The Saddest Thing I Know about the Integers

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 :)

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?

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)

> 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.

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..

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...)

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.

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.

>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.

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).

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.

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.

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:

http://www.wolframalpha.com/input/?i=sin%28x%29+%2B+sin%28x+...

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

http://www.wolframalpha.com/input/?i=sin%28x%29+%2B+sin%28x+...

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.

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 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.

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):

(http://www.violinmasterclass.com/en/masterclasses/intonation)

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...

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.

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

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 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.

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%.

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 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'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.

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

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

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)

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 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.

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

https://plus.google.com/u/0/+DanPiponi/posts

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

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.

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.

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

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.

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.

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.

http://sethares.engr.wisc.edu/html/soundexamples.html

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.)

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.

"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).

Some high-end models do

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.

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!

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.

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.

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.