
The Saddest Thing I Know about the Integers - ColinWright
https://blogs.scientificamerican.com/roots-of-unity/the-saddest-thing-i-know-about-the-integers/
======
vortico
IMO the perfect fifth interval sounds boring. Equal temperament fifths at
2^(7/12) = ~1.4983 are close enough (and actually amazingly close to 3/2) to
be recognized as a fifth, _and_ they offer a nice chorus/unison effect that
many synthesizers even have a specific knob for. Slight error in close ratios
gives beautiful movement in complex chords, otherwise you'd be hearing a
repeating waveform with a period of lcm(1, ratio_1, ratio_2, ...). Well-
designed temperaments don't try to use nice fractions. Most of them don't use
fractions at all, and this sounds nice because slight dissonance sounds
beautiful. See [http://huygens-fokker.org/scala/](http://huygens-
fokker.org/scala/), the state-of-the-art tool for creating 12-tone
temperaments, as well as xenharmonic and microtonal tunings with non-12-tones.

~~~
edejong
It sounds boring but... I don’t know how to bring it into words, perhaps holy?
Especially on synths without any other modulation going on, but still rich
overtones it feels extremely ‘solid’ and pure. Perhaps primary (as in color).
Especially in melody the function of each note is more pronounced.

Still boring though...

~~~
DennisP
I once attended a piano concert in which the piano was tuned to just
intonation. It was anything but boring. I guess partly it's because I've
played piano all my life, and some of the chords were downright startling.

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781
Has anybody tried dynamically changing the tuning of an electronic instrument,
so that when you play a fifth or another chord, the individual notes are
exactly tuned so that the chord is itself perfectly tuned? Basically a per-
chord adaptive tuning kind of thing. Or would that sound bad in another way.

~~~
skywhopper
No need for electronics. Fretless string instruments, trombones, the human
voice, and most any wind instrument with a good player can bend the notes
sufficiently to achieve whatever precise pitches you want and hit these
perfect intervals. A typical piano doesn't have a way to bend the pitches on a
whim.

All that said, perfect tuning can only happen with certain types of chords,
and music that used only those perfectly tuned chords would be quite boring in
general. There are so many more combinations that still sound "good". Part of
what makes music interesting is building and releasing tension over time, and
one way to do that is with chords that don't hit these perfect intervals.

As for the piano and other instruments like it, centuries of good composers
have found all sorts of ways to take advantage of its particular
idiosyncrasies, both alone and with all sorts of combinations of instruments.
And it's not just old classical artists: in the 20th and 21st centuries, there
are plenty of geniuses who have and continue to find new ways to bring out
what makes the piano such an interesting instrument. Debussy broke all the
conventional wisdom of his day to make the most beautifully resonant piano
pieces. Check out Nancarrow's player piano pieces on YouTube to see the
instrument taken to a ludicrous extreme; John Cage's piano pieces often go all
the way in the other direction; Steve Reich's _Piano Phase_ is a brilliant
exploration of how we hear piano music; and finally, compare John Adams's
_China Gates_ (a solo piece) with _Century Rolls_ (a concerto) for very
different approaches from the same composer.

~~~
analog31
Indeed, when you get very far outside of keyboard instruments, there is no
"temperament," but merely a scale that's good enough for most purposes,
evolved over time. And a better instrument might have a better scale. Players
"lip" the notes up and down. Brass instruments have little tuning slides --
it's fascinating to watch a master play the tuba. Whether or not you activate
these devices depends on how fast the passage is, and whether anybody is
likely to notice.

~~~
copperx
Are you saying that you can sing a little bit out of tune and still sound
good?

~~~
analog31
If the passage is fast enough, yes. Of course "a little bit out of tune" for a
master is still pretty darn accurate.

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cousin_it
And the saddest thing about the saddest thing is that you can't even get the
white keys in tune! For example, if you tune the C major scale (C D E F G A B)
in the most natural and singable way, the "perfect fifth" between D and A will
be 40/27 and the "minor third" between D and F will be 32/27\. And that's
unavoidable - there's no way to tune the C major scale so all perfect fifths
are 3/2, all major thirds are 5/4 and all minor thirds are 6/5\. Basically if
the D note is in tune for the G major chord, it will be out of tune for the D
minor chord and vice versa.
[https://en.wikipedia.org/wiki/Ptolemy%27s_intense_diatonic_s...](https://en.wikipedia.org/wiki/Ptolemy%27s_intense_diatonic_scale)

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FabHK
He should be happy that (3/2)^12 is so close to 2^7 that it works at all.

Just as I'm happy that 2^10 is close enough to 10^3 that the whole kB/KB/kiB
issue rarely matters much.

~~~
fhars
The name “Evelyn“, the image and the repeated references to the girl scouts
may make your use of the pronoun „he“ suspect.

~~~
Wildgoose
I think the picture of the author (clearly female) is the give-away, but the
name "Evelyn" isn't because it is one of those names (e.g. "Lee") that are
applied to both men and women.

~~~
adrianratnapala
Yep.

Though admittedly it belongs to the category: names I'd have thought were
exclusively women's except I heard of just this one dude. My list is:

* Evelyn Waugh

* Vivian Richards

* Marion Barry

------
jdietrich
As it happens, pianos are deliberately tuned slightly flat in the bass and
slightly sharp in the treble, to compensate for the inharmonicity of vibrating
strings.

[https://en.wikipedia.org/wiki/Stretched_tuning](https://en.wikipedia.org/wiki/Stretched_tuning)

------
mikorym
Another aspect of tuning (and the piano in particular) that is interesting is
the duality of intervals.

A fifth is dual to a fourth (a fourth is a fifth going downwards and taken the
octave). You can the apply duality also on chords: a major is dual to a minor.

If you would like this to be more rigorous, there are a few first steps. For
example, you can look at the cyclic group of 12 elements. Lets call them
{0,1,...,11}. Then, 7 has inverse 5. And upon translation of this back to the
piano you'll see that 7 is the fifth and 5 is the fourth. (7 halftones and 5
halftones.) To see that a major chord is dual to a minor chord you similarly
use the cyclic group of 8 elements.

~~~
mhh__
What does dual mean in the context of a chord? If it means parralel then why a
minor triad (Presumably implied)

~~~
mikorym
To answer that, you have to first answer what the dual means in the first
place. And the notion that you get to is dual _with respect to the octave_.

A usual chord is _with respect to a fifth_. Then a major third is a minor
third down. Starting position should be at the tonic. Otherwise, you get at
the fifth that when you go a major third down it is a minor third from the
tonic.

------
superpope99
True it is sad in the domain of fixed interval instruments, but ask a
Barbershop quartet singer if they're sad about the ratio 4:5:6:7 and you'll
get a wholly different opinion :)

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timvisee
For people that don't have 'free articles' left:

[https://outline.com/https://blogs.scientificamerican.com/roo...](https://outline.com/https://blogs.scientificamerican.com/roots-
of-unity/the-saddest-thing-i-know-about-the-integers/)

------
red-indian
> Imperfect octaves are pretty unacceptable to any listener

All pianos use stretch tuning.

Also having all completely beatless intervals on perfectly harmonic overtoned
(aka theoretical and not existing in nature) timbres isn't the only criteria
for "correct" or good sounding tuning. It's not even a common criteria.

~~~
mrob
Wind instruments and bowed string instruments have perfectly harmonic timbres
during the steady-state sustained phase of the note. The piano is noticeably
inharmonic because it's a percussion instrument (the strings are hammered, not
bowed).

See:

[https://newt.phys.unsw.edu.au/jw/harmonics.html](https://newt.phys.unsw.edu.au/jw/harmonics.html)

~~~
red-indian
Your statement is incorrect, which your provided source clearly acknowledges.

Only idealized waveguides produce perfectly harmonic timbres. Idealized
waveguides do not exist in nature. The closest thing we have to the sound
produced by ideal waveguides is digital oscillators driven by a highly
accurate clock playing back a single-cycle waveform. The result is close
enough and is often described as an organ like sound.

~~~
mrob
"The inharmonicity disappears when the strings are bowed, but is more
noticeable when they are plucked or struck. Because the bow's stick-slip
action is periodic, it drives all of the resonances of the string at exactly
harmonic ratios, even if it has to drive them slightly off their natural
frequency."

See:
[https://newt.phys.unsw.edu.au/music/people/publications/Flet...](https://newt.phys.unsw.edu.au/music/people/publications/Fletcher1978.pdf)

~~~
boomlinde
That's true by a limited definition of inharmonicity. Maybe it is perfectly
harmonic in a musical sense. In reality any audible tone with a slow timbral
change will have sub-sound, inharmonic undertones. Depending on the context
that could or might not matter in practice. For example, the sub-audio
harmonics of a trombone might not matter if I'm just listening, but if you
ever record a trumpet you'll know that the pressure at the bell has a low
frequency bias that might affect the recording in a significant way, simply
because the air entering the system through the mouthpiece largely leaves
through the bell. The paper only takes overtones into account.

There is also no such thing as a steady-state for a person blowing or bowing.
The paper you linked to discusses the ideal conditions for mode locking, but
does not assert that these are ever actually satisfied during normal play. It
is only for an idealized bow action that the stick-slip is actually perfectly
periodic. It is clearly not the intent of the paper to assert that bowed
instruments are somehow inherently perfectly harmonic (if nothing else for
language like "slightly", "large", "noticeable" and "sufficient" without
further specification) but to provide a theoretical framework for discussing
this effect and the conditions under which it occurs. This is noted at the end
of the discussion section as well.

------
matchagaucho
Worth noting that Eastern microtonal music evolved just fine without equal
temperament. This was largely a Western phenomenon.

~~~
analog31
It's not even terribly important to Western music. In my view the main problem
is an instrument of a complexity that needs a professional technician to tune
it. And then you have to choose how you want it tuned. For instruments that
could be tuned by musicians (harpsichords had to be, because they went out of
tune easily), a variety of temperaments were used. String and wind instruments
have no temperament to speak of.

------
ken
> We can never get perfect octaves from a stack of fifths because no power of
> 3/2 will ever give us a power of 2.

Yep, none of them will ever be in the same ring. This is actually quite
similar to the reason you can't trisect an angle, square a circle, or double a
cube.

~~~
dr_dshiv
Do you have any references for this relationship/reason? In my understanding,
this question played a major role in the development of classical mathematics.
But proof of the impossibility came only in the early modern period.

------
8bitsrule
Having grown up in a nothing-but-Western-music culture, I was unsure what to
think when I first heard Nonesuch's gamelan album 'Music from the morning of
the world'.

[https://www.youtube.com/watch?v=AdJMesV7UP0](https://www.youtube.com/watch?v=AdJMesV7UP0)

Here, "nothing is right!" ... but I was delighted. It was quite a while before
I could understand why !!

------
js2
(2014). Big discussion four years ago around when this piece was published:

[https://news.ycombinator.com/item?id=8682782](https://news.ycombinator.com/item?id=8682782)

------
Gibbon1
> And sadly, I had to break it to the girls that these two facts mean that no
> piano is in tune. In other words, you can tuna fish, but you can't tune a
> piano.

I heard a sound as if two dozen girl scouts cried out.

------
YjSe2GMQ
FYI - this was already posted, 2014 comments:

[https://news.ycombinator.com/item?id=8682782](https://news.ycombinator.com/item?id=8682782)

------
golemotron
No, the saddest thing is that she hasn't accepted the miracle of Just
Intonation.

------
riccardopa
Sadly this article - seemingly an objective mathematical treatment of tuning -
is simplistic to the point of worthlessness. No mention is made of stretched
tuning, inharmonicity, or psycho-acoustic treatments of tuning. I expected
more from Scientific American.

~~~
jacobolus
It’s a blog post by a mathematician who knows more about pure mathematics than
music, and is riffing about a conversation she had with a Girl Scout troop.

We shouldn’t be expecting that to be equivalent to a PhD thesis about musical
harmony or whatever.

