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The Saddest Thing I Know about the Integers (scientificamerican.com)
396 points by lisper on Dec 1, 2014 | hide | past | favorite | 188 comments

Welcome to the world of Barbershop music (one of my primary hobbies) ;) Unlike a piano, which must be tuned and which has a temperament that is fixed from chord to chord in a piece, the human voice can make minute adjustments to come as close as possible to those nice integer ratios (both in the fundamental and in the upper partials produced by their voices). When we do, we are rewarded with (sometimes screaming loud) overtones caused by the constructive interference between the sounds being produced by each of the four parts.

As this article points out, it's mathematically impossible to perfectly tune some of these intervals, but depending on the relationships between the notes being sung, you can tune to one singer or the other. It takes a lot of practice and a good ear, but the resulting effect is pretty darned cool.

I went looking for examples of what you were talking about and discovered that, amazingly enough, a single human voice can even reliably produce overtones! I'm guessing these are very different, though, than what you're talking about https://www.youtube.com/watch?v=2i61_JNc_Nc&t=1m25s

I'd love to hear some examples of a barbershop n-tet doing it too, I'm sure it's even better. But when I search for 'barbershop overtones' it's all music by groups called The Overtones :)

If you're curious about overtone singing, listen to this -- it's insane: https://www.youtube.com/watch?v=vC9Qh709gas

The principle is actually the same -- she is modifying her singing apparatus to emphasize different upper partials that are already in her voice. Barbershoppers do this as well, though less explicitly (we do vowel matching, which helps emphasize upper partials to produce greater ring).

Here's a video from a perennial favorite quartet (the Gas House Gang): https://www.youtube.com/watch?v=pvYT_yWiLqU The top/tenor note is often the same note as the primary overtone produced by the other 3 parts (adding further emphasis to the overtone), but this effect is what gives barbershop the quality of sounding like more than 4 voices, and produces the "ring" in the sound.

Just had to add (since I'm a giant Barbershop Geek in addition to all of the other ways I'm a geek ;))...

Lest anyone think that barbershop is just for old folks: https://www.youtube.com/watch?v=XWmFfFx24zs (Vocal Spectrum, 2004 international collegiate quartet champions and 2006 international quartet champions)

Or just for Americans: https://www.youtube.com/watch?v=rFWhnOP_UVk (Ringmasters, 2012 international quartet champions, from Sweden) https://www.youtube.com/watch?v=T6HM-oLG_cI (Musical Island Boys, 2014 international quartet champions, from New Zealand)

Or always serious: https://www.youtube.com/watch?v=ihdtI0E_mmQ (Storm Front, 2010 International Quartet Champions and comedy quartet)

Or just for men: https://www.youtube.com/watch?v=tWqtFxW4kiE (LoveNotes, 2014 Sweet Adelines International Quartet Champions)

Or just for adults: https://www.youtube.com/watch?v=sREQu_AaUso (The Osmond Brothers... yup ;))

Is it an example of an overtone music? http://www.nealstephenson.com/anathem/music.htm

It's a traditional way of singing in Mongolia I think.

My favorite example of this is Huun-Huur-Tu, I hadnt even seen a non-tuvan do it before this thread. https://www.youtube.com/watch?v=i0djHJBAP3U&t=100

I can do it pretty well -- the higher one, not the deep version (I can approximate the growly version, but it'll have me coughing after about 10 seconds, so I'm clearly doing something wrong!) -- after seeing Huun Huur Tu in concert back when I was in school I just tinkered until I could get the overtones reliably.

I also did a bit of barbershop (we called ourselves The Overhead Projectors...) and playing with throat-singing was actually quite useful in that context. I'm not sure how much it affect my actual voice control (probably some), but mostly I just became keenly aware of how much tone could vary just based on little changes in positioning of mouth, throat, tongue, palate, etc..

Overtone singing itself isn't very common in barbershop music, but their harmonies certainly make use of overtones. Here's a performance of a song called Play That Barbershop Chord (a stable in the genre). It's not instructive, but it's got some great harmonies: https://www.youtube.com/watch?v=F5bYnRkwpco

Here's a fairly technical article about three different seventh chords, one (the harmonic seventh) being the so-called barbershop seventh chord. There are embedded YouTube videos with computer-generated audio samples. http://www.garygarrett.me/?p=1575

There are some multi-track barbershop videos on Youtube which are good examples of over-the-top overtones. The fact that it's the same person (with the same timbre, "overtone profile") doing all the voices seems to amplify the effect. Here's one: http://www.youtube.com/watch?v=QpU_X6VKqvA

Here's a great video of the Ambassadors of Harmony (an incredibly good barbershop chorus from St. Louis, MO) doing some exercises to tune overtones, and they're really loud: https://www.youtube.com/watch?v=sCdQVqQXkzc

Listen for when the director sings the overtone note alone, then listen to the chorus as they produce it.

Incidentally, cheesy musical puns are the norm for barbershop and other a capella groups. Aural Fixation, Decibelles, Chiefs of Staff, Vocal Majority, Clef Hangers, and so on.

Extremely interesting overtone singing:


I just went "WTF" out loud. WOW! I had no idea music (or at least vocals) had this much depth. Thanks!

Is there a solid example of loud overtones being generated by the "barbershop seventh" that you can point to, such as in a recording?

I don't mean to be skeptical in a negative way. It would be great if music has a built-in payoff for singing with just intonation.

I've recently heard some doubt, however, about whether this overtone effect is actually perceptible to listeners, in a way that would pass double-blind tests and stuff. It's not that the alternative is that barbershop singers are delusional or something: the alternative is that barbershop singers -- and some listeners -- have well-trained ears for just intonation, and the perception of the chord they're singing changes inside their brain when the intervals line up.

I don't doubt the power of overtones lining up. This effect is where the entire musical scale comes from, after all. But does the barbershop effect actually physically come from constructive interference (this would limit it to a small number of chords made of undertones, incidentally), or is it just a noticeable auditory illusion that occurs in trained ears?

It doesn't strictly come only from 7ths. Here you can see a chorus practicing producing these overtones, and it's both highly audible and demonstrably there: https://www.youtube.com/watch?v=sCdQVqQXkzc

A sonogram like the one in this app will confirm that the effect is physical and not just an illusion: https://play.google.com/store/apps/details?id=com.ntrack.tun...

I still want to know more.

What should I be listening for in the video? There's the moment at 0:45 when everyone stops and says "wow", but that could come from either explanation.

Should I be expecting to hear a pitch that's higher than all the notes in the chord, lower than all the notes in the chord, or a different phenomenon that's not a distinct pitch?

This effect is sometimes claimed to happen with a 4:5:6:7 dominant 7th chord. In that case, would I hear it at the 28th harmonic? The 30th? The 420th (which is the lowest overtone that all the notes have in common)? The 1st (the undertone they have in common)? Or is 4:5:6:7 actually a bad example?

(I really doubt it's the 420th, of course. That would be a very high pitch around 17 KHz, which would be difficult to hear, and wouldn't be preserved at YouTube quality anyway.)

The pitch is higher than the notes in the chord. At 0:32 I believe the instructor sings the overtone pitch in isolation and says "you should hear that over it," then they sing the chord again and you can hear that same overtone pitch fairly clearly.

I think we're looking for a higher pitched sound in that video. I've heard it can take time to hear. I'm gonna listen again as well

I have never yet found anyone who agrees with me, but I believe the terminology around music theory is horrible. I mean that it is unnecessarily confusing. Most people that use it have no problem, but I believe they developed their overall understanding almost independent of the terminology, so now just use it as descriptive in its context, and do not at all notice how bad it is if you are trying to use the language as your first point in comprehending the underlying notions.

I'm surprised you've not found more agreement. I studied music in university, and have run a music-theory centered website since 1998 or so, and I definitely find the terminology to be rather bad.

It sounds mathematical in some ways (there are certainly numbers in there); but when you look closer it's inconsistent and/or just plain weird.

When you start writing code working with scales, intervals, triads, seventh chords, etc., the oddities really become obvious. The base unit of movement is the half-step; but the full-step is almost a useless concept -- a major scale is a seemingly random pattern of half and whole steps, major and minor intervals included in it. Whole notes refer to note duration. A whole tone scale is about the intervals between pitches. Looping the pitches from G back to A, but most starting counting at the C (for the simplest major scale).

There are historical reasons for the oddities; but it wouldn't be too hard to come up with better... it's much like language that way, though. Think of all of the inconsistencies in English -- yes, we'd be better off in many ways if we could just fix all of the broken bits (e.g.: the past tense of "lead" is "led", but the past tense of "read" is written "read"... but pronounced "red"). But we can't -- because everyone's using the broken version, there is a huge corpus of knowledge written in the broken version, etc. etc..

>>>> But we can't -- because everyone's using the broken version, there is a huge corpus of knowledge written in the broken version, etc. etc..

In addition to the amount of existing literature, there's also the matter of developing the skill to read it. I'm tethered to "standard" notation because I developed fluent sight-reading ability as a kid, and I continue to practice it. I daresay that learning a new notation system would be virtually prohibitive at my age.

Every town has a cadre of musicians with similar skills, which in turn creates an incentive for composers and arrangers to continue using standard notation. And some notations are considered to be useful for one thing but not another. For instance, tabulature is used for teaching the electric bass, and documenting transcribed bass lines, but nobody can sight-read tab well enough for it to be an alternative to standard notation for performance use.

I actually had such an experience not too long ago. I play mostly jazz, which has its own notational traditions. But a bandleader hired me, who had her repertoire written out using the Nashville Number System. Though her charts were simple, my brain had to work in overdrive for that entire gig.

I'm not so sure "it wouldn't be too hard to come up with better..."

I read about a thing called hummingbird notation - uses the same clefs, but changes noteheads, staffs and sharps/flats. Your opinion may differ, but here is a group of folks who spent a lot of time coming up with a "system" and it does not strike me as an improvement.

Have there been other attempts?

The trouble with notation systems is that the primary thing they need isn't to be easy to learn, but rather to have a ton of people who have learned it. I've seen lots of alternative notation systems, particularly in pedagogy, and from my experience people tend to hack up their own systems for jotting down ideas or composing, but mainstream staff notation is very adequate and has an insane amount of momentum.

Translation system, maybe?

Yeah, its the same with QWERTY, the only way you could replace the existing system is if the new system was so much better in some way that it replaces it by default. Mobile phone keyboards are the only thing to come close to replacing qwerty, and most of them just re-implement it anyway.

Alternate notation systems are another subject from music theory terminology.

I have fewer problems with notation than with theory terminology & concepts, personally.

There are alternatives that get a lot of use -- esp. TAB and various types of chord charts; but they're not as general purpose as standard notation.

There's a thing called "Integer Notation", which is (from what I've seen) used a lot in the more mathematical regions of musical theory.

It's pretty much what you'd expect, zero-based integers that represent semitone intervals. Sometimes they write chords in curly brackets, like {0,3,7} for the major triad.

Integer Notation also happens to be pretty much equivalent to the 0-127 based MIDI notes--in the sense that it's also zero-based integers representing semitones.

https://en.wikipedia.org/wiki/Integer_notation actually redirects you to "Pitch Class". Also very useful and mathematically sound, it's the same idea but with modulo-12 arithmetic.

My main gripe is that intervals should be zero-indexed instead of one-indexed. A double octave is known as a fifteenth. Wouldn't it be nicer if the octave was called a seventh, and the fifteenth was called a fourteenth?

There's also a concept known as inverted intervals, which are formed by moving downwards instead of upwards. The inverse of a major third is a minor sixth, because that's the interval you get (modulo octaves) by moving down a major third from your root note. The inversion n is given by 9-n. With zero-indexing, inversions would be given by simple modulo seven arithmetic.

>Wouldn't it be nicer if the octave was called a seventh, and the fifteenth was called a fourteenth

How would that be better?

>The inversion n is given by 9-n. With zero-indexing, inversions would be given by simple modulo seven arithmetic.

Again, how module arithmetic is simpler than a simple deduction from 9?

> How would that be better?

It would be better because a fifteenth is a double octave, but eight times two is not fifteenth.

> Again, how module arithmetic is simpler than a simple deduction from 9?

Because you'd be able to base both concepts on octaves (well, renamed to sevenths) being seven steps on the scale.

I still don't see it being much better in practice. It's pretty rare to use ordinals to refer to intervals greater than 2 octaves. There is really virtually no cognitive load required to hear "major tenth" and realize that it's a major 3rd plus an octave.

There's virtually no cognitive load if you already know music theory. That's a very important distinction - music theory in its current state is completely undiscoverable for no good reason.

How undiscoverable? I grew up in a fairly music-oriented family and played a lot of instruments as a kid, but didn't formally study music theory. It wasn't until I read parent that I realized "fifth", "third", etc. are 1-indexed ordinals, and not some inscrutable scheme involving fractions. Everything suddenly makes way more sense.

I feel fairly confident that sane terminology would greatly ease this problem.

> That's a very important distinction - music theory in its current state is completely undiscoverable for no good reason.

I don't doubt that it could be improved, but I don't think it's all that bad, considering how many people do learn music theory.

Pretty irrelevant but that reminded me of this scene from Cannibal the Musical [0]. The whole video is worth watching, especially if you've seen the rest of the movie but I only linked to the music theory argument. The movie is one of Trey Parker's earlier works and is quite entertaining [1].

Coming from a family of musicians and artists, it's all pretty foreign to me. And I used to take lessons, and occasionally sit and work my way through a beginner guitar book. I've always heard music theory to be pretty obtuse and challenging. I'm sure if I put in some hours to understand the fundamentals it'd make a lot more sense, and read it at home with a little dedication it'd be a bit easier. It's truly interesting.

[0] - https://www.youtube.com/watch?v=pF8oIKLlBsI#t=132 [1] - http://www.imdb.com/title/tt0115819/?ref_=fn_al_tt_1

> There is really virtually no cognitive load required to hear "major tenth" and realize that it's a major 3rd plus an octave.

Maybe not for you, but it certainly took me a moment to convince myself that that is true.

Just use modulo-seven arithmetic...

Er, yes, I believe that's the point the grandparent was making (and the parent was disagreeing with)

I was just trying to show that changing the indexing from zero- to one-based makes no difference in this case. The same-octave interval will always be the multi-octave interval mod 7, regardless of whether you call the tonic 0 or 1.

Granted, but I was replying to a post stating that "it" is not better in practice, where "it" is using 0 base, mod 7 arithmetic.

Anyway, yes, you've shown correctly that mod 7 arithmetic is useful, proving half the great-great-somethingth-parent's point. Now, the remaining interesting cases are the inversion case and the multiplication case, where 0 based is simpler (for me) than 1 based.

Also, the idea of using the word "octave" for "mod 7" is already pretty broken, and is what caused me my initial hesitation.

There is some cognitive load, but it is certainly no more than it would be to hear "Major 9th" and realize that it's a Major 2nd plus an octave, as it would be in the zero-indexed case.

It's really no more than a matter of semantics, Any interval mod-7 gives you the number of the scale degree on which the interval lands within the current octave. This doesn't change if with zero-indexing.

A Major 10th mod-7 is a Major 3rd regardless of zero or one-based indexing, the only difference is which notes in the current key you're actually talking about (in the key of C it would be E or D for one-based and zero-based indexing respectively)

I don't know that it's any easier to remember "the tonic is 0" than it is to remember "the tonic is 1"

>It would be better because a fifteenth is a double octave, but eight times two is not fifteenth.

If you are concerned about that, then wouldn't calling an octave "a seventh" (your proposed fix) be equally concerning? Eight (octave) is not seven.

You seem to be missing the point. An octave is a difference of seven scale steps, just like a second is a difference of one scale step, and a unison (a "first") is a difference of no scale steps.

The fact that this interval of seven notes is called an "octave", named after the number 8, is entirely because of our 1-based system that double-counts the note you start on.

If we had a 0-based system, it wouldn't be called an "octave". It would be named after the number 7. Let's call it a 7-interval, to avoid confusion with actual music terms.

In a 0-based system, you could say, for example, that two 2-intervals make a 4-interval because 2 * 2 = 4.

In the 1-based system we really have, you have to say "two thirds make a fifth", and the mathematical connection is not obvious.

> How would that be better? The 7th, 14th, 21st, etc would all be octaves of the root. Currently it's the 8th, the 15th, 23rd, which is much less intuitive.

Minor nitpick: 22nd would be an octave not 23rd... mod-7 still works, its just that you get a 1 back because the root is currently defined as scale degree 1

You're right, good catch.

The same can be said of almost any field. Algebraic rings have little to do with rings in real life, tensors have little to do with the muscle in my calf, and so on.

What do you suggest? We could invent new words - great, instead of an octave we now have a zargablub. Not terribly helpful.

Or we could give concepts more explicit names - instead of octave, we now have interval-between-two-pitches-with-logarithm-base-two. But all the experts will soon abbreviate this to something else, and soon enough we'll have the same problem we started with.

Or we could try to find a modern word that explains the concept better than the legacy one, but that means that all material from before the change is going to become very hard to understand unless you know all synonyms for a given concept. And since language is always evolving, in 50 years people will complain that your once modern and easy to understand word is now obscure terminology.

Really, what do you suggest? Jargon is hard for the newcomers, but there's a reason it exists- it enables experts to communicate clearly and non ambiguously about specific concepts and ideas. It's infinitely better for things to be hard to get acquainted with in the first 6 months, and then you get to used it, than to have a field where no progress is made because no one can communicate.

It's kind of like software, actually - these days, it's all about ease of use and being able to pick up an application and use it in 5 seconds flat without having to read a manual. Which is fine if your software is Snapchat or Instagram, but terrible if you're building a synthesizer or video editor or statistics package. In some cases, it is worth having something that's hard to learn but is very powerful once you've gotten past the initial hurdle, because while you will only be a beginner for a short while, once you are past that you will be an expert for the rest of your life. (these ideas are developed in Jef Raskin's The Humane Interface)

Jargon doesn't have to be complicated. You praise jargon as "infinitely better", the alternative being "no progress" but there's no room for improvement? The jargon we use is not some perfectly designed DSL, it's a catalog of concepts accumulated over decades and centuries. It's not master planned, it's organic sprawl. And I think it's fair game to critique.

Also, it's totally fair to raise an issue without having a solution or suggestion. And the lack of one doesn't make the issue less valid.

And "zargablub"? Straw-man territory...

>The jargon we use is not some perfectly designed DSL, it's a catalog of concepts accumulated over decades and centuries. It's not master planned, it's organic sprawl.

Organic is usually better than "master planned". Evolution et al...

That's assuming that there is some sort of selection pressure that will cause improvement over time.

Yeah, usefulness (and market).

Also, it's totally fair to raise an issue without having a solution or suggestion

Sure it's fair, but is it helpful? I guess in some way it is helpful to raise awareness of the issue but then what? We all stand around with our hands in our pockets, whistling away.

A straw man argument occurs when someone misrepresents their opponent's argument. That clearly did not happen.

Defenders of jargon usually pick examples in which a better substitution would be hard to find, yet unnecessary jargon abounds. Then there's necessary jargon but which is in conflict with jargon from another field. Sigh.

I have as much of an instinct to dislike jargon as anybody, but when I am writing code (in Haskell usually,) I find that naming things proc, process, process', preprocess'', etc., to not be any less confusing...

So there is a certain usefulness to it, and I comfort myself by knowing that our words are mostly all arbitrary anyway.

In my opinion Haskell is horribly jargon-heavy. Its syntax is a disaster.

As a mathematician by training (who has done mathematical research), I can confirm that creating names for new objects and concepts is no easier in mathematics than in programming. Creating the right notation (when it is necessary to do so) can be even harder.

> I have never yet found anyone who agrees with me, but I believe the terminology around music theory is horrible.

I'll be your first, then. Agreed.

> ... I believe they developed their overall understanding almost independent of the terminology, ....

That certainly happened here. E.g., that (say) 2^(4/12) is pretty close to 5/4 is an interesting fact that requires no knowledge of music terminology to grasp. That we call a pair of notes whose frequencies are roughly in this ratio a "major third" is just trivia to memorize.

> That certainly happened here. E.g., that (say) 2^(4/12) is pretty close to 5/4 is an interesting fact that requires no knowledge of music terminology to grasp. That we call a pair of notes whose frequencies are roughly in this ratio a "major third" is just trivia to memorize.

For musicians, this is actually precisely opposite. Performing and composing musicians absolutely never use the ratio of fundamental frequencies. They constantly use thirds and think of them as the third note in the major scale, which is the most common scale.

Do you have an example of something you think is horrible? This particular article is extremely technical or "low level," and is beyond what most people would call "music theory," so I wouldn't make too many conclusions just from this article. Most people use "music theory" to refer to music notation systems and some theory about composition, harmony, and rhythm, but seldom diving so deep into the physics and psychology behind interval and pitch perception.

Personally, I find most music terminology to be fairly decent, but I grew up with it, so perhaps I'm the type of person you describe. There is some ambiguity in terminology, some of which is inherent in the dual usage of certain harmonies (is it an A minor 7th flat 5, or a C minor 6th?), and some of which is simply the historical result of competing notations (is it a diminished 7th or a minor 7th flat 5?). However, I think most of the terminology is at least somewhat self-hinting, and the weird random names are few enough to be memorized or looked up when needed (like all the Italian words to notate dynamics and tempo). It certainly doesn't seem particularly worse than the terminology used in, say, computer science or computer programming.

A big part of the problem is that most music theory was developed to describe preexisting practice, without a whole lot of knowledge about why tonal music is pleasing. The result is a music theory that's highly layered, with the layers corresponding roughly to the major quantum leaps in theory all mixed together, rather inconsistently. We could go back and fix it, but that would require a grand unified theory of harmony, something which hasn't quite emerged, although we've made some big steps on the past decade.

The other reality is that we tend to teach theory from a perspective that explains simple music and progresses to the more complex, and then chronologically. This may be helpful for practicing musicians, but it does little to explain how the fundamental tonal concepts for together (scale structure, harmonic progression/voice leading, timbre, and tuning).

> We could go back and fix it, but that would require a grand unified theory of harmony, something which hasn't quite emerged, although we've made some big steps on the past decade.

I'm intrigued, got any more info or links on what sort of big steps have been made in the past decade?

>Most people that use it have no problem, but I believe they developed their overall understanding almost independent of the terminology

Not really. At least in Europe, and at least for the last couple of centuries, musicians learned and developed their understanding of music through studying its formal theory and terminology.

The only exception was autodidacts, which through common in folk music traditions, was not common in urban environments in Europe for classical or contemporary popular music.

Slightly OT, speaking about wrong terminology, also the blood types are pretty confusing. It's 3 bits, isn't it? So why calling one bit +/- and the other two A/B? (fine, besides historical reasons...)

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.


To my ears the stretched versions are far more engaging, especially the example with the tenor Jussi Bjorling around 52:50.

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:


And an out of tune one has very audible dips and troughs in intensity:


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...

I once saw a study that said perfect pitch is far more prevalent in mandarin speakers due to it being an penal language.

>there are tonal languages, after all!

I believe most (if not all) natural tonal languages use relative pitch, not absolute, so you wouldn't need perfect pitch to understand them.

Huh; quite right. I've had it in my head for several years that Thai, at least, was tonal based on absolute pitch; but it looks like that's wrong.

I did turn up that a much larger percentage of people have perfect pitch in populations that speak tonal languages.

Can perfect pitch be developed with a lot of training, or is it really a biological thing?

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."

It is likely some amount of both.

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.

Around the age of 15, I acquired something an awful lot like perfect pitch: I can identify and (conversely) produce seven particular notes. n=1.

Wouldn't absolute pitch be enough to make you struggle to enjoy?

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).

Here's equal: http://vocaroo.com/i/s1a539BsTA8F

Here's just: http://vocaroo.com/i/s1bLClEDtLsO

The most apparent difference is the wobbliness in the first one.

(try opening both at the same time!)

EDIT: I mixed them up previously, now correct.

I hear more wobbliness in the second one (...TA8F)

Great example, though - and playing them both at once is an interesting exercise in aliasing.

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.

445? Ouch!

tune to the soloist no matter what

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.

Also, apparently pitch used to vary quite a lot during the baroque period. I've even heard claims of pitch varying by as much as a minor third, though that seems rather extreme: http://en.wikipedia.org/wiki/Historically_informed_performan...

this one was tuned down a whole step, Bb-F-C-B but they DMCA's his yotube vid: http://www.telegraph.co.uk/culture/music/classicalcdreviews/...

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)?

Pure sine wave? Not very good. A violin? Much better. My violin? Very high.

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.

[1] https://en.wikipedia.org/wiki/Stretched_tuning

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.

That's just changing the reference pitch, a different thing.

I don't know about that particular synth, but having a menu of temperaments is not uncommon on digital pianos, see p 41: http://download.yamaha.com/api/asset/file/?language=no&site=...

No, it the pitch change is different for each note. That's why it's a function (of the pitch number).

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.

http://en.wikipedia.org/wiki/Harmony (Historial rules, and Perception of Harmony sections are both relevant)

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.

Pedal steel! Do you live in the Bay Area and like playing country music? If so, wanna jam sometime? I've got a rehearsal space in SF.

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 live in Texas. It's a bit of a haul :)

As you could guess it's pretty tough to find pedal steel players around here!

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.

Good luck in your search!

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!

You know what this means - become one! Learn you the steel guitar! It's intimidating, but it's worth it.

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.

But between neighboring notes (half-steps) it would be a 14% difference, which is noticeable to almost anyone with ears.

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.

This thread did remind me to try Werkmeister and Kimberger tunings on my digital piano, p 41: http://download.yamaha.com/api/asset/file/?language=no&site=...

I shd practice viola, also

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.

Would it be possible for an electronic piano to dynamically adjust the temperament based on the keys currently being played? In such a piano, the C and G keys could be precisely 3/2 when they're played together, and the E♭* and G could be precisely 5/4 when they're played together at some other time.

Assuming an algorithm could be worked out, would it end up sounding worse due to notes shifting around ever so slightly to make all the ratios work?

* Edited.

It's already been done and tried by many folks in many different contexts. Search for adaptive Just Intonation.

Anyway, there's fundamental issues this approach can never solve. An E and G should NEVER EVER be 5/4. That would be E and G-SHARP (or E-flat and G). But this is a good example of the issue. E and G could be 6/5 or 7/6 or 19/16 or… (the first two make the most sense). So, how is the keyboard to know when I play E and G that I wanted it to be part of a C-major 4:5:6 chord versus part of an A7 4:5:6:7 chord? There's no way other than some input that can tell me or the thing adapting after-the-fact when I later add the rest of the chord.

I think the best overall tuning software for keyboards is http://www.tallkite.com/alt-tuner.html by the way, although adaptive stuff isn't the focus.

There's many others though. Cheers

You can still certainly do adaptive JI with manual composition, particularly with electronic music production.

This exists in a few synths, and is included in Apple’s Logic Pro for software (MIDI) instruments as “Hermode Tuning” (http://www.hermode.com/html/hermode-tuning_en.html)

It’s a really unusual experience using it, because the never-quite-tuned sound is part of a piano’s character. With Hermode tuning engaged, every chord has a ringing pure bell-like quality, especially with 7th/diminished chords (which sound immensely satisfying).

Since few instruments in the real world can tune so precisely, Logic amusingly has a slider to make the Hermode tuning “less perfect” if desired.

I actually worked on an iPad app when I was in school that does this: https://github.com/benweitzman/retune (please excuse the terrible code/project layout, I was just learning iOS at the time)

That's doable. You'd probably want to use something Luke organ pedals to set the active tuning rather than try to guess.

This isn't sad, because we don't simply compare pitch ratios when we listen to music. Real human consonance perception is much more interesting than that, and William Sethares has formalized a very good model: http://sethares.engr.wisc.edu/consemi.html

There's great opportunity for microtonal tunings without the dissonance often associated with microtonal music. And as others have already pointed out, traditional techniques already do this (eg. stretch tuning). The theory explains why it works, and how to expand it to arbitrary timbres, including those possible only with synthesizers. I'd love to see more musicians experimenting with it. There's a whole lot of unexplored musical novelty out there.

I love Sethares work, but to use it in its full generality, we'd want arbitrary timbres. In practice, most of our timbres are highly harmonic.

Speaking of which, another theorist, Dmitry Tymoczko, has some fascinating theories on harmony that you might like in his book A Geometry of Music. A few years ago, when I was reading his and Sethares's work, I really felt like their ideas put together would make a fascinating grand theory of tonal music.

> Imperfect octaves are pretty unacceptable to any listener

This is false, and in fact pianos are tuned in imperfect octaves. The reason is that due to nonlinearity, the harmonics of a vibrating string are not perfect multiples. They are sharp! And that actually contributes to the character (timbre) of the note.

When a piano is tuned, typically the middle range is set according to an electronic source (nowadays). The higher keys are tuned against the harmonics of their previously tuned lower octave counterparts. This means that those notes are slightly sharp. Likewise, notes in the lower register are a bit flat.

Dan Piponi has been blogging about this topic recently, which lots of serious math and code:


Yep, although you could probably say that given a timbre, octaves without optimal partial coincidence are pretty unacceptable.

This is hard to say; is it simply expectation bias? Because some instruments have elements with wacky harmonics, like bells, metallophones, xylophones, Jamaican steel drums and such. Notes do not have a fundamental which coincides with a harmonic of a note one octave below. That's just how the instrument sounds. If a piano reminds us of a metallophone :) then we conclude that it's awful and out of tune. Still, even that is appropriate to some genre, or for comic effect. A western saloon bar calls for a detuned piano. :)

With computers you can fix the problem. It's interesting to experiment with playing arbitrary frequencies instead of the standard ones. A computer could generate music with perfect pythagorean ratios relative to the dominant key at any given time. It could even follow those ratios as the dominant key changes, gradually drifting away from equal temperament.

>The difference between a Pythagorean fifth and an equal temperament fifth is not enough to bother any but the fussiest listeners, but it is detectable to some.

Thought experiment: Suppose someone with perfect pitch were to sing out all the notes on a scale which they feel sounds perfect, while someone else would measure the frequencies of each note they sing. Obviously, the result would not show a counter example to the fundamental theorem of arithmetic, but I'd be curious to know what it would show...

Mostly likely the person with perfect pitch would be singing the "correct" scale they were taught to sing, which almost certainly would be equitempered. But in principle it's not impossible to imagine someone (someone impossibly talented anyway) singing their way around the cicle of fifths and landing in a key that doesn't quit match the one they started in.

And the point in the article that temperment cannot be heard is sort of wrong anyway. You can write some quick code to generate both chords, and the difference is notable. Perfect tuning is something we actually hear regularly in things like tight vocal harmony. It's not alien.

In fact, I think an unaccompanied trained singer would tend to sing a scale in just intonation, with or without perfect pitch.

In the 1960s, La Monte Young wrote a piece entitled The Well-Tuned Piano, a riff on J. S. Bach's The Well-Tempered Klavier. It features an equally-tuned piano, so each interval is mathematically correct, rather than being the standard "good-sounding" intervals. It really throws you off at first, but it's well worth a good listen. (It's six hours long though, so don't listen through it in one sitting.)


Wow.. the article and the comments here are fascinating. I don't have much to add, but I think what you guys are calling overtones are something some of my friends and I used to do as kids. One person would sing or yell a single loud note and another would sing or yell a sliding scale of notes listening for the overtone where a "spooky third voice" would chime in. We made up stories about it being a ghost summoning ritual and such. :)

If you haven't seen Qiaochu's answer about this subject over at math stack exchange, it's easily one of the best answers on any stack exchange site: http://math.stackexchange.com/questions/11669/mathematical-d...

More a fact about human psychology than about integers, I'd say.

So how well can you approximate a fifth with a different number of notes in your equal-tempered scale?

    import itertools, math

    def rationalizations(x):
        ix = int(x)
        yield ix, 1
        if x != ix:
            for numer, denom in rationalizations(1.0/(x-ix)):
                yield denom + ix * numer, numer

    for frac in itertools.islice(rationalizations(math.log(3, 2)), 10):
        print '%d/%d' % frac,

    1/1 2/1 3/2 8/5 19/12 65/41 84/53 485/306 1054/665 24727/15601
I guess 8/5 is for the pentatonic scale? Then after 12 tones the next good approximation has 41 and we've run out of the alphabet.

This series is, in fact, really cool. It does in fact point out that the pentatonic scale and the 12-tone scale are things that you can almost make just by gluing together fifths, and then it points you to the 53-note scale, which despite being hard to work with is freaking awesome: https://en.wikipedia.org/wiki/53_equal_temperament

(The 41-note scale doesn't accomplish much, it turns out.)

The 53-note scale was supposedly discovered by Ching Fang (the same guy who discovered that the moon reflected sunlight) in the first century BC.

The equivalent of fifths in the 53-note equal-tempered scale are an unreasonably good approximation to just intonation, kind of like 355/113 is an unreasonably good approximation to pi. It also happens to contain good approximations to all kinds of other just intervals. And because 53 is prime, you don't just have a circle of fifths, you have a circle of any interval you want.

So you can use this scale as a reference point to compare different tunings, scales of world music, and scales invented by avant-garde composers. Or you can use it as a cool thing to play with.

Cool! Yeah, I considered pointing out that 84/53 is an especially good approximation for its complexity (since the next in the sequence uses much bigger integers) -- but how are you going to deal with 53 notes, or even 41? It's neat to hear that others were not stopped by that.

(It did occur to me you could almost label them with letters of the alphabet, with one left over: 53=26x2+1. But that'd be silly and invite confusion with the 12-tone note names.)

Trombones are in tune.

In theory, at least. I've never noticed it in practice.

Heh. My dad's a jazz pianist and the quip is ingrained: "You can tune a piano, but you can't tuna fish." Dramatic pause. "Sure you can. You use its scales."

Heh. But seriously, folks...

Posted today by Paul Barton, a video that lets hear piano sounds never heard before.

Like the moon that eclipses the sun, revealing solar flares, this mechanism helps hearing what's hiding behind the main notes hit.

Skip to 3'58'' to hear only harmonics sounding after dampening manually a C3.

Skip to 7'17'' for a demonstration of the sympathetic sounds.



here's a link to the print version, the floating 20% off horror for black friday and cyber monday are gone, thank god.

Do we perceive octaves as precisely 2:1 or is that an approximation? i.e. is it really 2.041256:1 or similar.

Is this problem then because we start with integer ratios? Is this problem solvable if we started a "true(?)" irrational ratio?

It's exactly 2. It's not so much about how we perceive them (well in the end it is I guess), but there is a real, physical relationship between the two notes.

Two notes at 2.041256:1 sound really annoying together.

You can try by putting 250 and 510 into the boxes here: http://onlinetonegenerator.com/binauralbeats.html

and revert to a nice-sounding octave with 250 and 500.

It depends on the timbre, the frequency makeup, of the sound. If all frequencies present are integer multiples of the fundamental frequency, a 2:1 ratio will line up all the frequencies nicely, which "sounds right."

This frequency relationship is approximately true for many, but not all, acoustic instruments.


The mp3 at the above website called "Challenging the octave" gives an example of a bell that sounds more in tune when the "octave" is a frequency ratio of 2.1 vs the usual 2.

If you have any interest in music theory, even if you've already studied traditional western music theory, read Dr. Sethares' work on the subject: http://sethares.engr.wisc.edu/ttss.html (He was mentioned elsewhere in the comments, but he's too awesome to risk missing.)

The problem is that we use approximate with irrational numbers. Naturally they are all integer ratios.

In the perfect world you have e.g.:

+ C major scale: C D E F G A B C, where frequencies of all notes depend on frequency of note C (like, “fifth” from C is G and it's exactly 3/2 of C)

+ and D major: D E* F♯* G* A* B* C♯* … D (and depend likewise). But now G is not necessary equal to G*, but they are close. So here comes the idea of equal temperament where octaves are strictly 2:1 as they suppose to be, but all notes between are equally scattered (on log scale) in between.

So TL;DR: nowadays it's all approximation. You can do it perfectly, but only for one root.

What if you stop using Hz and use something else?

This is about unit-less ratios, so the basic unit of measure doesn't matter.

You could, though the relative ratios would still be the same so the issue wouldn't change. Also, not using Hz hurts.

>The fact that 3/2, 2, and 5/4 are incommensurable makes me genuinely sad

But they're not...

I am also puzzled about the alleged connection to UFDs.

"incommensurable" is apparently being used in the sense that their logarithms are not in rational ratios (so no common step size generates them all in the relevant way). The connection to UFDs is that one way of seeing the incommensurability in this sense is through unique prime factorization (as exponent vectors over the primes 2, 3, and 5, the noted values are 3/2 = <-1, 1, 0>, 2 = <0, 1, 0>, and 5/4 = <0, -2, 1>, none of which are collinear).

I agree that the connection with UFDs is tenuous. Being not a UFD is nether a necessary or a sufficient condition to have powers of a single thing (or a finite number of things) generate all nonzero elements. The ring of integers modulo 7 for example is a UFD, but all nonzero elements are powers of 3. Also, the ring of integers of a number field is only sometimes a UFD, but it is never generated by powers of some finite number of elements.

Being a UFD in general is not enough, but being a UFD in which 2, 3, and 5 are (co)prime is certainly sufficient for incommensurability of 3/2, 2, and 5/4 in the relevant sense.

I agree that it's rather odd to start discussing UFDs in general, Gaussian integers, etc., just for this incommensurability result, but since unique prime factorization is the key to the whole thing, it's not entirely out of left field.

(I'm also not accustomed to considering the ring of integers modulo 7 a UFD, insofar as exponents in prime factorizations are never unique as integers in this context (only as integers modulo 6), but that's just a minor difference in the way we apparently use terminology)

"Being a UFD in general is not enough, but being a UFD in which 2, 3, and 5 are (co)prime is certainly sufficient for incommensurability of 3/2, 2, and 5/4 in the relevant sense."

Ok, this seems like kind of a contrived condition though, compared to, say, the condition that 2, 3, and 5 are prime (which also suffices).

Something that occurred to me after I made my last post (and possibly what you meant in the second paragraph?): perhaps the author didn't mean to imply that the UFD property is useful for characterizing when you have incommensurability in different rings, but rather that it's relevant because it's used as a step in the proof that these ratios are incommensurable in the case of Z. This makes sense to me but in my opinion it wouldn't hurt if the article were more explicit on this point.

I'm not familiar with a commonly-used definition of UFD for which the ring of integers mod 7 is not a UFD. Could you please point me to a reference that contains this alternate definition of UFD you are referring to? The definition I am using is the one in Abstract Algebra by Dummit & Foote, which happens to agree with the definition on wikipedia at the time of this posting.

Yes, I think what you're saying in your third paragraph is what I was trying to say in my second paragraph. And I agree that the article could stand to be much clearer on its motivations.

Re: the integers mod 7, I had a brainfart; of course the integers mod 7 are a UFD, but trivially so, as they are a field. My apologies!

Well I'm glad we are in agreement then. No worries about the UFD thing.

The ring of integers modulo 7 is also a field. While fields are (trivially) UFDs, the only non-field class of examples of UFDs having every nonzero element as a power of a fixed nonzero nonunit element is a (rank 1) discrete valuation ring.

" the only non-field class of examples of UFDs having every nonzero element as a power of a fixed nonzero nonunit element is a (rank 1) discrete valuation ring."

Ok, but the UFD criterion is redundant here because there are no non-fields for which every nonzero element is a power of a fixed nonzero nonunit element. So that doesn't make UFDs any more relevant than any other extra criterion you could add.

Well, if you're willing to allow a bit of wiggle room in the form of "up to associates" (which is a qualifier that I forgot to add), there are integral domains in which every nonzero element equals the product of a unit and a power of a fixed nonzero nonunit element. This qualifier isn't too much to ask since "up to associates" is the generalized version of "up to a plus or minus" when factoring integers (which has no real bearing on factorizations).

In particular, let F be any field, and consider F[[x]], the ring of formal power series[1] over F. Given any nonzero f(x) in F[[x]], we can "factor out as many x's as we can" to write f(x) = x^n g(x) where n is a nonnegative integer, g(x) is in F[[x]], and the constant term of g(x) is nonzero (here we explicitly define x^0 to be 1). Thus, g(x) is a unit[2] (ie invertible), and every nonzero element of F[[x]] is a unit multiple of a power of x.

[1]: http://en.wikipedia.org/wiki/Formal_power_series

[2]: http://en.wikipedia.org/wiki/Formal_power_series#Inverting_s...

Ok, but rings for which every nonzero element is a power of a fixed nonzero nonunit element up to associates are exactly discrete valuation rings. So your statement boils down to "The only UFDs that are discrete valuation rings are (rank 1) discrete valuation rings". And I'm not sure what you mean by rank 1 here, but I can deduce that all discrete valuation rings are rank 1 from what you've said, since all discrete valuation rings are UFDs.

So what you've said boils down to:

"The only UFDs that are discrete valuation rings are discrete valuation rings", which again doesn't make UFDs more relevant than any other condition you might add.

Correction: the _integral domains_ for which every nonzero element is a power of a fixed nonzero nonunit element up to associates are exactly discrete valuation rings.

So my final statement should read: "...which again doesn't make UFDs more relevant than any other condition you might add _that is stronger than the much more general condition of being an integral domain_"

mumm is mam

num i s yyh

"there are no non-fields for which every nonzero element is a power of a fixed nonzero nonunit element".

There are enough qualifiers here to go quite a bit further: Not only are there no non-fields of this sort, but, in fact, there are no rings of this sort whatsoever: if x is our fixed nonzero, nonunit element, we must have x + 1 as some power of x (x + 1 being nonzero, as x is not -1, as x is not a unit). Furthermore, this power cannot be zero (as x + 1 does not equal x^0 = 1, as x is nonzero). Thus, we would have x + 1 = x^(m + 1) for some m. Which is to say, 1 = x * (x^m - 1). But this makes x a unit, contra stipulation. Thus, there is no such ring.

[Of course, another way of looking at this is that, dropping the "nonzero nonunit" stipulation, we prove that the generating element x is either zero or a unit, and thus the rings with this property are, as noted, necessarily fields (or the trivial ring in which 0 = 1)]

It's seems a bit absurd to me to go further than non-fields, because fields have no non-zero non-units by definition.

That is a nicer argument than the one I had come up with though, which used the observation that all such rings must be quotients of the polynomial ring Z[x].

I'm not a music-making-guy, so I ask out of curiosity — do current electronic keyboards have selectable temperaments?

Yes, my Kurzweil K2000 (http://en.wikipedia.org/wiki/Kurzweil_K2000) supports fully customisable temperaments – you can even tune it to the Bohlen-Pierce (http://en.wikipedia.org/wiki/Bohlen%E2%80%93Pierce_scale) scale if you want!

That's a sweet, sweet synth.

Now I'm kicking myself for selling my gear. I had an Alesis Ion. So much fun.

Workstation-style keyboards do (the ones that have a synthesizer and a built-in sequencer as well as other features). This is implemented using the MIDI tuning Standard which has been around for years. Such functionality is usually not available on simpler synthesizer, which allow adjustment of concert pitch but not much else. There just isn't much demand for this outside of professional composition circles, for the same reason that Barbershop Quartet music (which relies very heavily on just intonation for its ringing harmonies) is an extremely niche musical taste.

It's useful if you want to write non-western music, eg Arabic or Indonesian or Indian styles which have different scalar intervals from Equal Temperment, but only if you're very concerned with authenticity. The fact is that if you're composing for a Western-music audience you can get most of the 'feel' of other musical traditions by selecting a suitable scale and playing in equal temperment. If you're composing within the musical tradition of a non-western culture you're less likely to be using something conceptually oriented around a piano keyboard in the first place. In a lot of musical traditions (and indeed country and folk music in the west) the emphasis is not so much on tonality and composition as having a good repertoire of stock musical phrases that everyone is familiar with, and virtuosity consists in skillful improvisation employing those musical tropes.

Some, yes. (Generally those geared more toward the synthesizer/MIDI crowd.)

Of course if your synthesizer is your computer, it's a question of what the synthesizer software does, not the keyboard; and the answer is the same.

Not many of them. They often let you change the fine pitch of the whole keyboard though (in old school analog keyboards this was because the whole instrument could go out of tune over time / as it warmed up I believe, but these days it's sometimes still present, maybe as a way to play with out of tune instruments). According to other responders, some keyboards do, but it is not standard.

But besides using the built-in sounds in your keyboard, you can use most modern keyboards to send MIDI notes to some external (or virtual) synthesizer to play.

I just looked into handling MIDI for different temperaments. MIDI is a way to digitally represent each note as an integer (so let's pretend Middle C is the number 80, the B directly below would be 79, etc). The MIDI standard itself afaik has no support for other temperaments, but the receiving synthesizer can. So some VST instruments (virtual instruments you run on your computer for either recording or live performance) do support other temperaments, but it's not across the board. If you want to write or perform in a different temperament, you probably are limited to using VST instruments that support it.

If you're sending MIDI to a synthesizer that doesn't support choosing temperaments you can mostly fake it by sending pitch bend signals.

Many of them do. Sometimes you can select from a menu of pre-defined temperaments, sometimes you can individually tune each pitch class (e.g. you can tune all the G's 4 cents sharp if you want), and sometimes you can tune all 88 (or so) notes individually.

In general, you run into problems if you want more than 12 notes per octave because a) standard keyboards are really inconvenient for that and b) midi wasn't designed to support microtonal music in any consistent way, so you have to employ strange kludges, like use a separate channel for every pitch class and use pitch bend for tuning.

My Roland digital piano has several temperaments it can use including meantone, just, pythagorean. It's a low-midrange model HP203, make to look and sound more like a piano.

Some high-end models do

mind = blown.

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