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Comment by whycombs | Hacker News Reader

Sorry, a lot of this is wrong. A major 7th is not 17:9 but 15:8. Our ear tries to represent all these rich resonances in as simple primes as possible (Think, combos of 2's 3's and 5's - and occasionally 7's in blue notes).

A minor 6th isn't just a note in between, but a simple 4:5, The minor sound that we associate with sadness is simply the reciprocal nature of the ratio - the more complicated prime is in the denominator (as opposed to the overtonal nature of the M7 above)

Short answer: read Harmonic Experience by W.A. Mathieu

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The ear does not care about ratios. Dissonance can be modeled using Plomp and Levelt's curve, which predicts the perceived dissonance of a pair of sine waves. Dissonance is minimum at unison, smoothly increases to a maximum at a narrow interval (dependent on absolute frequency), and smoothly decreases again as the interval increases.

Sethares extended this model to arbitrary sounds by decomposing them into their sine-wave partials with a Fourier transform and then taking an amplitude-weighted average of the predicted dissonance of each pair of partials. If you apply this procedure to intervals using harmonic timbres (timbres composed of partials with frequency an integer multiple of the fundamental frequency), the dissonance curve has minima that happen to be at small integer frequency ratios.

Harmonic timbres are most common in music, but there is an important class of musical instruments that are naturally inharmonic: tuned percussion. Unless substantial effort is put into designing them to approximate harmonic timbres, these sound most consonant when tuned outside small integer ratios. Indonesian gamelan music is a well known example of this. And with synthesized timbres you can have whatever inharmonicity you like, so timbres can be designed to suit any arbitrary tuning system.

Non-technical explanation of Sethares' model:

https://sethares.engr.wisc.edu/consemi.html

Full mathematical details:

https://sethares.engr.wisc.edu/paperspdf/consonance.pdf

> If you apply this procedure to intervals using harmonic timbres (timbres composed of partials with frequency an integer multiple of the fundamental frequency), the dissonance curve has minima that happen to be at small integer frequency ratios.

In practice that's pretty much "the ear cares about ratios" isn't it? Sure, it's caring about the ratios because they align with harmonic timbres in non-percussive instruments, but various human cultures do listen to a wide variety of instruments and voice with harmonic timbres, and perceive melody and chords based around [reasonably close approxiaations] of simple integer ratios sounding more consonant.

The implications for creating chordal music with synthesised inharmonic timbres designed to be consonant are interesting, but not widely applicable to what we actually listen to as music; and much of what we hear that goes outside simple ratios with the express intent of sounding dissonant...

Take one minute and try to sing a perfect 5th over a drone note. You'll waver around that beautiful sturdy 3:2. And if you're lucky, or if you practice you'll eventually lock in to that resonance/ratio and feel a universal truth as important as the right triangle.

All of music is based on resonance. And resonance is a ratio

Voice is a harmonic timbre. If you could sing with inharmonic partials that perfect 5th would not sound so consonant.

And this is relevant even in Western music. The piano is technically tuned percussion, and although it's very close to harmonic, it's inharmonic enough that you have to deviate from small integer ratios (beyond any deviation required for tempered scales) if you want the most consonant sound:

https://en.wikipedia.org/wiki/Stretched_tuning

Some people can do that all by themselves:

https://www.youtube.com/watch?v=vC9Qh709gas

> Sethares extended this model to arbitrary sounds by decomposing them into their sine-wave partials with a Fourier transform and then taking an amplitude-weighted average of the predicted dissonance of each pair of partials.

This is not entirely fair to Plomp & Levelt: they were the first to consider a timbre as a set of sine-wave partials and do the pair-wise partial average. Sethares' extension was to consider inharmonic sine wave series and reason about the implied relationship between timbre and consonance.

As others have noted, I think that that "The ear does not care about ratios" is pushing the implications of Plomp & Levelt's and Sethares' work a little too far. They have made a convincing case that our experience of dissonance and consonance is based on a more complicated set of relationships that just the harmonic series, but it's still fundamentally all about ratios (just more of them, and they cannot be considered independent of each other). Even the P&L dissonance curve is in some senses a ratio plot.

It's not always just about simplicity of primes. Some intervals correspond to harmonics too.

Major 2nd: 9th harmonic (9/8)

Major 3rd: 5th harmonic (5/4)

Perfect 4th: inverted 3rd harmonic (4/3)

Perfect 5th: 3rd harmonic (3/2)

Major 7th: 15h harmonic (15/8)

In fact one of my fav tunes Easy on Me by Adele has a suspiciously sharp perfect 4 in the chorus melody... I think she was tuning a bit towards the 11th harmonic which is sharper than a perfect 4. Someone measured her pitch and it's actually somewhere in between perfect 4 and 11th harmonic. So cool how it works!

https://youtu.be/U3ASj1L6_sY?t=145

Harmonics are the stacked ratios of primes. There's no 9th harmonic, there's the overtonal stacking of the two 5th harmonics. Or more literally, when you play a maj 9th on the piano, your ear is filling in the missing perfect 5th between them.

Every part of what you said confuses me so much!

How does 5+5=9? (Wrong I know but what is right?)

Isn't the 9th harmonic a distinct frequency in a musical note? I thought it was side effect of how physical vibrations ripple out or something...

Where can I learn more about what you're talking about? It's very interesting. Thanks.

The comment that confused you is largely wrong and confusing, so don't fret about it.

But the 5+5=9 part is correct. It's a weird artifact of note-counting. C D E F G is 5 letters. C to G is called "a fifth" (which is a horrendous use of language, but it's a perversion of saying "we got to the fifth letter"). G to D is also "a fifth". Stacking them means C to G and G to D, and you don't count the G twice, that's why 5+5=9 instead of 5+5=10. But if we had actually counted the steps through the letters instead of counting the starting note, it would be 4+4=8, and that's logical and correct, but you can't say that to musicians because that's not how music jargon works.

Although that makes sense, I actually just figured out what the other person meant.

A perfect 5th is a 3/2 ratioof the base frequency (3rd harmonic). If you square that fraction you'll get 9/4, which is a 9/major 2nd.

Yes, ratio math works without the strangeness. Except it's unfortunate that the 3rd harmonic is the fifth letter and the 5th harmonic is the third letter. Coincidentally, from harmonics 7 through 14, the harmonic numbers match the letter-count numbers. That's partly because 7:8 is the beginning of the harmonics being roughly whole-step sized, and 14:15 is where they shift into half-step size.

You read a strange subset of musical tuning literature and picked up a weird misunderstanding.

Harmonics means all the multiples of the fundamental. See https://en.wikipedia.org/wiki/Harmonic_series_(music)

People talk about primes because of their significance in mapping harmonic relationships.

Please don't talk about "overtonal stacking" that's not a thing. "Overtones" means any of the actual tones in a sound that are higher than the lowest one, whether the sound is harmonic or not.

Your piano claim is wrong also. What's happening is that the 4th harmonic of the major ninth matches the 9th harmonic of the lower note. And you do not fill in the "missing 5th" unless you happen to have trained yourself to imagine it.

Knowing the mathematical ratio of the root to it's Major 7th is of course far less important than knowing what they sound like, being able to hear the difference between a minor/major/diminished/augmented 7th, and knowing what effect using each has in getting to where you want to be.

The focus on this in these articles is missing the forest so you can focus on a single tree.

A lot of the ratios have not been fixed in time either, or can't be fixed correctly across all instruments.. the article doesn't get into that.

There's a simple reason for 15/8 being the major 7th. It's the 15th harmonic. The 15th harmonic is not changing anytime soon!

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