
The Saddest Thing I Know about the Integers - lisper
http://blogs.scientificamerican.com/roots-of-unity/2014/11/30/the-saddest-thing-i-know-about-the-integers/
======
depoll
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.

~~~
dandelany
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](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 :)

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

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

~~~
hobs
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](https://www.youtube.com/watch?v=i0djHJBAP3U&t=100)

~~~
jtheory
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..

------
sopooneo
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.

~~~
jtheory
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..

~~~
lmg643
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?

~~~
baddox
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.

~~~
saraid216
Translation system, maybe?

~~~
hobs
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.

------
billforsternz
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 ?

~~~
cynicalkane
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.

~~~
pmiller2
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.

~~~
ArkyBeagle
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.

~~~
williamcotton
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.

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

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

~~~
ArkyBeagle
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!

~~~
williamcotton
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!

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

------
function_seven
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.

~~~
quadrangle
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](http://www.tallkite.com/alt-
tuner.html) by the way, although adaptive stuff isn't the focus.

There's many others though. Cheers

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

------
mrob
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](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.

~~~
acjohnson55
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.

------
kazinator
> _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.

~~~
acjohnson55
Yep, although you could probably say that given a timbre, octaves without
optimal partial coincidence are pretty unacceptable.

~~~
kazinator
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. :)

------
tiglionabbit
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.

------
placebo
>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...

~~~
ajross
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.

------
jstanek
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.)

[https://www.youtube.com/watch?v=vzwgJq9vsDM](https://www.youtube.com/watch?v=vzwgJq9vsDM)

------
DEinspanjer
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. :)

------
calhoun137
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...](http://math.stackexchange.com/questions/11669/mathematical-
difference-between-white-and-black-notes-in-a-piano)

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

------
abecedarius
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,
    

produces

    
    
        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.

~~~
rspeer
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](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.

~~~
abecedarius
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.)

------
snarfy
Trombones are in tune.

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

------
joseph8th
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...

------
cellover
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.

[https://www.youtube.com/watch?v=zHWex94-mHg](https://www.youtube.com/watch?v=zHWex94-mHg)

------
throwawayaway
[http://blogs.scientificamerican.com/roots-of-
unity/2014/11/3...](http://blogs.scientificamerican.com/roots-of-
unity/2014/11/30/the-saddest-thing-i-know-about-the-integers/?print=true)

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

------
AndrewWorsnop
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?

~~~
adamnemecek
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.

------
jensenbox
What if you stop using Hz and use something else?

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

------
pervycreeper
>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.

~~~
Chinjut
"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).

~~~
tprice7
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.

~~~
Chinjut
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)

~~~
tprice7
"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.

~~~
Chinjut
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!

~~~
tprice7
Well I'm glad we are in agreement then. No worries about the UFD thing.

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

~~~
danieldyer
Yes, my Kurzweil K2000
([http://en.wikipedia.org/wiki/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](http://en.wikipedia.org/wiki/Bohlen%E2%80%93Pierce_scale))
scale if you want!

~~~
Florin_Andrei
That's a sweet, sweet synth.

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

------
tehwalrus
mind = blown.

