly good at expressing the music our own culture. The high open E string can be and is regularly played on the other 5 strings too, potentially at the same time! Natural and artificial harmonics, which string to play on and which fingers to use, and all sorts of right hand notation all feel very much bolted on cluttering the actual score. The continual coming and going of voices due to technical limitations means that music has to be experimentally decoded with much backtracking.
interfixus 5y ago
They tried to teach me various instruments when I was a boy. I didn't mind the playing, but Lord Almighty how I hated the notation stuff. It simply made no sense whatever, and was - as you describe it - clearly invented by morons. Only when I ventured into tech-land in my teenage years did any of it finally click: So that's what they mean by 'octave' - a doubling of frequency. Why couldn't they just have said so? Harmonies, chords, what have you - it was all just frequencies and sine waves. Didn't musicians know anything? These many years later, I still don't quite get it, but do of course realise there must be some sort of point to it all. High level language? Yes, I get it. Bach wrote in Python, not C. Time to revisit.

boothby 5y ago

Fiddlers don't always tune in fifths. There's a couple of tunes I play in DDAD, my favorite for its three octaves of awesome resonance. Classical violinists tend to sneer at such tunings, outside of rather esoteric baroque players, I believe. It's a shame, because different tunings can really change the tone of the instrument in interesting ways. As the comments allude, fifths make it easy to play a large range of harmonious music. And, no doubt. A lot of fiddle tunes are written for cross-tuning, and they're tortuous to play at speed in GDAE.

BoxOfRain 5y ago
>It's a shame, because different tunings can really change the tone of the instrument in interesting ways. I find this with guitar, but guitarists on the whole aren't snobbish about alternate tunings. DADGAD is my favourite guitar tuning for similar reasons.
- Zuider 5y ago
  DADEAE is another tuning that is good for resonant melodic playing without straining the neck. As you can see you have the D,A,E strings tuned in 5ths like a banjo or fiddle.
- aidenn0 5y ago
  According to my dad, guitarists were more snobbish about tunings in the past. By the late 60s it was cool to experiment with tunings (at least for rock guitarists), but he says there definitely was a point where you were looked down on, particularly for open tunings

xscott 5y ago

I have an electric guitar with 5 strings tuned fifths (C G D A E), and a tenor guitar with 4 strings in fifths (currently D A E B). I've been mucking around with both for a few years. Conceptually, they're nice. You get a great set of movable bar shapes, giving you spread chords with the first, fifth, and easy access to a 9th (2nd), minor third, major third, or 4th (11th). So it's super easy to get the common chord "colors" (sus2, maj, min, sus4). Similarly, you can add the next string to get a maj 6th, min 7th, and maj 7th with a 4 string movable chord. However, it all sounds a little thin if you're playing in an acoustic style. Bar chords in standard guitar tuning sound better to my ear, so I'm not sure I believe any of the responses talking about better overtones. Maybe that's different with a violin and the music that's played, dunno. If it was overtones, why wouldn't the "double bass" violins use it? Even with the shorter length of the tenor guitar (23 inches), playing diatonic scales is unpleasant. My pinky finger feels week when it has to stretch that distance for 4 notes per string. And I really don't like the stretched out (three notes per string) pentatonics. Because of this, I think the answer to the link above really is about the scale size and ergonomics. I'll probably leave the tenor guitar alone (it's a novelty for me anyways), but I'm thinking I'll switch to CGDGBE or go back to Drop D on the electric guitar. All fifths just isn't that enjoyable for me.

xscott 5y ago
There was a King Crimson link on HN the other day. I should mention Robert Fripp's "New Standard Tuning" is all fifths up the point where the highest string would be way too thin or way too tight (CGDAEG). Even as is, that high G seems unhappy, but he makes it work :-) I think someone used to sell string sets for that, but I can't find it on Amazon atm.

ithinkso 5y ago

This article, and the comments, made me realize why some friends that I tried to teach coding just couldn't grasp it yet for me it seems kinda obvious. I don't understand music one bit, none, zero. I can read descriptions and even understand what different notes are, and the symbols for them but it's a very dry knowledge. Nothing I do make it 'click' for me and it feels so disjointed and foreign that I just got overwhelmed and I know it will never make any sense to me I can read a math paper, and even though I might not understand it at first, at least I know that if I just take a book that explains the terminology then I'll be able to grasp it. With music it's a complete opposite - nothing no matter what I do will make it understandable for me Edit: To give you and example, I have been studying the famous clip [0] about the rhythmic displacement. I have watched it thousands times, read tons of analysis etc. and I know exactly what was done and when. I just don't hear it being done - I know people aren't lying to me but I just cannot hear whatever happened at that famous point and what changed after [0] https://www.youtube.com/watch?v=UinRq_29jPk https://www.youtube.com/watch?v=UinRq_29jPk

patrickthebold 5y ago
I actually don't like this attitude. In every case it's a mostly a matter of interest and effort. If you have those you can learn something.
- ithinkso 5y ago
  I agree, that's what I told my friends, to just keep learning because if I could why wouldn't you. The point of my post is that I have realized how unhelpful that comment is because if someone told me that about music today... all I see is an insurmountable mountain ahead even though I've read quite a bit about it already Mind you it's different with for example fluid dynamics or functional programming or even chemistry or biology, I have zero knowledge of it (nor interest) and all I know is that it's hard... but I know that if I spend years learning it I will succeed. I don't know how to explain this feeling I have with music - it's just foreign on another level
- jessermeyer 5y ago
  It's because they treat music like a math problem -- something that only necessitates reading and writing. That's how you play math, after all, but that's not how music is played.

dr_dshiv 5y ago

Music/math nerds: "The Science of Harmony", from 2019, has some major breakthroughs. Enjoy. https://spj.sciencemag.org/journals/research/2019/2369041/ https://spj.sciencemag.org/journals/research/2019/2369041/

klelatti 5y ago

It may be of interest that even in classical music there are occasional exceptions to the normal GDAE violin tuning. In Mahler’s 4th symphony the violin solo is played on an instrument tuned to AEBF# which gives it a very distinct quality. https://www.violinist.com/blog/stormviolin/20173/21062/ https://www.violinist.com/blog/stormviolin/20173/21062/

bjowen 5y ago

There’s probably a PhD length answer that involves the transition from Renaissance to Baroque music, the plagues of the mid 17th century and the Little Ice Age. Violins have always been identified as being tuned in 5ths to the best of my knowledge, but the features mentioned above helped them out-compete other stringed instruments over a century when the fashion in music favoured those features. Some of them are - the construction favours better projection over the kind of mellow sound of a viol, the rounded bridge allows for better virtuosic playing rather than chords. Music from the late 16th to the time of Monteverdi had a tradition of florid embellishments (diminutions) which favour a more responsive instrument, like cornetto, recorder or violin. The plague of 1630 in Venice is thought to be a major factor in the decline of the cornetto, so the violin had an opportunity to fill that gap then. There’s also a change in the theory of harmony around this time from the Guidonian hand and hexachords to a more complete chromaticism. As someone’s already pointed out, fifths make for a very convenient arrangement of tones and semitones in order to play a scale. I suspect hexachords fit more naturally (!) across strings tuned to fourths - you would have ut-re—mi on the hard, soft and natural hexachords at your fingertips. As music theory moved away from this understanding of harmony, it would have also been more suited to the violin over the similar instruments tuned to fourths and thirds.

musicale 5y ago

We don't. :) https://hubguitar.com/music-theory/why-the-guitar-is-tuned-in-fourths https://hubguitar.com/music-theory/why-the-guitar-is-tuned-i...