Philosophy
TheoryKamil Szczerba

What is beauty? Can we mea­sure it? If so, how?

When pon­der­ing such ver­tig­i­nous ques­tions, it is al­ways use­ful to turn to the mas­ters. Aquinas, fol­low­ing Aris­to­tle, de­fines beauty by:

1. clar­ity. A one-line math­e­mat­i­cal proof is more el­e­gant than an equiv­a­lent one re­quir­ing two pages.
2. pro­por­tion/​har­mony. Mu­si­cal chords are a per­fect ex­am­ple of this. Some are pleas­ant; oth­ers, jar­ring. More be­low.
3. in­tegrity. A church look­ing like a church is more beau­ti­ful than a church look­ing like a wash­ing ma­chine.

The un­der­ly­ing cause is that beauty re­veals the on­tol­ogy, the true na­ture of what is. That is Christo­pher Alexan­der’s fas­ci­nat­ing the­sis, most clearly ex­plained in The Na­ture of Or­der: An es­say on the art of build­ing and the na­ture of the uni­verse, book four: The Lu­mi­nous Ground.

We ap­ply this es­sen­tial­ist phi­los­o­phy to colours 🌈here.

Both agree that it is a sub­jec­tive per­cep­tion of an ob­jec­tive qual­ity. Our goal is not to de­ter­mine whether it is true, but to ex­am­ine the ram­i­fi­ca­tions of that as­sump­tion:

1. A Fourier trans­form model of the brain
   The mech­a­nism of that per­cep­tion
2. Aes­thetic ex­pe­ri­ence as the­ory build­ing
   How it mea­sures beauty
3. Ex­am­ples
   Ex­am­ples to clar­ify our un­der­stand­ing
4. Quan­ti­sa­tion of beauty
   An ob­jec­tive and math­e­mat­i­cal de­f­i­n­i­tion of that mea­sure
5. Ap­pli­ca­tion to chords
   An ap­pli­ca­tion to cre­ate aes­thetic works

The de­f­i­n­i­tion of in­tel­li­gence still dis­puted to­day, so let us take the de­scrip­tive path in­stead of the pre­scrip­tive one.

Imag­ine that you hear a chord (here, C and G):

The pi­ano has been kindly sam­pled by Alexan­der Holm.

Its sound wave is ap­prox­i­mately de­fined by:

$signal_1( t ) = \sin\left( t \right) + \sin\left( 1.5 \times t \right )$

And it looks like this:

Find­ing the orig­i­nal fre­quen­cies, $1$ and $1.5$, is dif­fi­cult with only a glance.

When the sig­nal reaches your ear, it is en­coded through the fir­ing in­ten­sity of your neu­rons: the more in­tense the sig­nal (a spike on the graph), the faster they fire.

How­ever, the next neu­rons trans­mit­ting the sig­nal also have their rates, such that we can in­fer that a sec­ond neu­ron, fir­ing nat­u­rally at $signal_2\left( n, t \right) = \sin\left( n \times t \right)$, will fire at a to­tal rate of $signal_{total}\left( n, t \right) = signal_1\left( t \right) \times signal_2\left( n, t \right)$.

To find how much a neu­ron will fire de­pend­ing on its rate, let us look at the in­te­gral:

$activity\left( n \right) = \lim_{t \to \infty} \int signal_{total}\left( n, t \right) dn$

This is an ap­prox­i­ma­tion to make it vis­i­ble; there should only be two ver­ti­cal bars at $1$ and $1.5$.

What this graph teaches us is that a sim­ple mech­a­nism, neu­rons fir­ing at their own rate when they re­ceive a sig­nal, en­ables them to de­com­pose a com­plex sig­nal into its ex­act fun­da­men­tals! It is called the Fourier trans­form. Each layer of neu­rons finds the rate at which the pre­vi­ous layer changes its rate, find­ing al­ways more global pat­terns.

The brain re­duces sig­nals to fun­da­men­tal har­mon­ics. Since a sig­nal is pe­ri­odic, the mech­a­nism is in­trin­si­cally pre­dic­tive: the brain al­ways has a the­ory of what is com­ing next. When that the­ory is dis­proven, the brain learns some­thing new and ad­justs its the­ory. That learn­ing is in­her­ently pleas­ant, as it pro­duces plea­sure-in­duc­ing sub­stances in the brain to strengthen neural con­nec­tions.

• Play­ing al­ways the same note is easy to pre­dict. Each new note brings no in­for­ma­tion: it be­comes bor­ing.
• At the op­po­site end, play­ing ran­dom notes also has an easy the­ory: ran­dom­ness it­self! Like­wise, it be­comes bor­ing.
• To be pleas­ant, a work must be be­tween both, be­ing pre­dictable and just sur­pris­ing enough.

Here are the first 30 sec­onds of Bach’s Kyrie as an ex­am­ple:

To make it sim­pler, let us sup­pose it was writ­ten in A mi­nor in­stead of B mi­nor. It starts with the fun­da­men­tal of the key, A. On the sec­ond mea­sure, it moves up­wards, B. We are al­ready build­ing a the­ory: it is mov­ing up­wards on the scale, and the next note will be C. How­ever! Bach de­stroys our the­ory and moves even fur­ther up­wards, out­side our scale, to C sharp. This un­ex­pected note has a great ef­fect.

To test our the­ory, here are two ex­am­ples (both ver­sions are ro­man­tic arrange­ments):

The chord pro­gres­sion is C → G → A → E → F → C → F → G → C. You can draw it be­low, as if you wanted to plot the path with­out lift­ing your pen. At every step, can you try to vi­su­ally pre­dict the next note?

Once notes do not bring new in­for­ma­tion (af­ter the E, the pro­gres­sion is very pre­dictable: there is a ver­ti­cal sym­me­try axis), your emo­tion should de­crease. An­other ex­am­ple:

Here, the chord pro­gres­sion is A → D → G → C → F → B → E → A.

Much harder to pre­dict, is it not? To­wards the end, you fi­nally dis­cover the logic: aha! It is a star! In­deed, the sym­me­try is cen­tral this time. Here, you ac­cu­mu­lated more chords be­fore shat­ter­ing your the­ory, and it was more pleas­ant. Its name is the cir­cle of fifths. Per­son­ally, I call it the sub­lime pro­gres­sion. Even im­pro­vis­ing sim­ple arpeg­gios (sep­a­rat­ing the notes com­pos­ing the chord) on that very math­e­mat­i­cal pro­gres­sion is pleas­ant:

More rig­or­ously, the goal is to max­imise the amount of in­for­ma­tion in the min­i­mal amount of data. To show how you do it in­tu­itively, try to draw the most beau­ti­ful ‘H’:

There is a good chance you chose $70\%$. Why? You can en­code your ra­tio through two ways:

1. Width / to­tal height
2. Half-height / width: in­deed, a ‘H’ is made of two stacked rec­tan­gles!

In other words, you must store the ra­tio plus its mean­ing. How can we avoid it? By mak­ing (1) and (2) equal!

$\frac{width}{height} = \frac{0.5 \times height}{width} \Rightarrow \frac{width}{height} = \frac{1}{\sqrt{2}} \approx 0.7$

This avoided us 1 bit to store the choice be­tween (1) and (2). We can ob­serve that the quan­ti­sa­tion of beauty is pos­si­ble with the the­ory of in­for­ma­tion. Math­e­mat­i­cally, we want to max­imise Shan­non’s en­tropy:

$-\int p\left( x \right) \log_2 p\left( x \right) dx$

$p\left( x \right)$ is the prob­a­bil­ity of ob­serv­ing $x$. In our ex­am­ple, $x \in \{ choice_1, choice_2 \}$.

How can we ap­ply this the­ory to find, for ex­am­ple, beau­ti­ful chords? In the twelve-tone equal tem­pera­ment, the rel­a­tive fre­quen­cies are:

$frequency\left( note \right) = 2^{\frac{note}{12}}$

$bits\left( denominator \right) = \log_2\left( denominator \right)$

The frac­tion of E is much sim­pler than C#‘s. In­deed, E’s de­nom­i­na­tor is $4$, and its nu­mer­a­tor re­quires only $\log_2\left( 4 \right) = 2$ bits, while B’s de­nom­i­na­tor is $8$, and its nu­mer­a­tor re­quires $\log_2\left( 8 \right) = 3$ bits. Same in­for­ma­tion (note of a chord), but more data. We can ex­pect, there­fore, the first chord to be more pleas­ant than the sec­ond one.

Even with­out know­ing the the­ory, you hear that it sounds odd. Does it mean that com­bi­na­tions like C + B are for­ever ban­ished? No, on the con­trary:

When we wanted a two-note chord, B was too dif­fer­ent from C for them to be re­lated. How­ever, when you com­pose a mu­sic, you want more notes. And only re­peat­ing the same easy chord will bore the lis­tener quickly (no new in­for­ma­tion!). How­ever, if you com­bine clev­erly dif­fer­ent notes, they can be pleas­ant to­gether.

Pre­vi­ously, we saw that stor­ing B rel­a­tively to C re­quires 3 bits. How­ever, stor­ing E or G is more com­pact (2 and 1 bit). And stor­ing B rel­a­tively to E or G re­quires 1 or 2 bits too. In other words, E and G bind C and B to­gether. We have more notes = more in­for­ma­tion with fewer bits. Com­pres­sion is higher and the chord is more pleas­ant!

For a deeper study of in­for­ma­tion the­o­ry’s en­tropy and har­monic waves as the root of the way our in­tel­lect works, see the free en­ergy prin­ci­ple and con­nec­tome-spe­cific har­monic waves: an in­tro­duc­tion and an aca­d­e­mic ar­ti­cle.