I am hoping someone can answer the following genuine question(s) from a mystified layman: Is our perception of sound really so sensitive to precise ratios ? Is a frequency ratio of 1.5000 somehow inherently more pleasing than 1.5001, or is it more the case that appropriately trained or gifted individuals can detect the difference ?
I have experience of performers, when not accompanied by a piano, choose to perform intervals closer to 1.50000 than 1.50001, so it seems they prefer them.
However, the answer is more complicated than that. Quoting directly from my acoustics textbook:
"Research on the intervals played or sung by skilled musicians shows substantial variability in the intonation from performance to performance. These variations are frequently larger than the differences between the various tunings. Studies of large numbers of performances have shown that deviations from equal temperament are usually in the direction of Pythagorean intervals (Ward 1970).
"Several investigators have found that performers tend to stretch their intervals (including octaves), even when unaccompanied by a piano. Many choral conductors prefer the third in a chord slightly raised, especially in sustained chords or cadences, to avoid any suggestion of 'flatting' the chord, a particular nemesis of choirs. How much of the apparent preference for sharpened intervals is due to constant exposure to pianos with stretched turning is difficult to determine."
The Science of Sound, Third Edition, by Rossing, Moore, and Wheeler, p. 187.
(Ward 1970) is "Musical Perception" by W.D. Ward, in Foundations of Modern Auditory Theory, Ed. J. Tobias.
> Several investigators have found that performers tend to stretch their intervals
Johan Sundberg has excellent examples of this "stretched" tuning in some of his talks. He claims that musicians stretch intervals intentionally in order to add excitement to particular passages. The link below contrasts equally tempered and "stretched" versions of the same musical phrases.
Yes, because frequencies that don't fit together precisely produce 'beats', where the wavelengths align, and sound quieter when they don't. You're adding two waves, so an in tune one looks regular:
There's a degree to which it doesn't matter (either it's below what we're sensitive to or it's drowned out by overtones) but the beating sound is very audible.
Right. Sort of the audio equivalent of Moiré patterns. Two overlaid frequencies that differ very slightly are actually more noticeable than ones that differ by a greater amount because the size of the least common multiple—which determines how the pattern perceptually repeats—is larger in the former.
Also there is sympathetic vibrations. So for example if the rest of the orchestra is playing a note that is an open string on your instrument, that string will vibrate even though you are not touching it.
You also get a similar impact from overtones.
So what happens is at a certain point is that you might not perceive the pitch difference, but you can detect a change in the timbre of sound (quality, tone, resonance).
It does depend on the style of music, but it's easy to notice the clash between piano and violin tuning in, say, a slow movement of a Beethoven or Mozart piano concerto. In particular, it's the sharpness of the major third on the piano that can makes a noticeable contrast with what the string players are doing. (string players know all about this, pianists have to put up with it):
Perfect (absolute) pitch can sometimes be a blessing, sometimes a curse. I don't have it, but my partner did, and she struggled to enjoy some concerts that I enjoyed, simply because the notes from the various players weren't aligned well enough...
> she struggled to enjoy some concerts that I enjoyed, simply because the notes from the various players weren't aligned well enough...
Hmm, it depends from listener to listener. Obviously I can't comment for your partner, but I too have perfect pitch. Bad alignment of players isn't really a problem for me unless the players are dramatically out of tune. I'm not the greatest at determining tunings by ear either, as this kind of thing varies.
...I too have perfect pitch... I'm not the greatest at determining tunings by ear either...
Sorry I'm relatively ignorant when it comes to these topics; what does this mean? What are you able to do, upon listening to a particular musical performance?
Perfect pitch means knowing absolute versus relative frequency.
Most people have no (or very little) sense of absolute pitch - if you get asked to sing / hum / whistle a song, you'll start on any random old note. You'll get the relative pitches right - the change in pitch between notes - but not the absolute.
(To be pedantic, you'll get the ratio of the frequencies approximately right)
Someone with perfect pitch can identify "that note is the A# below middle C", for instance. But not necessarily that accurately. " not the greatest at determining tunings by ear " just means that he's not very good at it. So he may identify that A# as a C or something.
Reply to sibling comment -- perfect pitch can probably be developed with a lot of training (though I imagine it's harder if you're not a kid anymore... for kids, sure -- there are tonal languages, after all!).
But it's honestly not that valuable a skill, even for professional musicians. Having really keen relative pitch (and avoiding slipping pitch if you're singing, for example) would be a much better focus to take.
What is the relationship between perfect pitch and tonal languages? I've dabbled in Mandarin and there relative pitch seems to be good enough. But I realize that there are tonal languages other than Mandarin, and I know nothing about them...
It's believed to be inborn or somehow acquired spontaneously at a very early age. There are products that claim to teach it, but there's no scientific evidence. Years ago I tried a few, and my impression was that it's mostly snake oil, and even if some rudimentary progress is possible, it's definitely not worth the effort, better expended on improving relative pitch and other aspects of musicianship. Wikipedia agrees:
From http://en.wikipedia.org/wiki/Absolute_pitch
"[...] there are no reported cases of an adult obtaining absolute pitch ability through musical training; adults who possess relative pitch, but who do not already have absolute pitch, can learn "pseudo-absolute pitch", and become able to identify notes in a way that superficially resembles absolute pitch. Moreover, training pseudo-absolute pitch requires considerable motivation, time, and effort, and learning is not retained without constant practice and reinforcement."
Studies have shown that there is a higher prevalence of people with perfect pitch in countries where tonal languages--languages in which the same series of sounds made with distinct voice pitches can denote entirely different words.
However, the percentage of people with perfect pitch remains tiny, which is not what I would expect if it was due to training alone. It seems like it might be helpful enough for musicians that we'd have turned it into a method by now if training alone reliably produced good results.
I'm fairly certain that it is one of those things you either can hear, or cannot hear - it requires a certain sensitivity in the ear, and not all people have that. Not all people that can hear the difference have trained it to a degree where they're conscious about it, so for some people it's possible.
I made two example audio files in Matlab for a previous discussion. Both are of a major chord (root, major third and perfect fifth) made up of three sine waves. One is in equal temperament (frequencies f, f * 2^(4/12) and f * 2^(7/12)) and the other is in "just intonation" (f, 5f/4 and 3f/2).
You heard it right, the second one was the equal temperament (i.e. the wobbly one). I had mixed them up by accident, but now they are labelled correctly.
Some people are more sensitive than others. As an experienced orchestral violinist with perfect pitch, I can hear the difference between 440 and 440.2 Hz; and I avoid attending concerts from orchestras which tune to A=442 because it sounds horribly wrong to me. (My preferred A is 438 Hz, in part because that works best with my violin's natural resonances.)
I used to take cello lessons, and my teacher had perfect pitch as well. I used to tune beforehand to A=440hz, but he kept having me tune higher. I was puzzled as to why until I asked and he said he tuned to A=445.
One of the lessons I learned was that it's better to be relatively in tune than absolutely in tune (i.e. tune to the soloist no matter what), and it's moderately better to be sharp than flat if you're going to be out of tune.
Not necessarily. Some soloists (particularly violinists) deliberately tune slightly sharper than the orchestra they're playing with, because the difference in tone allows them to cut through more easily.
You may be aware that baroque instruments tune to A=415 -- even worse! :) But it's a half-step lower than A=440 so maybe it wouldn't sound wrong to you.
What's your accuracy on a blind recognition test with a single 440 or 440.2 Hz sine wave, with no "warming up" (let's say you haven't performed the test or listened to any reference frequencies for an hour)?
Right, my wife has "piano-only" perfect pitch -- she'll recognize the key played on a piano, but if you sing a pitch (or play it on any other instrument) her accuracy goes down -- it's just based on years and years of playing piano. She can kind of hack singing a pitch (for example) by imagining playing the note on a piano, then trying to sing the pitch in her head.
I doubt she'd notice if a piano was tuned to 442 vs. 440, though... which makes some sense, really; a violinist really benefits from a sharp ear for pitch, for accurate finger placement (no frets or anything like that on a violin neck) -- but a pianist has a fixed set of keys to choose from.
That's interesting. My closest hack to fake perfect pitch is to estimate the range of a note my humming it or singing it. It works because my vocal range is more or less absolute, and I can easily feel how difficult the pitch is to sing by trying an octave above or below it.
You're fighting a lot of imperfections here. Even a well-tuned piano isn't going to be in tune lowest C to highest C, steel strings just don't allow for perfection [1]. Guitar gets even worse, as you have to deform the string to fret it, leading to strange bridges and nuts like Earvana to compensate. If you want perfect notes, stick to synths.
At least in their piano patches, synths are programmed to replicate, or at least imitate, this inflation in tuning in order to sound like a proper piano. My mid-90's vintage Alesis QuadraSynth plus has a setting for this which basically applies a function to detune the pitch, across the keyboard range, by a programmable amount.
During history what it was perceived as consonant for most then became disonant, and the other way around. 6ths were the perfect 5th of their time.
Musics of the world uses different afination schemes, and thus different ratios than western musc.
As a jazz enthusiast I hear consonance in pieces that most people catalog as noise. (Abert Ayler comes to my mind).
My point is that, while there is an underlaying physics explanation for sound and harmony, the ultimate "perception of sound", specially music, is a human trait, where emotion and culture play a much more relevant role than the precision of any ratio between sounds.
I met a composer of minor note (pun intended) who avoided writing major 10ths in the melody, because the ratios in standard tuning were wrong enough to cause major discomfort for him. This interval is about 14 cents sharp, where a cent is defined as 1/1200 of an octave on a logarithmic scale, so 14 cents is about an 0.8% difference in pitch.
Curiously, major 3rds did not bother him, even though they have the same ratio problems.
To put that in perspective, pretty much anybody can distinguish a pitch difference of 25 cents. People with perfect pitch can distinguish differences of less than 10 cents.
Anecdotally, when I was taking lessons with my old cello teacher (who has perfect pitch), if he and I played a note on an open string simultaneously, I could always tell if we were in relative tune by the way the notes interfered. I don't think I'm particularly special in that regard.
Edit: Also, once I have one string tuned, I can tell if a neighboring string is correctly tuned because neighboring strings differ by fifths. Again, I don't think I'm particularly special; it's just a matter of learning what to listen for.
People with relative pitch can also discern < 10 cent differences. 10 cents is pretty obvious in the right context. People with perfect pitch don't need a reference; people with relative pitch do.
I play pedal steel, and string 6/G# drops a whacking 10 ten cents when press my A pedal and boy howdy can I hear it - and I don't have perfect pitch, just reasonably good relative pitch.
I don't play pedal steel, but I live in the city, and I've got a Martin 6-string and a fondness for bluegrass and acoustic music. I'd be interested in having a listen at a jam session, or sitting in if I'm at a level where I can contribute. I'm twitter.com/baddox.
I would actually think that to be untrue - NorCal has lots of pickers of many instruments. The process of finding musicians is just one of those hard problems.
It might be worth joining the (web-based) Steel Guitar Forum. Membership is $5.00 per year. There may also be a local steel guitar association. If you play another instrument, you may be able to volunteer to sit in backing other players at meetings. Also, find The Guy in the area who does steel guitar repair.
Oh no doubt the Bay Area has a ton of pickers, but pedal steel players are in short supply.
The main scenes tend to be related to the Grateful Dead and other jam bands. There's been a big rise in bluegrass and country inspired jam bands as of late although they're mostly living in the foothills or Tahoe.
Sweetwater in Mill Valley, Terrapin Crossroads in San Rafael, and Ashkenaz in Berkeley have a healthy amount of traditional American string players but I'm telling you, almost no one plays pedal steel. There's a few guys like Dan Lebowitz who are just phenomenal but they've all got their plates full.
San Francisco's got Amnesia and Veracocha and a few other smaller venues but I'd gotta say that Phil Lesh's Terrapin Crossroads is what the scene revolves around in the Bay Area.
I have no problem finding fiddle, banjo, or mandolin players! Just pedal steel!
Here's a second vote for the steel guitar forum-- they are quite helpful. Greets from Fredericksburg :D
If it is any consolation to your williamcotton, I'm also having trouble finding a band, but I haven't been at it so long so I usually end up playing bass....
It makes sense that major 3rds didn't bother him. With major tenths the fifth harmonic of the lower note is beating against the second harmonic of the upper note. With major thirds it is beating against the fourth harmonic of the upper note. The second harmonic is generally louder than the fourth, and so the effect is more noticeable.
I've been playing music all my life and even took theory courses in college and this whole thing is blowing my mind so I got out Excel and did a little math and discovered that the adjustments of equal temperament are significantly larger that 1.5000 to 1.5001. Here's my data:
A 55.00 55.00 55.00
E 82.50 82.41 110.00
B 123.75 123.47 220.00
F# 185.63 185.00 440.00
C# 278.44 277.18 880.00
G# 417.66 415.30 1,760.00
D# 626.48 622.25 3,520.00
A# 939.73 932.33 7,040.00
F 1,409.59 1,396.91
C 2,114.38 2,093.00
G 3,171.58 3,135.96
D 4,757.37 4,698.63
A 7,136.05 7,040.00
Starting with A1 (55Hz) the first column of numbers is simply multiplied by 1.5 12 times through the circle of fifths to get to an alleged A8. The last column I simply multiplied 55Hz by 2 7 times to get to the A8 to see the discrepancy.
The middle column of numbers I used to tweak the ratio for perfect fifths. I had to use 1.498307 to get the A8s to match up.
I suspect we are very sensitive to the introduction of beats which form when incommensurate frequencies are combined. That is to say when you add a frequency of 1KHz to 1.1KHz you get beats at 0.1KHz with an internal frequeqncy of 2.1KHz. This isn't the precise effect that takes place since the two frequencies are offset by some fixed amount, but a similar effect will take place for an offset that is not quite a perfect fifth.
I suspect that the problem is not just the subtle problem of pitch but the way waves phase in and out of sync with each other (especially on sustained notes) rather than remaining in consistent relation, so it should be possible for almost anyone to notice the effect if they know what to listen for. Not my area of expertise -- just an educated guess.
If two notes are played simultaneously and are off from that ratio a little bit, you can clearly hear "beats": a fluttering in the volume. This is audible not only for unison notes that are off, but for other intervals like fifths. The beats are faster (and easier to hear, and more annoying) the higher the note.
For instance if you play 1000 Hz against 1001 Hz (0.1% error), you will hear a 1 Hz beat. 100 Hz against 101 Hz also produces a 1 Hz beat, but the error is a lot greater at 1%.
the difference in a fifth between JI and Equal Temperament can be over 20 cents in at least 1 key, i.e. almost a quarter tone, or a lot, which i think most people can hear. So, yes, it can be a big difference.
And that's not accounting for the tuning peculiarities that every instrument has. A lot of older, perfectly playable pianos were always tuned below A=440 and can't be brought up now. When i started playing woodwinds, my band teacher specifically told me not to tune against piano in higher registers, which would have made the flute sharp, which is exactly what you don't want.
The discrepancies mentioned in the article are rather larger than 1 part in 15000 and can be heard quite easily. I'm not sure what the lowest difference the ear can detect is but with two notes played simultaneously you can get odd beat effects as the notes go in and out of phase. Say the two notes produce harmonics at about 5kHz, then if they are out by 1/15000 then they will go in and out of phase every three seconds and vibrate the ear drum more when in phase than when not so you may well notice.
Wanna see a cool visualization of the question you propose?
Google "Lissajous Pattern" and the like. Youtubes and images and wikipedia articles.
This is NOT what your eardrum looks like when you hear notes in tune, but it does provide a certain visual simulation of why simple integer-ish ratios sound better than random ratios.
The rough lower limit of pitch discernment is about one "cent" - the ratio 1.501/1.500. That's ten times the 1.5001 figure you gave. I am not sure that one tenth of a cent is perceptible.
Ten cents is a big difference, at least to me, but I'm somewhat trained.
