The Saddest Thing I Know about the Integers | Hacker News Reader

The Saddest Thing I Know about the Integers

blogs.scientificamerican.com

133 pointsColinWright61 comments

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/, the state-of-the-art tool for creating 12-tone temperaments, as well as xenharmonic and microtonal tunings with non-12-tones.

> Scala was developed in Ada...

I don’t understand why <head explodes>, can you explain more?

Just guessing, but there is also a programming language called Scala[0], and this person just had a type resolution failure.

[0] https://en.wikipedia.org/wiki/Scala_(programming_language)

Maybe because Scala is also the name of a programming language as well as the linked software tool? Don't know, I'm just a bystander guessing.

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...

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.

It’s anything but boring when you start adding in more harmony (like an equal temperament major third), and move away from pure sine waves to something with more complex timbre like a piano or a human voice.

IMO this is only half of the picture. Yes, maybe a perfect fifth sounds boring if the timbres have perfectly harmonic overtones and no undertones, but that is never the case with actual musical instruments in an acoustic space. The waveforms aren't periodic in the first place.

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.

Yes, there are dozens of adaptive tuning techniques that have been implemented.

One of these, Hermode, has been included by default in Apple Logic since Logic Pro 7 (2004). Steinberg Cubase has also implemented Hermode since Cubase 7 (2012).

"Three Ears" by William Sethares is an example piece for his book "Tuning Timbre Spectrum Scale" that demonstrates this:

http://sethares.engr.wisc.edu/mp3s/Three_Ears.mp3

"As each new note sounds, its pitch (and that of all currently sounding notes) is adjusted microtonally (based on its spectrum) to maximize consonance. The adaptation causes interesting glides and microtonal pitch adjustments in a perceptually sensible fashion."

This could be done, but as a question it's malformed. Musically, the same interval may appear in an ambiguous set of contexts: sure, it might be part of a chord, but it's also a counter point line that just happens to be crossing another in a way that sounds interesting. You can tune the chord right only by mucking with the scale of the counter point melody. Probably half of music is the practice of crafting ambiguities like this.

There is plenty of music that stays within a single key and exploits proper rational tuning -- go listen to barbershop, for example. Voices can do this trivially (so can a handful of other instruments -- trombones come to mind), it's only devices with fixed strings or holes that are tied to equal temperment.

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.

> A typical piano doesn't have a way to bend the pitches on a whim.

There are a few keyboard-ish instruments that allow this, and they're super fun and expressive:

- Roli Seaboard (https://roli.com/products/seaboard/)

- Haken Continuum (https://www.hakenaudio.com/continuum-fingerboard)

- Linnstrument (http://www.rogerlinndesign.com/linnstrument.html, more like a guitar layout than a keyboard)

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.

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

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

Perfectly tuned to what chord? What if you play a stevie wonder tune where there is non-funcitonal harmony i.e. all the little tunings would be jumping all over the place

See: barbershop quartets.

It's one of the things that makes acapella music in general sound so good. A good acapella ensemble is focused on making each chord perfectly in tune. Singers aren't thinking of the temperament of their scales, they're thinking of the resonance of their harmonies.

The melodyne software will do this with vocals, but I believe it requires a lot of manual intervention. Melodyne will "auto-tune" vocals and that sometimes sounds really weird in equal temperament since vocalists will fluidly switch between equal temperament and other modes of intonation depending on accompaniment context.

In Wendy Carlos' "Switched On Bach" albums the tunings are carefully constructed depending on context.

For a more radical approach see also her "Beauty in the beast", which uses nonstandard systems.

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...

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.

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

Wasn't there a study that demonstrated that most males would imagine a person always being male first - unless the opposite would be proven?

I'm not sure, but I (male) definitely do and my girlfriend does the opposite

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.

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

The only Evelyn that comes to my mind is “Waugh”...

Oops, looks like I was not only careless, but biased there! Sorry.

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

It matters more these days when we're more likely to be talking about 2^30 or 2^40 and the difference multiplies. (It's only 2.4% at a kilobyte, but nearly 10% at a terabyte.) In fact it has gotten quite a few people upset, like this reviewer on Amazon who said Samsung was falsely advertising their "512GB" MicroSD card that really only has 476GB as displayed by the reviewer's operating system:

https://smile.amazon.com/gp/customer-reviews/R2Q6PRT955GYE/

I wrote a reply to the review that explains the difference between the two numbering systems, as 512,000,000,000 / 1024 / 1024 / 1024 rounded down to an integer is the 476 that the OS or file utility displayed.

I suppose it's a shame that GiB and such never took off, but probably they would be even more confusing. (What does the "i" mean, and which is which?)

This may be a no-win situation.

> I suppose it's a shame that GiB and such never took off

It sort of did. This is what OSes use to report storage sizes.

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

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.

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

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.

This elaboration is late, and maybe will be read only in 10 years time by a unassuming bot scraping the web, but here goes...

The musical scale (and a piano) is a logarithmic device. A logarithm by definition exhibits an interplay between addition and multiplication. Addition is repeated increment by one, whereas multiplication is repeated increment by some number. Raising to a power is repeated multiplication by a number. Repetition, or increment by one, is the basis for addition, leading to multiplication and then to the next step of raising to a power. (You can go beyond that again, but I forget what it is called.) In the form of an equation, you can express this as

    lnx + lny = ln(x*y).

As it is a logarithmic device, in a musical scale you go up the scale in integer increments. What I mean by logarithmic is that, x +' 12 in the chromatic scale is equal in the actual frequency of the airwaves to 2 * x. I use +' here to denote addition along the scale. However, as we are working in the cyclic group of order 12 we have in fact that x +' 12 is congruent modulo 12 to x itself; i.e., the octave is the same note as the tonic. We are forgetting for a moment that some people would call the tonic for instance A and the octave A2 in favour of a simpler system for our current discussion.

To recap, we already have a connection (by definition) between multiplication and addition in the structure that we call a logarithm and in music we can model the sound wave frequencies as a logarithm.

Now, the next step is a more interesting step perhaps for a human that likes to listen to music.

We already know as mentioned above that 7 chromatic steps up is a fifth and 5 chromatic steps down is a fifth as well. This is embodied by the cyclic nature of our model (a mathematical group structure). In the same way, in an analogue clock display, 12 + 7 hours is equal to 7 o'clock. 12 - 5 hours is also equal to 7 o'clock. However, time (if you experience it like me) moves forward linearly and indiscriminately. Even though they are inverses in a group, between the hours of 5 o'clock and 7 o'clock themselves, there is not a special physical connection. They are inverses simply due to the way we decided to partition the seconds in our days. (Incidentally, the human hearth rate is usually close to 60 bpm, but that phenomenon is perhaps for discussion another day.)

So back to the scale: Within music there is a very special connection between the 7 step chromatic interval and the 5 step chromatic interval. The 7 step interval is a fifth and the 5 step interval is a fourth. There have been countless musical pieces that comprise only the tonic chord, the fourth and the fifth. You can apply duality on chords as well: using the cyclic group of order 7 we have that the major third is dual to the minor third (with respect to the fifth = 7 steps chromatic) giving that a major chord is dual to minor chord.

One can continue in this fashion, using cyclic groups, and elements and their inverses will have special significance for music. Of particular note is that we can hear some of these connections, perhaps not rigorously or even explicitly, but indeed we know 1-4-5 is an important chord progression. I am tempted to say that this embodies a physical human interpretation of multiplication and addition, but one can equally argument that this is a physical human interpretation of duality, of group theory, or a very many other mathematical phenomena. We can speculate, but we do know for a fact that songs and progressions that we regard as aesthetically significant indeed turn out in the process of modelling to exhibit mathematical significance.

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 :)

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

https://outline.com/https://blogs.scientificamerican.com/roo...

> 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.

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

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.

"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...

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.

"Criteria" is plural. The singular is "criterion". I shall assume by the above that you will welcome the correction.

That’s because piano strings are inharmonic. I suspect that the same octave stretching would sound pretty bad with square waves.

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

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.

> 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.

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.

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

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

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

https://news.ycombinator.com/item?id=8682782

> 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.

FYI - this was already posted, 2014 comments:

https://news.ycombinator.com/item?id=8682782

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

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.

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.

I think your expectations are too high.

they're accepting submissions now -- get writing, then!

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