Related: in a grand piano, the hammers strike the string at a node of the 5th harmonic (thus avoiding excitation), which reduces the beating from thirds that do not match the harmonic series (due to equal temperament)
It seems that no one objected to the (slightly off-topic) claim that "the vast majority of western music uses equal temperament". As I understand it, equal temperament is just a way of tuning keyboard instruments so that they sound the same in every key; a string quartet or a choir doesn't use equal temperament. (And some people claim that J. S. Bach, who insisted on tuning his own keyboards, didn't use equal temperament, either.)
It's a way of tuning so that all semitone intervals are an equal [2^(1/12)] frequency ratio. The vast majority of western music, today, uses equal temperament. This includes most recorded performances of JS Bach's pieces.
It wasn't particularly common in Bach's time, though, and the math to do it 'properly' wasn't even known outside of China.
Today, though, it's everywhere. Modern Woodwinds are equal temperament due to the placement of the holes. To match that, the entire orchestra is in equal temperament, regardless of the requirements of the instrument.
Guitars are in equal temperament due to the placement of their frets -- to do otherwise would require frets that bend between strings.
In isolation, people will [attempt to] sing in equal temperament, because that's what 'all' accompaniment is like.
Barbershop quartets, acapella groups, and to some extent string quartets have a tendency to shift harmony notes toward simple fractions of the root note of the chord, on the fly, even as the overall melody runs in equal temperament. This isn't a different 'tuning', per se, but it is a thing.
Barbershop is notable for using the harmonic seventh, which doesn't exist in the standard 12 note scale. Its frequency is 7/4 times the frequency of the root, so a harmonic 7th above C is somewhere between A and B-flat.
From that page: the harmonic seventh is '"sweeter in quality" than an "ordinary" minor seventh'. Well, as an ex- trumpet and trombone player,[0] what it sounds is out-of tune, and is generally avoided, although the out-of-tuneness can be compensated for by adjusting the trombone slide. It doesn't sound particularly bluesy to my ear. Sure, sliding around the flat 7 and flat 5 areas sound bluesy, but just that note as a melody note sounds very flat.
[0] Playing brass instruments is a matter of using your lip to find the various harmonics, and using the valves to flatten them by various amounts.
I don't think there's anything inherently wrong with natural harmonics in a melody. The opening and ending of Britten's Serenade for Tenor, Horn and Strings, Op. 31 uses them, and it sound good to me. Because it's notated as equal temperament, the score is ambiguous, and there are two ways of playing it:
I suspect that brass players are unusually sensitive to deviations from equal temperament because they have to work so hard to overcome them if they want to play with other instruments (including other brass instruments of different sizes).
>I don't think there's anything inherently wrong with natural harmonics in a melody.
Not sure what you mean by 'natural harmonics', i.e. which ones you mean. I was just talking about how the Ab harmonic (on trombone/trumpet) sounds very flat. It's avoided because it sounds out of tune. I'm not sure what something being 'inherently wrong' would even mean. In music, if it sounds good, it is good.
>I suspect that brass players are unusually sensitive to deviations from equal temperament because they have to work so hard to overcome them if they want to play with other instruments
I don't know what you're referring to there - I've never had or even heard of that problem. And I hadn't noticed or heard, and don't believe, that 'brass players are unusually sensitive to deviations from equal temperament'.
By "natural harmonics" I mean notes at the resonant frequencies of the instrument. Several of them are obviously different from equal temperament, which means they will sound bad when played with other instruments. When brass players talk about "good intonation", they often mean adjusting these notes into equal temperament with valves/slides/embouchure/hand stopping. Because brass players actually have to pay attention to the tuning, I wouldn't be surprised if they were more sensitive to it than people who played fixed pitch instruments like piano. The Britten piece is notable for the parts where the horn player is instructed not to adjust the pitches to equal temperament, and because it's a solo it still sounds good.
True temperament is meant to address guitar intonation problems regardless of the temperament. It's basically impossible to get a guitar intoned correctly over the fret board using straight frets because of the different string gauges. So it's not specifically meant for just intonation.
I would amend this to say that modern woodwinds are supposed to be equal temperament, but this is more aspirational than reality. Some instruments better than others, and in general they are all vastly improved over their 18th- and 19th-century ancestors, but a lot of the burden of making the pitches "equal" is left to the player.
(Source: I play bassoon, probably the worst of the lot.)
> It's a way of tuning so that all semitone intervals are an equal [2^(1/12)] frequency ratio
Or so I thought until I programmed an AVR. The F, I think it was, is terribly out of tune. I haven't found the wikipedia article again, that had the correct Frequencies noted in Hz, with F decidedly not in the scheme that you allege.
Correct! Most live performed music does not use equal-temperament. Many keyboard instruments do not use equal-temperament either, especially organs/clavichords etc.
One of the interesting things about organ tunings is that because you can choose which pipes to perform with, different pipe-sets may be tuned according to slightly different needs on the same instrument. So, you might have one set of pipes that is well suited for equal-tempered, highly chromatic pieces. Another set of pipes may be intended for use with choirs, and another set still tuned to be ideal in an orchestral setting. However, the specific tuning of the instrument is usually maintained over the course of its life, which is part of why organs can be so distinct and characteristically themselves. Most organs are tuned by ear with tooling assists, and generally land in a Wohltemperiert that compromises between equal-temperament and pure 5ths.
Pure 5ths are are especially important on an organ, because many of the pipes produce highly sinusoidal fundamentals, making un-pure 5ths especially obvious and "out of tune" sounding. The complexity and bloom of the harmonics in a piano helps to mask these effects.
"a string quartet or a choir doesn't use equal temperament."
But they are still fairly deeply influenced by the fact all our ears are generally tuned to equal temperament, so I think it's still fair to say they use equal temperament + some local deviations, rather than some other tuning. Music that is "truly" not based on equal temperament sounds "wrong" to most people nowadays, or at least requires a significant adjustment period.
While they may be influenced by it, the serious choir I was in was adept at producing perfect harmonies rather than the slightly-off equal temperament. To say that 2 solid players can't produce a perfect fourth or fifth together due to this equal temp influence, that's incorrect.
The main issue comes when you have other instruments playing with you that are equal tempered. You will sound off if you sing or play with them, and you will sound off if you don't. That's one reason a capella music can sound really great, it's not slightly off but rather perfect intervals.
"To say that 2 solid players can't produce a perfect fourth or fifth together due to this equal temp influence, that's incorrect."
No, of course not. What I'm saying is the underlying tune is very likely still equal temperament. Probably one end of your perfect fifth is pinned to the equal temperament frequency, and the other end is tweaking just a bit to be the perfect fifth. That's still at least "influenced heavily" by equal temperament, if not simply "equal temperament with a couple tweaks". I've heard plenty of vocal/string works like this. I have not heard a lot of vocal works on a Pythagorean tuning or something, or something more exotic like shifting the base key of the tuning on the fly.
By contrast, historically, music was simply in another tuning. All of it, including vocal. It sounds strange to modern ears in a way that "equal temperament plus slight tweaks for a perfect 4th/5th" doesn't sound weird to anybody.
Close but not right. String players have to learn to place their fingers in locations that are consistent with equal temperament tuning, or they will sound of out key. It would be fine if a string quartet played on its own, but even then, most would play in ET not JT because otherwise everybody would complain it sounds funny.
I have a lovely blues album by a guy who plays a keyboard in JT. most people who hear it ask me to turn if off "because it sounds wrong" and that includes musicians! it sounds fine to me because I grew up listening to ragas and developed an appreciation for alternate tunings.
it's the tuning system that was most commonly used before equal temperament in the west. It's attractive due to the geometry underlying the pitch ratios, but it's hard to work with if you have a bunch of different instruments that youy have to keep in tune with each other.
A choir doesn't need equal temperament because they have a continuous tuning versus a piano. A string quartet (and a trombone!) can do the same thing, they can adjust the tuning based on the instrument, the octave, and what everyone else is playing because they all have continuous tuning versus discrete tuning (I am a trombone player, so my individual instrument has its own quirks at different partials, but I can use the tuning slide to adjust and be in tune exactly).
For the record: I am a professional bassoon player. I have been working professionally for the last ten years in various orchestras in Sweden and germany.i currently hold the position as first bassoon in Norrköpings Symphony Orchestra.
Yes and no. It is true that we in orchestras "listen" and intonate in tune (with low major thirds, slightly high fifths and so on), but playing strictly in tune quickly becomes impractical.
If you play many harmonic sequences or modulations "in tune" and then return to you original key through a different path you will often end up nowhere near where you started. We all have the equal temperament as our base and even though we play one or two chords perfectly in tune we adapt tuning not to end up far from the standard equal temperament.
Then we have the problem of melodic intonation. If you play a melody with the intonation of the harmonic series you will sound _very_ low whenever you play a third.
In his lecture series on Bach, Robert Greenberg discusses temperament quite a bit and I found it fascinating. He relays all these descriptions that Bach had of different keys that are basically lost on us now as we cannot refer to the specific tunings he utilized (and thus cannot hear a lot of the music he wrote as he wrote it).
He also makes the claim that equal temperament became wide spread _because_ of the piano specifically - not just to sound the same in every key, but to sound the same on each piano. And so equal temperament would not have saturated European music until the early 19th century when the piano became ubiquitous. The reason being (as mentioned) that many other instruments can be tuned by the musician on the spot, whereas a piano requires some standardization.
The question of what temperament, exactly, Bach meant by "Well-Tempered" is a hotly contested question, with the majority assuming it is one of the Werkmeister variants, and a vocal minority insisting it was Bach's own custom temperament, with a couple of Bach scholars even hypothesizing that Bach encoded his temperament scheme into the decorative squiggle on the cover page of the original printing of the book.
But everyone agrees it wasn't equal temperament. The point of the book was to showcase the unique character of each key, uniqueness that is lost in equal temperament.
Have people listening recommendations to hear music without equal temperament? Having played the violin for over 30 years - and sung a bit of barbershop - this is the first time I've come across it, and I'd like to hear the difference to recognize it.
Byzantine chant divides an octave into 72 comas. The intervals for the diatonic first mode are 10-8-12-12-10-8-12, although this changes when descending. For more information, see http://www.kelfar.net/orthodoxiaradio/Diatonic.html
Related: If you have a piano you can check that by using my online tuner that shows you the power of all the 88 notes a piano has. This means you can look at the fundamental frequency and its harmonics too.
The piano keyboard (usually) has 7 octaves. Frequency doubles every octave, therefore if the lowest key is C, then the highest C all the way on the right is
C*2^7
You can also follow fifths and arrive at the same note after 12 fifths. A fifth of a note is 3/2 of its frequency. Thus, starting from the same C, you arrive at
C*(3/2)^12
But...
(3/2)^12 = 129.7, while 2^7 = 128.
And that's (roughly) the problem that Bach's temperament addresses by ever so slightly adjusting the frequency so that in any key it sounds "right".
I don't follow this. The answer contained in the link seems more relevant as it has to do with the fact that the harmonics of real metal strings are not ideal and so the detuning at both ends compensates for this.
It seems like you are talking about what would happen if you tuned a piano using something other than equal temperament. Why is the iterated fifths relevant at all in this case?
Generally, Pythagorean tuning uses perfect fifths, as it's the easiest to hear and to match using harmonics. That example is just showing that if you using consecutive intervals (such as fifths) then other intervals (such as octaves and thirds) won't line up.
This doesn't address the question at all. You're talking about how it's tuned "out of tune" within an octave, not at the extremes of a piano keyboard. In any temperament, C1 should have exactly half the frequency of C2, but that's apparently not how pianos are actually tuned.
Even electric (sampled) pianos are tuned that way. The difference from the well-tempered tuning is given by the Railsback Curve (https://en.wikipedia.org/wiki/Piano_acoustics#The_Railsback_...). For the highest and lowest notes it is as much as 30 cent. It comes from the inharmonicity of the strings.
Aside from sampled instruments, there's the entire world of synthesis, including the endless number of virtual instruments that have become staples in modern popular and electronic music. As to your question: they are not generally stretched tuned. Many instruments support alternate temperaments through the use of .tun files, but temperament (with a few outlying exceptions) only speaks to the interval deltas of the notes within an octave, not of the entire range of the instrument.
However, it doesn't matter. The reason for the stretch-tuning of a piano is due to the physical limitations of a vibrating string being unable to produce the same harmonic intervals at the extremities of the instrument's range, and does not apply to electronic instruments which, particularly in additive synthesis, can generate whatever harmonics are desired, including harmonic configurations that no physical instrument would be able to produce.
I ran across this Red Bull video the other day; right at the beginning Mark Verbos shows how his new synth design has (if I understand correctly) hardware sliders to allow the user to adjust individual harmonics ... and circuitry (Hadn't seen the like of that before.)
Stretch tuning would make sense for small niches of synthesized instruments that lie beyond the typical musician or sound designer's skills and tastes:
- physical models that strive for realism enough to reproduce a defect
- deliberately inharmonic additive synthesis
- carefully designed FM/PM synthesis
Conversely, popular synthesis methods, particularly subtractive synthesis, are usually perfectly harmonic and do not need stretch tuning.
A lot of keyboards are based on recorded samples of acoustic pianos – they might use a Steinway or something as the source of the samples, so the keyboard and piano would sound ostensibly similar and good together.
The best guitar players also do a form of stretch tuning, depending upon the range of the piece that they are playing. Just as on a piano, the high end of the instrument will sound flat if they didn't. This is a limitation of all stringed instruments.
Inharmonicity increases with string thickness and decreases with string length (or effective string length when you're fretting it). It's most noticeable if you play high pitched notes on the E2 string. However, slight inharmonicity is just part of the sound and not a flaw:
Indeed - I've been playing piano and guitar (badly) on and off since childhood. I'm surprised I never wondered about this. It also explains why tuning guitars purely by harmonics doesn't "work" - you end up with the bottom and top E strings sounding out of tune.
Guitars are even worse, due to fretting. The act of fretting a note stretches the string, which raises its pitch, and the amount varies by string and by fret. So guitars are never truly in tune. Not only are harmonics not sufficient, but neither are electronic tuners! A guitar that is "in tune" according to an electronic tuner is always sharp in practice.
And it gets worse from there! The tuning of a guitar string varies as it decays. This is true of piano strings too, but not to the same degree. A freshly struck string is vibrating more widely than a gentle or decaying note, so it's stretching itself sharp. You can see this on a fast digital tuner - the note will go sharp at first, and then settle to a slightly lower pitch. So when you tune to an electronic tuner, are you tuning the initial pitch, or the decayed pitch? This variation might be 20 cents or more.
Because of this, how to tune a guitar well is very much a matter of taste, context, and experience. I use harmonics and electronic tuners to get myself in the ballpark, then start fine-tuning based on the guitar itself (each one has its own quirks), and the material I'm planning to play. On acoustic guitar, I tend to focus on getting the B string in tune with the D and A strings first, by the quality of octaves for open C and D chords (which also gets the A and D in tune with each other). Then I focus on getting the low E in tune with an octave E on the D string. Then get the high E in tune in unison with E on the B string. Finally, get the G string in tune with G on the low E, an octave down. This means my G string is usually a bit flat relative to the D and B strings, but that's okay - it's in tune for G chords, and being a little flat is good for E major and D chords. I might adjust a little if I'm playing in C/Am.
James Taylor has an excellent YouTube video about tuning guitars consistently with electronic tuners. It's very much to his taste and the specific guitars he uses, but his principles are sound. And if you try it on an acoustic with good intonation, you'll immediately hear that "James Taylor" sound.
You're welcome! The most important takeaway, I think, is that to a certain degree, "in tune" is a matter of opinion (for fretted instruments, anyway). You can significantly change the tonal quality of the instrument while remaining "in tune". That's the neat thing about the James Taylor video... follow his method, and your guitar suddenly gets that James Taylor sound, very rich and resonant.
Try experimenting with modal tunings like DADGAD, too. It's much easier to get them "in tune", and you hear this beautiful resonance that guitars can make.
I brought this up on Facebook, and a friend who is an excellent player responded with his own tuning method. He tunes the A string to a reference (tuner, piano), and then tunes every other string to a fretted A note that is in tune with the open A string. This is probably more "in tune" than the highly resonant approach that I use.
Music is based off of ratios in between each other. An Octave is 1:2 (For example, A1 = 440 Hz, A2 - 880Hz, etc.). But the fundamental ratio for modern music is the Fifth. Thr first eight fifths of F is C, G, D, A, E, B. Re arranged, that is also C, D, E, F, G, A, B (a Major Scale). If you go to 12 Fifths from F, you get the chromatic scale. This will work for any starting note, and is why we have 12 Major Scales.
However, we typically say a fifth has a 1:1.5 Ratio. This is an approximation due to the overtone series. When a string vibrates (or a horn vibrates, take your instrument), it vibrates at a fundamental frequency and several harmonics above it as well. I cannot find it exactly, but I believe it is closer to 1:1.48 to make overtones work. This means the math does not work out between a Fifth and an octave. To solve this issue, we use the approximation on the piano so that the fifth and Octave line up exactly for the ratios (we tune fifths slightly sharp and octaves slightly flat).
Now remember how I said music is all based off of ratios? Due lot this, we need a common fundamental frequency that everybody can agree to tune to. The musical world as decided on the A=440 Hz (which is close to the middle of the piano). Due to the fact that the reference frequency is in the middle of the piano, as you get out further, the approximation rears its ugly head.
This is why pianos are tuned "out of tune" at extremes.
> However, we typically say a fifth has a 1:1.5 Ratio. This is an approximation due to the overtone series. When a string vibrates (or a horn vibrates, take your instrument), it vibrates at a fundamental frequency and several harmonics above it as well. I cannot find it exactly, but I believe it is closer to 1:1.48 to make overtones work.
I think you got this backwards. A perfect fifth is precisely 1:1.5 — or, put differently, the third harmonic is 3x times the root/first harmonic's frequency, and, transposed down an octave, that gives us the fifth.
Modern instruments use equal temperament tuning, where the frequency ratio between any two consecutive steps is the same, and you go from any note to a note at double the frequency in 12 equal steps — this means each step has to correspond to multiplying the base frequency by the 12th root of 2. In this tuning, the fifth corresponds to 7 steps, which corresponds to a multiplier of 2^(7/12) = 1.498.
I very well could have, I just got annoyed at Edomund's explanation. It isn't a limitation of the piano, it's the physics of how music is (Cunningham's Law at play).
I can look it up when I get home and dig up my music theory books.
You are right that there is an unavoidable mathematical compromise. A pure 5th would be a ratio of 3:2 or 1.5:1. But in equal temperament, semitones are 2^(1/12), so a 5th is 2^(7/12), which is about 1.4983:1, so just slightly off from 1.5:1. This issue applies to all instruments.
BUT, while all of that is true, it isn't the reason a piano is tuned how it is. It's because the strings don't behave like the mathematical ideal of a string. The harmonics aren't at exact integer multiples of the fundamental frequency.
My source for this is asking a piano technician a bunch of annoying questions while he tuned my piano. He helpfully responded to by explaining the reasons behind stretch tuning, and then showing me the process of using his Sanderson Accu-Tuner to measure the inharmonicity of my piano and compute an individualized stretch tuning for it.
Firstly, you got the ratio thing backwards. A perfect fifth is 1:1.5. an equally tempered fifth is 1:1498...
Secondly the explanation is good. A string under tension (such as in a grand piano) has all kinds of weird things going on with the harmonics (mostly, they all go sharp). When tuning pianos this is called stretching.
Edit: apparently I got the tension thing wrong. Lower tension and higher stiffness are what makes the lowest octave of the grand piano exhibit inharmonicity. To quote the wikipedia article on inharmonicity:"The less elastic the strings are (that is, the shorter, thicker, smaller tension or stiffer they are), the more inharmonicity they exhibit "
Hmm, not really. If we look at the "Railsback curve" someone linked below ( https://en.wikipedia.org/wiki/Piano_acoustics#The_Railsback_... ), it looks clear that around 440 Hz an octave is realized by a nearly perfect 1:2 ratio. Of course, it means the fifth (C-G, etc.) is never perfect 2:3, but that's an orthogonal issue.
If piano strings were perfect, then we could easily extend the pattern to lower/higher pitches by halving/doubling frequency. The fact that it doesn't work at the extreme is due to the physical reality, as the stackoverflow answer correctly pointed out.
Both explanations are correct factors in the overall messy compromise that is tuning. The explanation they gave was attempting to describe string inharmonicity, which is worse for metal strings than more flexible ones and only applies when they are plucked or struck. So violins do not have this problem, because the bowing constantly corrects the frequencies as it slips and grabs. I love your description of the Pythagorean comma, though. The difficult compromise of mathematical purity is what led me to an obsession with pedal steel guitar, since it's one of the only things that can play full chords in all keys with perfect intonation.
Edit: apologies for adding yet another mention of string inharmonicity before reading the rest of the replies.
As a fun fact too, the overtone series is also why you have different size pianos. The longer string gives you a different overtone, and if i recall right, the overtone is "more in tune" overtone series (if there are any piano folks in here and I got that wrong, please forgive me, I am a mere trombone player).
Inharmonicity is a trade off between the ratio of string length to fundamental and stiffness (which is related to thickness.)
You could make a piano with really, really long bass strings, and the overtones would be more linear. There would also be a lot more fundamental, which is almost non-existent on the lowest octave or so of a grand piano.
Native Instruments actually sells a sample pack called The Giant which is taken from an experimental long-string piano.
I don't think it sounds all that good, because there's something just right about the colour of the "imperfect" bass strings on a fine grand. (That could just be acculturation, but I'm not completely convinced that's the case.)
Contrariwise, uprights tend to sound boxy and constrained in the bass because the strings are shorter than on a full-sized grand, and the overtones are even louder and even less linear.
You can't do much at the top end, because nicely linear strings - like the ones on a steel guitar - would have to be very thin and they'd be too fragile to survive piano hammers.
The closest approximation would be a hammer dulcimer, which has a much sweeter and more open top end than the percussive plink of a piano, but doesn't go quite as high.
Deeper tones have a longer wavelength. If the resonator is too small compared to the wavelength, it can't transmit the sound efficiently. I guess it has something to do with the sound wave wrapping into itself and causing distortions or something like that.
Deep tones sound better on a concert grand piano than on a small upright where they tend to be somewhat blurry.
Compare with big subwoofers for low-frequency sound reproduction.
Caveat emptor: I'm not really in the business, this is just off the top of my head. :)
