Post · bonfire.mavnn.eu

@RonJeffries@mastodon.social replied · 5 hours ago

@johncarlosbaez
I love this! Thanks! Now I need to find a way to hear the tones as I read.

1

@johncarlosbaez@mathstodon.xyz replied · 5 hours ago

@RonJeffries - yeah, I wish I were smart enough to make a webpage of this talk where you could just click on things and hear them.

Here are some good fifths (3/2) and some "wolf fifths" (3/2 divided by the Pythagorean comma). Can you tell which is which?

https://www.youtube.com/shorts/7lWINJ9Dxxg

Wolf Fifth

YouTube

1

Ron Jeffries

@RonJeffries@mastodon.social replied · 4 hours ago

@johncarlosbaez
I assume the ones that hurt my ears are the wolf ones, but some seemed less awful than others. I'll browse some channels.

Good stuff, I'm delighted!

1

John Carlos Baez

@johncarlosbaez@mathstodon.xyz replied · 4 hours ago

@RonJeffries - I hope the ones that hurt your ears are the ones that hurt mine! The ones that hurt my hears are the wolf fifths. Instead of a frequency ratio of 1.5, they should have a ratio of about 1.48. That's enough off to be terrible.

the roamer

@the_roamer@mastodonapp.uk replied · 7 hours ago

@johncarlosbaez

Fascinating thread, deeply enjoyable. Thank you!

Vincenzo Tibullo

@enzotib@mastodon.uno replied · yesterday

@johncarlosbaez A very interesting post with a long thread of comments
https://mathstodon.xyz/@johncarlosbaez/115774416563837297

rtn

@rtn@chaos.social replied · 2 days ago

@johncarlosbaez Holy shit. This information is EXACTLY what I needed right NOW! Thank you John!

1

John Carlos Baez

@johncarlosbaez@mathstodon.xyz replied · yesterday

@rtn - thanks, that's great! You might like my blog articles about this stuff... or maybe it's too much.

For more on Pythagorean tuning, read this series:

https://johncarlosbaez.wordpress.com/2023/10/07/pythagorean-tuning/

For more on just intonation, read this series:

https://johncarlosbaez.wordpress.com/2023/10/30/just-intonation-part-1

For more on quarter-comma meantone tuning, read this series:

https://johncarlosbaez.wordpress.com/2023/12/13/quarter-comma-meantone-part-1/"

For more on equal temperament, read these:

https://johncarlosbaez.wordpress.com/2023/10/13/perfect-fifths-in-equal-tempered-scales/

Stylus

@stylus@social.afront.org replied · 2 days ago

@johncarlosbaez

Hi! A couple of thoughts from someone who is fascinated with musical tuning but in a fairly superficial way.

First, in looking at the hexagonal grid pages about 2/3 of the way through I wondered: You touch on scales with more than 12 tones, but only in the guise of equal temperament (29-TET). Would it be possible instead to develop a family of "just" intonations with 29 tones? Would it be described by a similar kind of grid, just with more elements, or would the shape be different?

Second, I went looking in OEIS for the list of "best approximation to 3/2" and found https://oeis.org/search?q=5%2c7%2c12%2c29%20id:A060528 -- but it goes 1,2,3,5,7,12,29,... and I can't figure out what difference between the two statements of the problem leads to 3 being/not being on the frontier of best approximations.

Finally, just an observation: It's unusual to see a discussion of tuning without an appeal to the overtone series somewhere. I assume this was a conscious decision.

5,7,12,29 id:A060528 - OEIS

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John Carlos Baez

@johncarlosbaez@mathstodon.xyz replied · yesterday

@stylus - people do indeed develop just intonation scales that are more complex than the historically important one I explain.

In Pythagorean tuning, all the frequency ratios are of the form

2ᵃ 3ᵇ

so this is also called 3-limit tuning.

In the just intonation I explain, they're all of the form

2ᵃ 3ᵇ 5ᶜ

This is 5-limit tuning.

People have gone on to study 7-limit tuning, 11-limit tuning, etc.

You're making me wonder if one of these, preferably involving not too many of these, could fit well with a 29-note scale. Maybe, as you suggest, 5-limit tuning or even 3-limit tuning could do the job!

This is a great math question, and I haven't seen anyone discuss it. But I've mainly been interested in the historically widely used systems. To explore the wild universe of mathematically possible systems, this is a good resource:

https://en.xen.wiki/w/Main_Page

Maybe the answer to your question is lurking in there.

"Finally, just an observation: It's unusual to see a discussion of tuning without an appeal to the overtone series somewhere. I assume this was a conscious decision."

Yes: my talk will be just 50 minutes long. After a run-through of basic harmony concepts to get the non-musical mathematicians in my audience up to speed, I think I'll just barely have time to explain Pythagorean tuning and just intonation. I want to explain them slowly and clearly enough so it sticks.

If I had two hours I could say a bit about *why* rational numbers are a big deal (this is more physics than math), and also delve into the wonderful world of quarter-comma meantone:

https://johncarlosbaez.wordpress.com/2023/12/13/quarter-comma-meantone-part-1/

I like to think someday I'll write a short book about this stuff.

timitii

@timitii@mementomori.social replied · 2 days ago

@johncarlosbaez i do NOT understand math or music, but still enjoyed reading all of this. Thanks :D

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John Carlos Baez

@johncarlosbaez@mathstodon.xyz replied · yesterday

@timitii - glad you liked it!

hambier

@hambier@mastodon.opencloud.lu replied · 2 days ago

@johncarlosbaez Hey that's great. I don't know the first thing about music theory. But one of the kids is learning an instrument and I've got a math background... This is very welcome, thanks! And happy holidays.

1

John Carlos Baez

@johncarlosbaez@mathstodon.xyz replied · 2 days ago

@hambier - great! Hope you and your kid enjoy it.

Vadim Zaliva

@vzaliva@mastodon.acm.org replied · 2 days ago

@johncarlosbaez For people curious about the subject, I want to mention the lovely 2023 ICFP FARM talk by Gloria Cheng https://youtu.be/k_htPyDDIwk?si=TpvGiQL1Nq-dE4Kl

[FARM'23] Keynote: Perfectly Imperfect: Music, Math and the Keyboard

YouTube

Vadim Zaliva

@vzaliva@mastodon.acm.org replied · 2 days ago

@johncarlosbaez Would love to see the video recording of the talk, if any.

Graeme 🏴󠁧󠁢󠁳󠁣󠁴󠁿

@pa27@mastodon.social replied · 2 days ago

@johncarlosbaez That's really good! Thanks!

Fragarach

@Fragarach@social.vivaldi.net replied · 2 days ago

@johncarlosbaez
Wow! Thanks for this, it's fascinating!

Walrus 🏴󠁧󠁢󠁷󠁬󠁳󠁿

@Walrus@toot.wales replied · 2 days ago

@johncarlosbaez

I'm sure you're well aware of all those other notes there are, that appear in, for example, Ottoman scales, Indian music, and so on...

https://www.doctor-dark.co.uk/rebetiko/hydra2004fri.html

Rebetiko - Hydra Conference 2004 Friday

Tony Finch

@fanf@mendeddrum.org replied · 2 days ago

@johncarlosbaez i like this web page https://aatishb.com/dissonance/ and this video https://youtu.be/tCsl6ZcY9ag which explain how harmony and dissonance are related to the overtones of notes

especially because that helps to explain how different tuning systems arise in other musical traditions: western instruments make sounds wih overtones on the harmonic series, but that isn’t true for bells or bars

https://aatishb.com/dissonance/

The Physics Of Dissonance

YouTube

Skewray Research

@skewray@mathstodon.xyz replied · 2 days ago

@johncarlosbaez (Typo in first post...missing the 'h' in 'http' for the slides link.)

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John Carlos Baez

@johncarlosbaez@mathstodon.xyz replied · 2 days ago

@skewray - thanks!

AIienated

@ghoti@infosec.exchange replied · 2 days ago

@johncarlosbaez Recently got curious about temperaments - long story related to Rob Reiner and "This Is Spinal Tap". Found some fascinating information including a standard for defining tunings called Scala - see https://huygens-fokker.org/scala/ if curious.

Scala Home Page

Weekend Editor

@weekend_editor@mathstodon.xyz replied · 2 days ago

@johncarlosbaez

But not Tmas?

The world is, after all, Minkowskian. That 4th time-like tree is essential, because otherwise how will you know when it's the holidays?

Marcus M. 滅拉マルクス🐧

@mocm@mastodon.social replied · 2 days ago

@johncarlosbaez I was expecting to see the 12th root of 2 somewhere in there.

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John Carlos Baez

@johncarlosbaez@mathstodon.xyz replied · 2 days ago

@mocm - patience, patience! If I were giving this talk out loud, I've given the first minute so far.

1

Marcus M. 滅拉マルクス🐧

@mocm@mastodon.social replied · 2 days ago

@johncarlosbaez Sorry

Hendi

@dg3hda@aschaffenburg.social replied · 2 days ago

@johncarlosbaez Math, music and time travel - nice!

Alexander Knochel

@Quantensalat@scicomm.xyz replied · 2 days ago

@johncarlosbaez One if my favorite topics. The impossibility of a "perfect" tuning has always fascinated me. Kind of like Gödel for musicians :)

John Carlos Baez

@johncarlosbaez@mathstodon.xyz replied · 2 days ago

If you look at a piano keyboard you'll see groups of 2 black notes alternating with groups of 3. So the pattern repeats after 5 black notes, but if you count you'll see there are also 7 white notes in this repetitive pattern. So: the pattern repeats each 12 notes.

Some people who never play the piano claim it would be easier if had all white keys, or simply white alternating with black. But in fact the pattern makes it easier to keep track of where you are - and it's not arbitrary, it's musically significant.

(2/n)

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Daniel Lakeland

@dlakelan@mastodon.sdf.org replied · 2 days ago

@johncarlosbaez
As a guy who went out of his way to create a Chromatic keyboard, I can tell you definitively, its easier to play improvised music, and with landmarks provided by blue tape in my case, but could be colors or textures, I dont think it'd be harder to play written.

On the chromatic keyboard, a major key always looks the same, its 3 successive notes from one row followed by 4 from the opposite row. This symmetry makes everything incredibly easier.

A chromatic MIDI keyboard. There is one black key between each adjacent white keys. Blue painters tape marks C,D,E in each octave as well as G which is a Black key, just as a landmark.

Fuse Views

@TimePencil@infosec.exchange replied · 2 days ago

@johncarlosbaez

88 keys on a grand piano keyboard.

88 constellations recognised by the IAU.

Coincidence?
Yeah... probably... unless there's some additional meaning to the phrase "music of the spheres"...

John Carlos Baez

@johncarlosbaez@mathstodon.xyz replied · 2 days ago

Starting at any note and going up 12 notes, we reach a note whose frequency is almost exactly double the one we started with. Other spacings correspond to other frequency ratios.

I don't want to overwhelm you with numbers. So I'm only showing you a few of the simplest and most important ratios. These are really worth remembering.

(3/ n)

3

pedro 🍉

@ubik@fedi.turbofish.cc replied · 2 days ago

@johncarlosbaez hey, really nice talk, so far! Just a small suggestion: I think it's worth spending some time explaining where those frequency ratios come from. Basically, how the harmonic series expands into infinity and all the notes we are used to are transpositions of the harmonics of C/Do. E.g. the octave is just n=2, the 5th n=3 and so on... As someone else also mentions in the comments, explaining the log nature of frequency perception is important in order to build that mathematical understanding.

1

John Carlos Baez

@johncarlosbaez@mathstodon.xyz replied · yesterday

@ubik - Alas, I won't have time to explain where the importance of frequency ratios comes from in my one-hour talk.... since my goal is to delve into the math of what you can *do* with those ratios.

But I hope someone in my audience of mathematicians asks about this. Psychophysically it's really about how the overtone series of one note interacts with that of another. So this guy was able to create a timbre that sounds dissonant at an interval of an octave!

https://www.youtube.com/watch?v=wg5QcF2akzQ

Can Octave Sound Dissonant?

YouTube

Jeremy Gibbons

@jer_gib@functional.cafe replied · 2 days ago

@johncarlosbaez Why only "almost" exactly double? I know that most intervals on the piano aren't perfect, but I thought that octaves were.

John Carlos Baez

@johncarlosbaez@mathstodon.xyz replied · 2 days ago

We give the notes letter names. This goes back at least to Boethius, the guy famous for writing The Consolations of Philosophy before he was tortured and killed at the order of Theodoric the Great. (Yeah, "Great".) Boethius was a counselor to Theodoric, but he really would have done better to stay out of politics - he was quite good at math and music theory.

Boethius may be the reason the lowest note on the piano is called A. We now repeat the names of the white notes as shown in the picture: seven white notes A,B,C,D,E,F,G and then it repeats.

[Whoop, Lisa is making me get up, make breakfast and go to the gym. I'll continue this later.]

(4/n)

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Martin Escardo

@MartinEscardo@mathstodon.xyz replied · 2 days ago

@johncarlosbaez writes "We give the notes letter names. [...] A,B,C,D,E,F,G"

Thanks for the nice thread.

I think I've already said this here a few years ago. I find it very difficult to get used to use "A,B,C,D,E,F,G" automatically without thinking. I grew up playing musical instruments with "do, re, mi, fa, sol, la, si" corresponding to "C,D,E,F,G,A,B".

Question. I know the origins of the two notational systems. But why does one start with "A/la" and the other with "do/C"?

You kind of mention this in (5/n).

But when did this actually occur?

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John Carlos Baez

@johncarlosbaez@mathstodon.xyz replied · 2 days ago

@MartinEscardo - in 5/n, I think I answered all your questions. The popularity of the Aeolian mode (now called the "minor scale") waned, and by very very roughly the early 1700s the Ionian mode (now called the "major scale") became dominant. A easy way to accommodate this shift was to start the scale on C rather than A.

(They could have also renamed C "A" at this point, but presumably that would have been incredibly confusing, especially since the shift took about a century.)

Jakob

@jdw@mathstodon.xyz replied · 2 days ago

@MartinEscardo @johncarlosbaez And then in German we refer to the note called *B* in English by the letter *H* 🥲

TobyBartels

@TobyBartels@mathstodon.xyz replied · 2 days ago

@johncarlosbaez

In the picture, you say that Boethius used the letters A through O for the notes that a theoretical male singer could sing. Since I and J were not distinguished in his time, this is 14 letters, making two octaves. Since he didn't repeat letters, it seems strange to stop at the high G and not continue with an extra high A to finish off the scale, but perhaps he thought of scales differently (or just did not have faith in his male singers).

1

John Carlos Baez

@johncarlosbaez@mathstodon.xyz replied · 2 days ago

@TobyBartels - interesting. There's always this tension in music between whether you end where you started or just before, e.g. 1 through 7 (the "heptatonic scale") versus 1 through 8 (the "octave"). Maybe Boethius was a 1 through 7 guy.

(In theory there's also the option of 0 through 7, but I guess Western music notation solidified before zero was fully accepted.)

John Carlos Baez

@johncarlosbaez@mathstodon.xyz replied · 2 days ago

So the scale used to start at A, using only white notes. But due to the irregular spacing of white notes, a scale of all white notes sounds different depending on where you start. Starting at A gives you the "minor scale" or "Aeolian mode", which sounds kinda sad. Now we often start at C, since that gives us the scale most people like best: the "major" scale.

(Good musicians start wherever they want, and get different sounds that way. But "C major" is like the vanilla ice cream of scales - now. It wasn't always this way.)

(5/n)

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John Carlos Baez

@johncarlosbaez@mathstodon.xyz replied · 2 days ago

From the late 1100s to about 1600 people called describe pitches that lie outside 7-tone system "musica ficta" ("false" or "fictitious") notes. But gradually these notes - the black keys on the piano if you're playing in C major - became more accepted.

To keep things simple for mathematicians, I'll usually denote these with the "flat" symbol, ♭. For example, G♭ is the black note one down from the white note G.

(Musicians really need both flats and sharps, and they'd also call G♭ something else: F♯. I'll actually need both G♭ and F♯ at some points in this talk!)

(6/n)

1

John Carlos Baez

@johncarlosbaez@mathstodon.xyz replied · 2 days ago

Since starting the scale with the letter C takes a little practice, I'll do it a different way that mathematicians may like better. I'll start with 1 and count up. Musicians put little hats on these numbers, and I'll do that.

For example, we'll call the fifth white note up the scale the "fifth" and write it as a 5 with a little hat.

(7/n)

2

1+ more replies (not shown)

John Carlos Baez

@johncarlosbaez@mathstodon.xyz replied · 2 days ago

Now for the math of tuning systems!

The big question is: how do we choose the frequency of each note? This is literally how many times per second the air vibrates, when we play that note.

Since 1850, by far the most common method for tuning keyboards has been "12-tone equal temperament". Here we divide each octave into 12 equal parts.

What do I mean by this, exactly? I mean that each note on the piano produces a sound that vibrates faster than the note directly below it by a factor of the 12th root of 2.

But we can contemplate "N-tone equal temperament" for N = 1, 2, 3, .... - and some people do use these other tuning systems!

(8/n)

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John Carlos Baez

@johncarlosbaez@mathstodon.xyz replied · 2 days ago

Here's a picture of the most popular modern tuning system: 12-tone equal temperament. As we march around clockwise, each note has a frequency of 2^{1/12} times the note directly before it.

When we go all the way around the circle, we've gone up an octave. That is, we've reached a frequency that's TWICE the one we started with.

But a note that's an octave higher sounds "the same, only higher". So in a funny way we're back where we started.

(9/n)

1

John Carlos Baez

@johncarlosbaez@mathstodon.xyz replied · 2 days ago

But now for a big question: why do we use a scale with 12 notes?

To start answering, notice that we actually use three scales: one with 5 notes (the black keys), one with 7 (the white keys) and one with 12 (all the keys).

As mathematicians we can notice a pattern here.

(10/n)

1

John Carlos Baez

@johncarlosbaez@mathstodon.xyz replied · 2 days ago

What's so good about scales with 5, 7 or 12 notes?

A crucial clue seems to be the "fifth". If you go up to the fifth white note here, its frequency is about 3/2 times the first. This is one of the simplest fractions, and it sounds incredibly simple and pure. So it's important. It's a dominant force in western music.

(11/n)

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John Carlos Baez

@johncarlosbaez@mathstodon.xyz replied · 2 days ago

We can make a chart to see how close an approximation to the fraction 3/2 we get in a scale with N equally spaced notes.

N = 5 does better than any scale with fewer notes!

N = 7 does better than any scale with fewer notes!

N = 12 does better than any scale with fewer notes! And it does *way* better. To beat it, we have to go all the way up to N = 29 - and even that is only slightly better.

(12/n)

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John Carlos Baez

@johncarlosbaez@mathstodon.xyz replied · 2 days ago

Here's a chart of how close we can get to a frequency ratio of 3/2 using N-tone equal temperament.

See how great 12-tone equal temperament is?

There are also some neat patterns. See the stripes of even numbers and stripes of odd numbers? That's not a coincidence. For more charts like this, and much more cool stuff along these lines, go here:

https://johncarlosbaez.wordpress.com/2023/10/19/perfect-fifths-in-equal-tempered-scales-part-2/

(13/n)

5

John Carlos Baez

@johncarlosbaez@mathstodon.xyz replied · 2 days ago

Here's the "star of fifths" in 12-tone equal temperament!

I'll quit here for now, and continue on Xmas.

(14/n)

4

John Carlos Baez

@johncarlosbaez@mathstodon.xyz replied · yesterday

Merry Xmas! My dream of the internet includes a lot of people giving each other gifts. Mastodon comes close to that. I'd like to thank everyone on Mastodon for all the you've gifts you've given each other over the year. Here are some presents in return....

Yesterday I talked about 12-tone equal temperament: the tuning system where we chop the octave into 12 equal parts.

This has been the most popular tuning system since maybe 1810, or definitely by 1850. But it's mathematically the most boring of the tuning systems that have dominated Western music since the Middle Ages. Now let's go back much earlier, to Pythagorean tuning.

When you chop the octave into 12 equal parts, the frequency ratios of all your notes are irrational numbers... *except* when you go up or down some number of octaves.

The Pythagoreans disliked irrational numbers. People even say they drowned Hippasus at sea after he proved that the square root of 2 is irrational! That's just a myth, but it illustrates how people connected Pythagoras to a love of rational numbers. In Pythagorean tuning, people wanted a lot of frequency ratios of 3/2.

In equal temperament, where we chop the octave into 12 equal parts, when we start at any note and go up 7 of these parts (a so-called "fifth"), we reach a note that vibrates about 1.4981 times as fast. That's close enough to 3/2 for most ears. But it's not the Pythagorean ideal!

As we'll see, seeking the Pythagorean ideal causes trouble. It will unleash the devil in music.

(15/n)

2

John Carlos Baez

@johncarlosbaez@mathstodon.xyz replied · yesterday

Start at some note and keep multiplying the frequency by 3/2, like a good Pythagorean. After doing this 12 times, you reach a note that's close to 7 octaves higher. But not exactly, since the 12th power of 3/2 is

129.746338

which is a bit more than

2⁷ = 128

The ratio of these two is called the 'Pythagorean comma':

p = (3/2)^12 / 2⁷ = 3¹² / 2¹⁹ ≈ 1.0136

This is like an unavoidable lump in the carpet when you use Pythagorean tuning.

It's good to stick the lump in your carpet under your couch. And it's good to stick the Pythagorean comma near the so-called 'tritone' - a very dissonant note that you'd tend to avoid in medieval music. This note is halfway around the circle of fifths.

In Pythagorean tuning, going 6 steps clockwise around the circle of fifths doesn't give you the same note as going 6 steps counterclockwise! We call one of them ♭5 and the other ♯4.

Their frequency ratio is the Pythagorean comma!

(16/n)

2

John Carlos Baez

@johncarlosbaez@mathstodon.xyz replied · yesterday

In equal temperament, the tritone is exactly halfway up the octave: 6 notes up. Since going up an octave doubles the frequency, going up a tritone multiplies the frequency by √2. It's no coincidence that this is the irrational number that got Hippasus in trouble.

In Pythagorean tuning, going 6 steps up the scale doesn't match jumping up an octave and then going 6 steps down. We call one of them ♭5 and the other ♯4. They're both decent approximations to √2, built from powers of 2 and 3.

Their frequency ratio is the Pythagorean comma!

(17/n)

1

John Carlos Baez

@johncarlosbaez@mathstodon.xyz replied · yesterday

The tritone is sometimes called 'diabolus in musica' - the devil in music. Some say this interval was actually *banned* by the Catholic church! But that's another myth.

It could have gotten its name because it sounds so dissonant - but mathematically, the 'devil' here is that the square root of 2 is irrational. If we're trying to use only numbers built from powers of 2 and 3, we have to arbitrarily choose one to approximate √2.

In Pythagorean tuning we can choose either

1024/729 ≈ 1.4047

called the sharped fourth, ♯4, or

729/512 ≈ 1.4238

called the flatted fifth, ♭5, to be our tritone. In this chart I've chosen the ♭5.

No matter which you choose, one of the fifths in the circle of fifths will be noticeably smaller than the rest. It's called the 'wolf fifth' because it howls like a wolf.

You can hear a wolf fifth here:

https://www.youtube.com/watch?v=7lWINJ9Dxxg

🐺

(18/n)

4

John Carlos Baez

@johncarlosbaez@mathstodon.xyz replied · yesterday

If you're playing medieval music, you can easily avoid the wolf fifth: just don't play one of the two fifths that contains the tritone!

A more practical problem concerns the 'third' - the third note in the scale. Ideally this vibrates 5/4 as fast as the first. But in Pythagorean tuning it vibrates 81/64 times as fast. That's annoyingly high!

Sure, 81/64 is a rational number. But it's not the really *simple* rational number our ears are hoping for when we hear a third.

Indeed, Pythagorean tuning punishes the ear with some very complicated fractions. The first, fourth, fifth and octave are great. But the rest of the notes are not. There's no way that 243/128 sounds better than an irrational number!

In the 1300s, when thirds were becoming more important in music, theorists embraced a beautiful solution to this problem, called 'just intonation'.

I'll explain this in some later installments. Maybe tomorrow. The British call it 'Boxing Day', because after Christmas with your relatives you sometimes feel like punching someone. But since I'm having a quiet day with Lisa, tomorrow I'll probably just explain just intonation.

(19/n)

3

John Carlos Baez

@johncarlosbaez@mathstodon.xyz replied · 11 hours ago

It's an amazing fact that in western composed music, harmony became important only around 1200 AD, when Perotin expanded the brand new use of two-part harmony to four-part harmony.

This put pressure on musicians to use a new tuning system - or rather, to revive an old tuning system. It's often called 'just intonation' (though experts will find that vague). We can get using a cool trick, though I doubt this is how it was originally discovered.

First, draw a hexagonal grid of notes. Put a note with frequency 1 in the middle. Label the other notes by saying that moving one step to the right multiplies the frequency of your note by 3/2, while going up and to the right multiplies it by 5/4.

(20/n)

1

John Carlos Baez

@johncarlosbaez@mathstodon.xyz replied · 11 hours ago

Next, cut out a portion of the grid to use for our scale. We use this particular parallelogram - you'll soon see what's so great about it.

(21/n)

1

John Carlos Baez

@johncarlosbaez@mathstodon.xyz replied · 11 hours ago

Now, multiply each number in our parallelogram by whatever power of 2 it takes to get a number between 1 and 2.

We do this because we want frequencies that lie within an octave, to be notes in a scale. Remember: if 1 is the note we started with, 2 is the note an octave up.

(22/n)

1

John Carlos Baez

@johncarlosbaez@mathstodon.xyz replied · 10 hours ago

Now we want to curl up our parallelogram to get a torus. If we do this, gluing together opposite edges, there will be exactly 12 numbers on our torus - just right for a scale!

There's a problem: the numbers at the corners are not all equal. But they're pretty close! And they're close to √2 - the frequency of the tritone, the 'devil in music'.

25/18 = 1.3888....
45/32 = 1.40625
64/45 = 1.4222....
36/25 = 1.44

So we'll just pick one.

(23/n)

1

John Carlos Baez

@johncarlosbaez@mathstodon.xyz replied · 10 hours ago

When we curl up our parallelogram to get a torus, there's also another problem. The numbers along the left edge aren't equal to the corresponding numbers at the right edge. But they're close! Each number at right is 81/80 times the corresponding number at left. I've drawn red lines connecting them.

So, we just one from each of these 4 pairs. We've already picked one number for all the corners, so we just need to do this for the remaining 2 pairs.

(24/n)

2

John Carlos Baez

@johncarlosbaez@mathstodon.xyz replied · 10 hours ago

So, here are the various choices for notes in our scale!

For the tritone we have 4 choices. That's okay because this note sucks anyway. That is: in Western music from the 1300s, it was considered very dissonant. So there's no obviously best choice of how it should sound.

For the 2 we have two choices, and for the ♭7 we also have 2 choices. So there's a total of 16 possible scales here.

Regardless of how we make our choices, we'll get really nice simple fractions for the 1, ♭3, 3, 4, 5 ,♭6, 6, and 8. And that makes this approach, called 'just intonation', really great!

(If you like math: notice the 'up-down symmetry' in this whole setup. For example the minor second is 16/15, but the reciprocal of that is 15/16, which is the seventh... at least after we double it to get a number between 1 and 2, getting 15/8.)

(25/n)

2

John Carlos Baez

@johncarlosbaez@mathstodon.xyz replied · 10 hours ago

Here's a chart of all possible just intonation scales: start at the top and take any route you want to the bottom. There are 16 possible routes.

A single step between notes in a 12-tone scale is called a 'semitone', since most white notes are two steps apart. In just intonation the semitones come in 4 different sizes, which is kind of annoying.

Notice that if we choose our route cleverly, we can completely avoid the large diatonic semitone. Or, we can avoid the greater chromatic semitone. But we can't avoid both. So, we can get a scale with just 3 sizes of semitone, but not fewer.

How should we choose???

(26/n)

1

John Carlos Baez

@johncarlosbaez@mathstodon.xyz replied · 9 hours ago

This is the most commonly used form of just intonation, I think. It has a couple of nice features:

1) It has up-down symmetry except right next to the tritone in the middle, where this symmetry is impossible.

2) It uses 9/8 for the second rather than 10/9, which is a bit nicer: a simpler fraction.

3) It completely avoids the large diatonic semitone, which is the largest possible semitone.

These don’t single out this one scale. I’d like to find some nice features that only this one of the 16 possibilities has.

But let’s see what this scale looks like on the keyboard!

(27/n)

2

John Carlos Baez

@johncarlosbaez@mathstodon.xyz replied · 9 hours ago

Here's the most common scale in just intonation!

The white notes are perhaps the most important here, since those give the 'major scale'. The fractions there are beautifully simple.

Well, the second (9/8) and seventh (15/8) are not so simple. But that's to be expected, since these notes are the most dissonant! Of these, the seventh was more important in the music of the 1300s, and even today. It's called the 'leading-tone', because we often play it right near the end of a piece of music, or a passage within a piece of music, and its dissonance leads us up to the octave, with a tremendous sense of relief.

(28/n)

1

John Carlos Baez

@johncarlosbaez@mathstodon.xyz replied · 9 hours ago

Here's the really great thing about the white notes in just intonation. They form three groups, each with frequencies in the ratios

1 : 5/4 : 3/2

This pattern is called a 'major triad' and it's absolutely fundamental to music - perhaps not so much in the 1300s, but certainly as music unfolded later. Major triads became the bread and butter of music, and still are.

The fact that every white note - every note in the 7-note 'major scale' - lies in a mathematically perfect major triad is a GIGANTIC feature in favor of just intonation.

(29/n)

1

John Carlos Baez

@johncarlosbaez@mathstodon.xyz replied · 9 hours ago

But let's take a final peek at the dark underbelly of just intonation: the tritone. As I mentioned, there are four choices of tritone. You can divide them into two pairs that are separated by a ratio of 81/80, or two pairs separated by a ratio of 128/125.

These numbers are fundamental glitches in the fabric of music. They have names! People have been thinking about them at least since Boethius around 500 AD, but probably earlier.

• The 'syntonic comma', 81/80, is all about trying to approximate a power of 3 by products of 2's and 5's.

• The 'lesser diesis', 128/125, is all about trying to approximate powers of 2 by powers of 5.

If these numbers were 1, music would be beautiful in a very simple way. But reality cannot be wished away.

And as we'll see, these numbers are lurking in the spacing between notes in just intonation - not just near the tritone, but everywhere!

(30/n)

1

Fedor Indutny

@indutny@mean.engineer replied · 2 days ago

@johncarlosbaez G# always sounds so tasty to me!

Edgar Chavez

@edgarchavez@mas.to replied · 2 days ago

@johncarlosbaez

I was pickled by curiosity because in Spanish, and other Romance languages, the notes have other names (Do, Re, Mí, Fa, Sol, La , Si). I found they were introduced for singing training using the Hymn to St. John the Baptist, "Ut queant laxis". They were the first syllables of the verses (except Ut and Sa which were changed to Do and Si for being easier to uterate).

We name the singing training "solfeo".

Nothing to do with math, sorry for the diversion, thank you for the post.

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TobyBartels

@TobyBartels@mathstodon.xyz replied · 2 days ago

@edgarchavez @johncarlosbaez

We use those names in English as well, except with Ti instead of Si. (If you know your English-language musicals, then you may have heard them used in The Sound of Music.) But the difference between these and the letters is that the letter notes are at (or at least near) specific frequencies, while solfege (as we call it) is relative. That is, A is always about 440 Hz (modulo a power of 2), but La could be anywhere (with the others arranged at the same relative frequencies).

Aditya Khanna

@adityakhanna@mathstodon.xyz replied · 2 days ago

@edgarchavez @johncarlosbaez in Hindustani classical, the notes are {sa, re, ga, ma, pa, dha, ni} and the alphabet is called "sargam", for the first 4 names