Discussion

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🐧

@[email protected] · 7 months ago

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

@[email protected] · 7 months ago

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

Marcus M. 滅拉マルクス🐧

@[email protected] · 7 months ago

@johncarlosbaez Sorry

Hendi

@[email protected] · 7 months ago

@johncarlosbaez Math, music and time travel - nice!

Alexander Knochel

@[email protected] · 7 months ago

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

@[email protected] · 7 months 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)

@[email protected] · 7 months 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

@[email protected] · 7 months ago

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"...

@[email protected] · 7 months 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)

pedro 🍉

@[email protected] · 7 months 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.

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

@[email protected] · 7 months ago

@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!

Can Octave Sound Dissonant?

YouTube

Jeremy Gibbons

@[email protected] · 7 months ago

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

@[email protected] · 7 months 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)

Martin Escardo

@[email protected] · 7 months 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?

@[email protected] · 7 months 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

@[email protected] · 7 months ago

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

TobyBartels

@[email protected] · 7 months ago

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).

@[email protected] · 7 months 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.)

@[email protected] · 7 months 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)

@[email protected] · 7 months 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)

@[email protected] · 7 months 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)

Counting Is Hard

@[email protected] · 7 months ago

@johncarlosbaez Your use of only flats has prompted one of those niggling annoyances back into my mind: why is the flat 2 called the minor 2nd when it doesn't appear in any of the three "normal" minor scales.

I assume the answer involves the phrygian mode

Paul Schwahn

@[email protected] · 7 months ago

@counting_is_hard @johncarlosbaez I think the question should rather be: why are these particular scales called "major" and "minor" in English? This is not the case in every language.

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

@[email protected] · 7 months ago

@counting_is_hard's question is not hard to answer. Musicians DON'T usually call ♭2 the 'minor second'. They call it the 'diminished second'. Similarly the ♭5 is the 'diminished fifth' and ♯4 is the 'augmented fourth'.

@pschwahn wrote:

"I think the question should rather be: why are these particular scales called "major" and "minor" in English? This is not the case in every language."

What are they called in some other language?

'Major' and 'minor' sound perfect to me, by the way, since the three main minor keys each have some notes lower than the corresponding notes in the major key.

(These three minor keys are natural minor, harmonic minor and melodic minor.)

Diminished second - Wikipedia

Glerk Pizza

@[email protected] · 7 months ago

@johncarlosbaez No, thats not right - The diminished second is ♭♭2 (as is mentioned in the wikipedia article you linked), so it's enharmonic to the 1 (in 12TET at least). Musicians DO indeed call♭2 the minor second. (the rest is correct, i think)

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

@counting_is_hard @pschwahn

Semitone - Wikipedia

TobyBartels

@[email protected] · 7 months ago

@johncarlosbaez @counting_is_hard @pschwahn

I was taught to call it the minor second, not the diminished second. The Wikipedia page that you linked agrees with me, calling the diminished second ‘enharmonically equivalent to a perfect unison’ in equal temperament.

But the question isn't (or at least shouldn't be) whether ‘major’ and ‘minor’ are good terms for these two modes once you've picked them out, but why the two terms should be used for *those* two modes. The truly minor mode is phrygian.

@[email protected] · 7 months ago

@TobyBartels @counting_is_hard @pschwahn - oh whoops, yikes, I take it back!!! So diminished second is one semitone lower than minor second. I should have known that.

I will not tell y'all what "the question" should be. There are lots of good questions, and each one should be itself.

Ulrik Buchholtz

@[email protected] · 7 months ago

@johncarlosbaez “What are they called in some other language?”
In Latin they are called “duris” and “mollis”(hard and soft), which survives into German as “Dur/Moll” (and similar in Danish “dur/mol” and other related languages). @counting_is_hard @pschwahn

@[email protected] · 7 months ago

@buchholtz @counting_is_hard @pschwahn - okay, thanks! Now I remember I've heard Bach's mass in H-moll, which features two ways in which German talks about music differently than English.

Jakob

@[email protected] · 7 months ago

@johncarlosbaez @buchholtz @counting_is_hard @pschwahn And a minor (pun intended) nitpick: It's not H-moll but h-Moll 😅 In H-Dur the H is capitalzed.

https://de.wikipedia.org/wiki/Doppelkreuz_%28Notenschrift%29?wprov=sfla1

@[email protected] · 7 months ago

@jdw @buchholtz @counting_is_hard @pschwahn - oh, so it's a "minor" h in h-Moll. 😆

Is it really true, as I just read on Wikipedia, that in German B flat is "B" while B and B sharp are "H"? Why? And why isn't H after G?

Jakob

@[email protected] · 7 months ago

@johncarlosbaez @buchholtz @counting_is_hard @pschwahn Yes, the English -> German translations are:
B -> H
B flat -> B
B sharp -> His (on a well-tempered piano this is the same as C)

It's also interesting that while (except for H, the flat version of which is B) the flat version of a note is indicated with the -es ending (C flat -> Ces), the flat sign ♭ itself is called b in German 😅 I guess the sign ♭ is also a stylized b. So for example you would say that the B-Dur (B flat major) scale has two b, namely B (B flat in English) and Es (E flat in English). https://de.wikipedia.org/wiki/B_%28Notenschrift%29?wprov=sfla1

So the name B for the English H-flat makes a little bit of sense, because it is somehow the most canonical flat note (the first one in the circle of quints, appearing already in the F-major scale).

I hope this wasn't too confusing 😅 I don't know about the history unfortunately.

Jakob

@[email protected] · 7 months ago

@johncarlosbaez @buchholtz @counting_is_hard @pschwahn And then we Germans say Cis/Ces for C-sharp/C-flat etc. And if we put two sharp-signs in front of C we say Cis-is (I'm not sure how that's called in English).

@[email protected] · 7 months 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)

@[email protected] · 7 months ago

@dlakelan @johncarlosbaez

What's the relevance of 1850 here? I know the equal division of the octave was invented in the 1500's but of course wasn't adopted immediately. did something happen in 1850 that really "killed" other tuning systems? Hoping you have some interesting history reference there.

Dannyboyguyman

@[email protected] · 7 months ago

The wiki page suggest this 12 TET became prevalent somewhat earlier...

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

Equal temperament - Wikipedia

@[email protected] · 7 months ago

@pinsk @johncarlosbaez

1584-1585 for the invention of the system and calculation of the intervals... But I think the tuning system became widely adopted because of fretted stringed instruments (lutes and guitars) and I'm not clear on when that really got going. The article doesn't make it super clear either, though I think it's earlier than the mid 1850s? Guitars have been around since like forever (lutes for well over 3000 years) but equal tempered frets must have been post 1585

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

@[email protected] · 7 months ago

@pinsk @johncarlosbaez

It wouldn't surprise me if the guitar transition took maybe a hundred years or more and might have started in the 1600's or 1700's. Unlike keyboards, changing spacing between frets on a guitar wouldn't feel very "weird" and guitars are a common microtonal instrument today:

That video suggests microtonal guitars were first made in 1840's or so

History of the Microtonal Guitar

YouTube

:uffda: Kacey

@[email protected] · 7 months ago

It's probably beyond the scope of your talk, but there's a bit more nuance than that, since pianos are typically tuned with stretched octaves ( https://en.wikipedia.org/wiki/Stretched_tuning ) that are a little bit wider than a mathematically perfect octave. This is because our experience of dissonance depends on the physical harmonics rather than idealized mathematical harmonics ( https://www.youtube.com/watch?v=tCsl6ZcY9ag ) and overtones are a bit sharper than they "should" be due to string tension

The Physics Of Dissonance

YouTube

Stretched tuning - Wikipedia

@[email protected] · 7 months 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)

@[email protected] · 7 months 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)

Speculations. Nothing new just held up at a different angle than what you’ve seen before. I want to write about something interesting you can do with a single pentatonic scale. I’m just going to begin with what I noticed that got me thinking about this.

@[email protected] · 7 months 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)

graeme

@[email protected] · 7 months ago

@johncarlosbaez I've been thinking about this question for a while, I think ever since I read the this week's finds post that pointed out the correspondences between music theory and group theory. In particular one thing I got interested in is just how quickly you can generate all 12 tones from a single pentatonic scale, and what doing this actually looks like on an instrument, rather than on paper. I wrote about it here, you might be interested https://graemephi.github.io/posts/twelve-tones/

Twelve tones are inescapable

@[email protected] · 7 months ago

@graeme - nice, sorry to take a while to respond but this is really cool! I'm gonna read it carefully.

Tom Forsyth

@[email protected] · 7 months ago

@johncarlosbaez Nit-pick - a "fifth" goes up FOUR notes. Because... uh... reasons. Bad ones. Same bad reason the thing is called an "octave" even though it's only got 7 notes in it.

@[email protected] · 7 months ago

@TomF - yeah, I really meant "go up to the fifth" in the sense shown below. This sort of terminology was pretty much standardized before people accepted the number zero. If I start trying to reform music terminology next I'll have to get started on astronomy.

Tom Forsyth

@[email protected] · 7 months ago

@johncarlosbaez First you have to fix floor numbering in the USA!

@[email protected] · 7 months ago

@TomF - I still get confused here in the UK, though in theory I like starting at zero.

I changed my post a bit. I'd been trying to avoid the term "fifth", which I hadn't defined. (I'm definitely *not* going to say "go up four notes" - that'll make everyone start thinking about this stuff, which is not the point here.)

glangmead

@[email protected] · 7 months ago

@johncarlosbaez I get it!

@[email protected] · 7 months ago

@glangmead - 😉

@[email protected] · 7 months 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)

jan Ki | 奇 :nonbinary_flag:

@[email protected] · 7 months ago

@johncarlosbaez
31edo ftw!

Chip Unicorn

@[email protected] · 7 months ago

@johncarlosbaez FYI --

The scale made up of the five black notes on the keyboard isn't the same as the scale that would happen if we divided the octave perfectly into five notes.

Same thing with 7 white notes.

(You can see this easily with the white notes -- the difference between "E" and "F" sounds very different from the difference between "F" and "G".)

@[email protected] · 7 months ago

@Chip_Unicorn - Indeed, the 7 white notes on a piano aren't equally space, nor are the 5 black notes. So these 7- and 5-note scales are only rough approximations to 7-TET and 5-TET. And yet they're close enough that I believe they inherit some of their good qualities.

wlmb

@[email protected] · 7 months ago

@johncarlosbaez
I made a similar chart to explain why 12 notes (or 2, 5, 7) in a scale,, but before introducing equal temperament, which came afterwards: how many fifths till you return approx. to the starting note. In 'Usted es desafinado' (you're out of tune), in Spanish, https://acmor.org/articulos-anteriores/usted-es-desafinado-parte-1
https://acmor.org/articulos-anteriores/usted-es-desafinado-parte-2-de-2 Regards

Ver | Usted es desafinado (parte 2 de 2)

| Academia de Ciencias de Morelos, A.C

Ver | Usted es desafinado parte 1

| Academia de Ciencias de Morelos, A.C

https://www.youtube.com/watch?v=g6w3a2v_50U

@[email protected] · 7 months ago

@wlmb - nice! "Desafinado" makes me think of this:

Desafinado by Joao Gilberto

YouTube

wlmb

@[email protected] · 7 months ago

@johncarlosbaez 😊 Great song

andeux

@[email protected] · 7 months ago

@johncarlosbaez
So more or less what we get from the continued fraction sequence for log2(3/2), which can be seen in OEIS A254351 and A060528.

("More or less" because 2/3 is slightly closer to log2(3/2) than 1/2 is, but 2^(2/3) is slightly farther from 3/2 than 2^(1/2) is, and you are using the latter criterion.)

@[email protected] · 7 months ago

@andeux - aha, so *that's* why they do it differently. I think their way is better since I'm taking a difference of powers of numbers and it's better to take a ratio of powers, or a difference of their logarithms.

Aditya Khanna

@[email protected] · 7 months ago

@johncarlosbaez oh that is such a cool explanation of finding the "minimum of a sliding window". Superabundant numbers are defined similarly but the ratio being extremized is related to number of factors.

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

@[email protected] · 7 months 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:

(13/n)

@[email protected] · 7 months ago

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

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

(14/n)

telephon

@[email protected] · 7 months ago

@johncarlosbaez in the thread you asked about further literature on the topic. A composer who has written on this is late Clarence Barlow (https://clarlow.org/texts/) in his book on Musiquantics. A theorist who is very involved in the topic is William Sethares (the book is Tuning, Timbre, Spectrum, Scale). The main takeaway from these theories is that the consonance of scales depends on the spectral structure of the basic sound. The aesthetics of timbre and scale are culture specific.

@[email protected] · 7 months ago

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)

Rob Hughes

@[email protected] · 7 months ago

@johncarlosbaez This is a fascinating thread!

While my college choir was rehearsing some early music, the conductor said, "Don't sing the tempered scale. It's the downfall to the devil."

@[email protected] · 7 months ago

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)

@[email protected] · 7 months ago

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)

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

@[email protected] · 7 months ago

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:

🐺

(18/n)

Janne Moren

@[email protected] · 7 months ago

@johncarlosbaez
Probably a result of our very varied musical landscape these days, but to me that interval sounds *spicy* , not bad. Something you'd save up for a particularly interesting transition or thematic mood change in a piece.

Tony Vladusich

@[email protected] · 7 months ago

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

Do you know about the tritone paradox?

Tritone paradox - Wikipedia

https://johncarlosbaez.wordpress.com/2024/07/27/tritone-substitutions/

@[email protected] · 7 months ago

@troy_s @TonyVladusich - no, I hadn't heard of the tritone paradox. Thanks! I wonder what kind of person I am. Is there a YouTube video or website that will play the Shephard tone, so I can find out?

Nor have I heard of sine wave speech, nor the thermal grill illusion.

I'm still trying to wrap my head around tritone substitution.

troy_s :praxis_100:

@[email protected] · 7 months ago

@TonyVladusich Nice! Another example of the multistability of audio. Similar to Sine Wave Speech! And Thermal Grill illusion.

Dan Riley

@[email protected] · 7 months ago

@johncarlosbaez The Fisk-Nanney organ at Stanford Memorial Chapel is an interesting instrument, with both fifth-comma and well-tempered tunings (in a pipe organ!). The “Brustpositive and the Brustpedalia are fixed in meantone and offer two sub-semitones, or split sharps, per octave, D sharp/E flat and G sharp/A flat”

https://cbfisk.com/opus/opus-85/

https://orsl.stanford.edu/memorial-church-organs

(My Mom, a student of Herb Nanney, loved playing this organ.)

Office for Religious & Spiritual Life

Memorial Church Organs

https://cbfisk.com

Opus 85

https://www.concertina.net/forums/index.php?%2Ftopic%2F21123-my-experience-with-15-comma-mean-tone-tuning%2F

@[email protected] · 7 months ago

@dan131riley - Wow! That sounds really interesting to play. So there were separate rows of keys for the two tuning systems?

Fifth-comma meantone is new to me: I've spent a lot of time studying quarter-comma, and I've see third-comma and sixth-comma. But I now see someone has blogged positively about fifth-comma:

"Well-tempered" raises the question: which of the many well-tempered systems?

@[email protected] · 7 months ago

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)

@[email protected] · 7 months 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)

@[email protected] · 7 months 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)

@[email protected] · 7 months 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)

@[email protected] · 7 months 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)

@[email protected] · 7 months 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)

julesh

@[email protected] · 7 months ago

@johncarlosbaez A discrete manifold equipped with an approximate scalar field? That sounds like it could be useful for some other stuff too

@[email protected] · 7 months 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)

Andrew Gallagher

@[email protected] · 7 months ago

@johncarlosbaez typo? 65/45 -> 64/45 😬

@[email protected] · 7 months ago

@andrewg - definitely a typo, the number 65 has no place in this game! I'll fix it! Thanks. This will spare me some embarassment at my talk.

@[email protected] · 7 months 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)

@[email protected] · 7 months 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)

su

@[email protected] · 7 months ago

@johncarlosbaez doesn't step 7 and 8 have the same kind of semitone next to each other?

@[email protected] · 7 months ago

@semnosao - If you mean the steps 7-8 and 8-1, yes you're right! The second one takes us above the scale I was considering. So yeah. I wasn't thinking about going past the 8. I was just looking at the picture here.

I don't really think this business of never having two notes of the same size next to each other is musically important. I was just trying to find some properties that pick out the most popular of the 16 just intonation scales. Maybe there's a better way to do it.

su

@[email protected] · 7 months ago

@johncarlosbaez and b5-5 -> 5-b6 are both diatonic semitone

@[email protected] · 7 months ago

@semnosao - okay, so my idea was just crap! 😆

So, we'd need some other theory to justify this particular scale.

@[email protected] · 7 months 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)

@[email protected] · 7 months 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)

Kernel Bob

@[email protected] · 7 months ago

@johncarlosbaez The I, IV, and V triads, aka tonic, subdominant, and dominant. This explains, to me, why those chords dominated (ahem) harmony at the end of the just intonation era. Thanks!

@[email protected] · 7 months 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)

Heribert Schütz

@[email protected] · 6 months ago

@johncarlosbaez
I always felt that the gap between a "kilobyte" of data in memory (actually a kibibyte = 1024 bytes) and a true kilobyte (1000 bytes) of data going over a network is like a comma in music. But I was only aware of the Pythagorean comma. Now I learned that the ratio 1024/1000 = 128/125 also has a name: "lesser diesis" (and according to Wikipedia also "enharmonic comma" or "augmented comma").

@[email protected] · 6 months ago

@hcschuetz - wow, that's really cool. So it's something people have been running into for millennia when seeking nontrivial solutions of

2ⁿ5ᵐ = 1

(namely 2⁷5⁻³ comes close).

@[email protected] · 7 months ago

Look! The four kinds of semitone in just intonation are related by the lesser diesis and syntonic comma!

In this chart, adding vectors corresponds to multiplying numbers. For example, the green arrow followed by the red one gives the dark blue one, so

25/24 × 81/80 = 135/138

Or in music terminology: the lesser chromatic semitone times the syntonic comma is the greater chromatic semitone.

And so on.

The parallelogram here is secretly related to the parallelogram we curled up to get the just intonation scale. Think about it! Music holds many mysteries.

(31/n)

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

@[email protected] · 7 months ago

Just intonation is great if you're playing in just one "key", always ending each passage with the note I've been calling 1. But when people started trying to "change keys", musicians were pressed into other tuning systems.

This is a long story, which I don't have time to tell right now. If you're curious, read my blog articles about it:

For more on Pythagorean tuning, read this:

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 well-tempered scales, read this series:

https://johncarlosbaez.wordpress.com/2024/01/11/well-temperaments-part-1/

And for more on equal temperament, read this series:

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

It's sad in a way that this historical development winds up with equal temperament: the most *boring* of all the systems, which is equally good, and thus equally bad, in every key. But the history of music is not done, and computers make it vastly easier than ever before to explore tuning systems.

What will come next? It's up to us. I hope next year you explore more of the wonders of music.

(32/n, n = 32)

Konrad Hinsen

@[email protected] · 7 months ago

@johncarlosbaez I wonder if (and to what degree) the math has influenced musical preferences in "the West".

Elsewhere, there are musical scales in which some notes don't fit into any nice mathematical scheme. E.g. the mode "rast" in Arabic music has two "off" notes (sometimes called e-half-flat and h-half-flat but they aren't really halves). I have been told by music historians that similar scales were used in Europe in the past, but given up in the move to polyphony. Did math play a role?

downbeatdan

@[email protected] · 7 months ago

@johncarlosbaez Thank you for this! Unpacks music for me a little more... Can I ask if you can back up the truck a little, and explain WHY we have to use a power term to approximate the harmonic fraction we are aiming for for each of the notes? Is it because it's a series, and we have to continue it on?

julesh

@[email protected] · 7 months ago

@johncarlosbaez This is definitely true for keyboard instruments and most ensembles, but most non-keyboard instruments have a "natural" key (eg. G for violin, which causes the lowest string to resonate on the fundamental and the next string to resonant on the 5th), and I reckon some very pedantic people playing solo instrumental music must tune a few cents off the standard tuning to match that

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

@[email protected] · 7 months ago

@julesh - Yes, I've been taking a perhaps overly keyboard-centric approach in this talk. (It makes sense because you can't do everything in an hour, and I play the keyboard, as do a lot of theorists.) Violinists live in another universe. A good violinist will apparently use equal-tempered, Pythagorean tuning or just intonation depending on context!

I saw a good video about this yesterday. I find it a bit mind-boggling and can't help but wonder if they're faking it. But there are actually lots of videos about this, so if they're faking it they do so en masse.

Michael Newton

@mavnn · 7 months ago

@[email protected] @[email protected] see also guitarists who play fretless guitars and adjust tuning on the fly. Humans do some pretty amazing things on occasion.

Zack Weinberg

@[email protected] · 7 months ago

@johncarlosbaez Do you know if anyone has ever done much with, er, *not* rolling the parallelogram up into a torus? Instead continuing the sequences of exact 3/2 and 5/4 steps indefinitely, and accepting that you will never get back exactly to an octave? Exact 1:2 ratios could still be used for harmonies within chords.

Jencen

@[email protected] · 7 months ago

@johncarlosbaez looking forward to more on this.
My partner restored the Ellis Harmonical from Bate collection in Oxford. Intended to be an instrument to demonstrate Just Intonation.

Alexander Knochel

@[email protected] · 7 months ago

@johncarlosbaez looking forward to the just intonation desserts

Grégoire Locqueville

@[email protected] · 7 months ago

@johncarlosbaez "lump in the carpet" is a great analogy

Kinene⭐🐻

@[email protected] · 7 months ago

@johncarlosbaez Early tunings (up to 1600±) were based on the concept of pure intervals; perfect 5ths, to start with. The documented history comes from European church music; early folk music may have been different, but there is no way of knowing. But when keyboard instruments became usable, around 1500, tuning became a problem. Working through a circle of perfect 5ths to tune an early harpsichord or organ produced the Pythagorean comma: Starting on middle C and tuning 7 perfect 5ths, the ending C is not a perfect octave. If you use a system of perfect 3rds, then you wind up dealing with the syntonic comma.

So various tempered systems developed to work around the comma. <insert ½ semester on early keyboard tunings> I have tuned instruments in meantone and Werkmeister III systems. The idea of everything except octaves being "not pure" was a compromise to allow music to be played in any key.

The piano mechanism we know was developed by Erard in 1821, and improved through the early 19th century. In the 1840s, builders added a heavy iron frame to support high tension strings. <insert section on the history of keyboard construction>

The final decision to use A440 as a standard was only settled upon in 1939. Early music performers sometimes use A415 or A435.

https://www.hpschd.nu/index.html?nav/nav-4.html&t/welcome.html&https://www.hpschd.nu/tech/tmp/kirnberger.html

@[email protected] · 7 months ago

@c_merriweather - My talk will get bogged down in the delightful intricacies of just intonation - I'm giving a talk in the math department and I think they'll like that - so it'll never reach quarter-comma meantone or well temperaments. But I had a lot of fun studying those, and should probably return to the latter.

I learned a lot from the website of the harpsichord maker Carey Beebe - maybe you know him:

He offers distilled descriptions of different tuning systems in a way that a mathematician like myself can quickly absorb.

Kinene⭐🐻

@[email protected] · 7 months ago

@johncarlosbaez All the math comes later, the people at the time tuned instruments by ear, as have I. I even built a monochord to illustrate the fractions involved in the overtone series.

The perfect intervals were in many ways easier to tune. And if in the proper key, the keyboards were able to match the singers, who were, and are, not limited by fixed tuning. To hear medieval and Renaissance music performed with historical tunings is amazing in its clarity.

Christoph Maier

@[email protected] · 7 months ago

@johncarlosbaez
Maybe you should stay pure,
but only go two fifth up and two fifths down, i.e.,
{♭7, 4, 1, 5, 2}
Kind of weird pentagram, that.

dingodog

@[email protected] · 7 months ago

@johncarlosbaez
Once you have settled on 12, there is a much more mathematical way to look at scales and modes.
Just count the number of half steps, then consider where they land.
The minimum number of half-steps is 2. There are 3 distinct ways we can put 2 half-steps in the scale: (1 is a half step, 2 is a whole)
1122222 (call this A)
1212222 (B)
1221222 (C)

You can also pick your starting note. So in this notation, regular major (Ionian) is C1, starting with the first (zero-based) interval.
Locrian is C0,
Dorian C2,
Phrygian C3,
Lydian C4,
Mixolydian C5,
Aeolian C6.

It seems like A and B are much rarer. Neopolitan Major is A1, Lydia Minor is A4, Locrian Major is A5, Super Locrian is B0.

And the Melodic Minor is B when ascending, C when descending.

Then, of course, you can add more half steps.
The Blues scale is 12212112. And trivially the chromatic scale is 111111111111.

Anyway that's how I would have defined everything if I had gotten to decide. 😂

Den of Earth

@[email protected] · 7 months ago

@johncarlosbaez
Found this...
https://youtu.be/3O5J6A4uh30?si=cqE0yt1Nr3BB5GkG

Piano TEMPERAMENT comparison.

YouTube

@rozeboosje @johncarlosbaez

@[email protected] · 7 months ago

@DenOfEarth - Hey, I had seen this and was looking for it. Thanks! I'm going to use it in my talk. Its sample of "just major" (the second one) is completely unfair - it's using this tuning system in a way nobody was ever meant to use it, so it sounds awful. But I'm hoping my audience will at least notice that it sounds different from "equal temperament" (the first one).

Den of Earth

@[email protected] · 7 months ago

@johncarlosbaez
I'm guessing this is being played on an electronic piano and the person is just going through the menu options one by one. If it was done on an acoustic piano manually retuned for two or three different temperaments, I wager it would sound much better.

Pino Carafa

@[email protected] · 7 months ago

@johncarlosbaez I don't know much about music but I am guessing these issues arise because we need to divide octaves into individual frequencies on the basis that this is necessary to create, for example, "keys" on instruments like pianos. You can't have an infinite number of keys in an octave on a piano.

But what about producing music through a computer? How the notation for such a thing would work, I have no clue but in theory I imagine every "chord" could be perfect?

Charles 𝄢 H

@[email protected] · 7 months ago

A lot of computer music doesn't exactly have notation. Instead it has a user interface.

Let's say you have a computer and want to make a very simple chord over A. You can go look for the frequency of the 5th note in the scale or you could just multiply 440 by 3/2. This direct use of fractions is called "just intonation".

The problem with doing this is that the third note in the scale in 12tet is way off from the ideal just intoned fraction. So if you have a perfectly in tune chord on C and then your melody goes to E, that E is going to sound really wrong.

So you could also just make all the notes in the scale just intoned, but if your music goes onto a key change, it can also sound off, unless you retune all your notes to be the just notes relative to the new key. All your As are now a new frequency. This can sound great, but its fiddly.

Changing all the notes in every chord relative to the bottom note in the chord also often sounds wrong and kind of unmoored. Its often better to stick with the scale that belongs to your key because the notes are more stable and you can more easily tell when you're getting to an important note.

There are many composers who write this way but its extra work so most people stick with 12tet.

Weekend Editor

@[email protected] · 7 months ago

One of the things that was hardest for me to get my head wrapped around was that we perceive log frequency. Hence the octave is a doubling of frequency, and everything else, where people talk about "adding a semi-tone" is actually *multiplying* by something. (In modern scales, multiplying by a power of 2^(1/12).)

Music theorists are not a mathematical tribe, and for many years I just *could* *not* understand the word salad they wrote about this!

But then some book or other whose title involved "music" and "physicists" explained it all with just a couple equations.

@[email protected] · 7 months ago

@weekend_editor - I hope that my talk will make sense to mathematicians. When I actually give it, I'll probably find that some things that seem obvious to me aren't obvious to others.

I had a lot of trouble explaining to my sister the concepts of "note" and "key". They sound obvious until you try to explain them - both those words mean at least two things.

Fedor Indutny

@[email protected] · 7 months ago

@johncarlosbaez G# always sounds so tasty to me!

Edgar Chavez

@[email protected] · 7 months ago