Charles Céleste Hutchins – Les said, the better II

The moral panic of ‘Adolescence’

They are not just gangs of kids anymore. They are often the kinds of kids that are called superpredators — no conscience, no empathy. We can talk about why they ended up that way, but first, we have to bring them to heel.
Hilary Clinton in 1996, about people roughly my age.

If Adolescence had been made a few years previously, it would have been against a backdrop of fear of video gaming. Or before that, rap music. Or before that, Dungeons and Dragons. Whatever it is that makes kids today more alarming and violent than kids of previous years. As they always are.

US trends in violent crimes. source https://ojjdp.ojp.gov/publications/trends-in-youth-arrests.pdf The UK doesn’t make such nice graphs so easy to find.

When I was 13, in 1989, many, if not most, 13 year old boys felt hatred and entitlement towards the bodies of their classmates that they perceived to be women. They were horrible little thugs.

They grew out of it.

Mostly.

So are we to believe that the 1980s was a more enlightened time for feminism for young teens? Or the 1970s, then? The 1990s? In exactly what year of pre-internet were boys respectful and well-behaved towards girls while seeing them as intellectual equals? One of the creators of Adolescence is trying to get phones taken away from kids, so there must be a year in which boys respected girls for their intellect, did not ever see them as meat, and in which students had equal opportunity across all genders. A year that we can return to, if only kids didn’t have phones!

Or not.

I do not mean to suggest that everything happening online is fine – far from it. But the same mobile phone and social media account that allows boys to access objectionable YouTube videos allows trans teens to find the community and support they need to keep going. So what can we change to prevent radicalisation into hatred without harming vulnerable youth?

Regulatory changes that might actually help

Automated, algorithmic news feeds take whatever demographic data they have and serve back what, statistically, the user will most likely interact with. Australian researchers found that for entirely blank profiles of “young men” – new email addresses of fictitious people – what those algorithms served back was intense misogyny. This is actually a problem. Keeping kids away from this until they turn 16, even if it were possible, would not meaningfully solve this problem.

A 17 year old boy being bombarded with sexism will still be impacted by it and, by extension, so will the girls and women around him. The solution is not to hide away intensely damaging content recommendation systems until people reach a threshold age, but to change legal regulation into how feeds are constructed and presented to people. Showing people things they’ve subscribed to in chronological order is fine. Showing them things we surmise they may like, based on algorithmic predictions. cannot be left out of human control.

It turns out that pandering to our basest instincts is not good for us. Computers cannot be held responsible, so nothing should happen with them without some human review. Not just for children, as everyone is vulnerable to radicalisation.

Indeed, YouTube’s radicalisation engine guides users towards more and more extreme versions of their interests. Someone who watches a video about jogging will be gradually lead down a path to ultramarathons. Running is innocuous enough, but the implications here with political content, sexism and health and wellness is highly concerning. My father might still be alive if he hadn’t started getting “health” information from YouTube. Again, children are only one category of vulnerable users. Information outlets, like libraries, used to curate their information.

I’m not suggesting that YouTube employ humans to look at every uploaded video, but their recommendation engine must be under human control. If it’s not vetted, it shouldn’t be pushed at us. Videos advocating for sexism could be made to be against the terms of service. They could be excluded from platform-provided recommendations. Google should have humans in the loop, to tie their brand identity to their curated lists. If they want to be the brand of sexism, they can be, but they need to have a human sign off on that and the rest of us can decide if we want our free time and our own videos associated with that.

What’s wrong with banning kids from social media?

Let’s start by talking about the impact on adult users of social media.

Right now, I can open a web browser and go sign up for a social media account with minimal fuss on a gigantic corporate network, or a tiny independent website. Usually, I have to give an email address, but I usually don’t need to supply a real name, a phone number or any other meaningfully traceable information.

This is valuable to me, because I may want to discuss a health problem anonymously, an identity I’m exploring, or anything else I’d rather that advertisers or bad actors not trace back to my actual home address.

If I wish to post to some of the amateur porn sites, however, I have to send them a picture of my face next to my passport so they can verify my identity. This has to be kept on file. This means that if upload a video that is not otherwise identifiable as me because of cropping or whatever, it’s still forever tied to my government ID. If the site gets hacked, all of my images could suddenly be very tied to my actual home address. These rules in most countries are only applied to adult content. It’s for age verification.

This is how age verification works in practice. If I wanted to make intimate videos, the de-anonymisation of age verification would be very likely to dissuade me. If this became required for all social media use, I would be also dissuaded if I was seeking help for an embarrassing or private medical condition. Or if I was exploring my gender identity, or trying to write heartfelt poetry about a girl who got away, or any number of things a person ought to be able to do with others without it being tied to their government ID.

Humans sometimes need to be able to access community anonymously. Because they’re dealing with addiction. Or they are questioning their gender or sexuality, and would be unsafe if some people in their lives learn of this; people seeking help trying to leave an abusive spouse or family member; anybody who wants or needs privacy. This isn’t all queers and people cheating on their wives – it’s also people who are in danger and are trying to get to safety. These are people we should support in a free country.

Meanwhile, kids under 16 are also sometimes trying to find out how to escape abuse; question their gender or sexuality; or just want and need some level of agency and privacy. There are things we should protect them from, but finding community and safety with others should not be on that list.

Good entertainment makes bad law

A lot of people like the massively overwrought series and I’m glad they found something they think is nice to watch. However, just because somebody is inspired by kitchen sink dramas does not mean that we should all lose significant freedom as a result. Especially the kids.

Bullying wasn’t invented with the internet. In 1988, “slam books” were made out of spiral notebooks, polaroids, glue and hatred. (I wasn’t popular enough to be allowed to look at them.) Unpleasant rumours, unkind judgements and even inappropriate photos weren’t invented yesterday.

What was invented yesterday were safe communication networks for agender furries.

Don’t take that away from them.

Take Action for Trans people

If you want to skip to the form, click here.

Nobody on the Right Actually Cares about Women Athletes

The Trump administration has moved to deny visas to trans athletes ahead of the 2028 Olympic Games. The US State Department does not have the right to dictate policy to international sports organisations. But what’s more, the way the memo is written does not limit the ban to athletes.

Most of the MAGA movement is openly hostile to women. They ended the right to abortion in the US. They are actively working to make it harder for women to vote. It defies reason that people who want to entirely shut women out of public life actually see any value in women’s sports. They don’t care about women athletes. But they do care about hurting trans people.

This rule change has serious implications for trans people’s freedom of movement. It labels trans people’s documentation as fraudulent, which also has serious implications for any interaction a trans person may have with the federal government. The purported target is foreign athletes, but the victims will overwhelmingly be trans Americans. The Trump administrations is using the Olympics as an excuse and a first step to seriously impacting the ability of trans people to exist.

The International Olympic Committee has yet to comment on this development. The latest thing on their website is an announcement of a sponsorship deal. This is certainly not what the sponsors signed up for. But Anheiser-Busch InBev, as the world’s largest brewer, has acquired this problem. They also have influence. I am therefore asking you to write to them. The form below gives suggested text. When you hit the “send” button, it will put the message into your email program. I will not collect or have access to your addresses.

Email Form

Your name:

Your Country:

Subject:

Dear Madam or Sir,

I am writing with regards to your sponsorship of the 2028 Olympic games in light of Trump’s decision to deny visas to trans athletes.

You signed up to associate your brands with the Olympic Spirit - to build a more peaceful, better world through mutual friendship and understanding. But Trump’s actions twist that noble goal into its opposite. His version of the Games uses them to promote hate.

This is not what you signed up for. I’m therefore writing to ask that you use your influence to try to either change that policy or to move the 2028 Olympics to a venue that will not deny admittance to some of the athletes based on unscientific, ideological lines.

And while this is not what you signed up for, it’s what your brands are now associated with. Should the 2028 games go ahead in the US, with this ban in place, all of its sponsors will be subject to boycott. This Trump plan is turning what should be a proud moment for your brands into a terrible liability.  I hope you decide to take action.

Best Regards,

User_Name
User_Country

Technical Issues

The form should open in your email client. If it did not, it did not send. You can still email manually:

In Europe: eu_media.relations@ab-inbev.com

In North America: media@anheuser-busch.com

Africa: mediarelations@sab.co.za

Asia Pacific: abimedia@cn.ab-inbev.com

South America: accomext@ambev.com.br

Middle America: media.relations@ab-inbev.com

Global headquarters: media.relations@ab-inbev.com

Note: I am not collecting your data.

But surely the face-eating leopards won’t eat my face!

Many people who have invested a lot of time and energy into corporate platforms don’t want to walk away from that, which is entirely understandable. And anyway, they muse, how bad could it possibly be?

Let’s look at some risks:

Radicalisation – but that’s fine because I’m immune to propaganda.

Um, maybe? Some people are more susceptible than others. However, people are on the platforms they’re on because of the “network effect”. You’re there because your contacts are there and your contacts are there because you are. Some of them are susceptible and by staying in propaganda-filled environments, you’re helping expose them.

You’re also exposing yourself and your contacts to greater risk of hate speech. Zuckerberg went to the inauguration, stood next to a guy who gave Hitler salutes, and is also doing his part for Trump by allowing greater hate speech. Sticks and stones may not break your bones or hurt your feelings, but it’s highly damaging to some people. And also changes the political discourse and will impact our rights. Again, you and your friends are holding each other hostage via the network effect.

Multiple studies have shown that boys are being groomed into extreme misogyny via algorithms on video-based social media. Again, the corporate platforms are a problem.

Whether or not you’re personally immune (you’re not, sorry), society as a whole certainly isn’t.

Non-state physical threat – I’ll be fine

In the US, it’s extremely clear that radicalised actors face a physical threat to minoritised groups. My own opinion is that migrants (and the children of migrants) who actually are in quite a lot of danger. Facebook has previously been a major platform for planning and coordination for at least one genocide.

Women and LGBT people are also at high risk of doxing, via Meta. “The Facebook platform makes doxing particularly easy and rewarding for doxers.” Facebook has also leaked personal information to people pretending to be police, as no warrant is required in emergency situations.

Maybe you’ll be fine, but some groups are in serious danger.

State-based threat – I’ll be fine!!

Americans generally don’t really have very many data rights. Europeans have many more, but Meta routinely ignores them (getting fines that would be massive for a less-profitable company). The problem is not just that they mishandled data, but they’ve been accused of collecting excessive data they had no right to and which users hadn’t and couldn’t consent to.

That data includes information like gender, race, sexuality, orientation, trans status, susceptibility to addiction – it’s far ranging. Indeed, Meta collects and utilises data about users likely race and other protected characteristics. Some of this data, say, identifying Hispanic Spanish-speakers, may be useful for Trump’s campaign promise of putting people into camps.

And, if a court orders Meta to share any of their collected data with the cops, they have to comply. Which is how they came to participate in helping prosecute abortion care.

If they stored less data, or kept messages encrypted, they would not have had access to this data to share. But instead, they also track as much information as possible from their own apps and from other, unrelated apps listed in your phone’s app store.

I searched my phone’s app store for “period tracker” and the top result leaks data to facebook. Apps with Facebook trackers collect “off meta activity” to use to show you adverts. Or to share with anyone who has a court order demanding they do so.

They also track your relationships on and offline, via apps, partly by tracking location.

WhatsApp’s message content are encrypted and thus not visible to the company, but they know who you message, how often, at what times, where you are when you send them. This is called metadata and in some ways it’s more valuable than the message contents. Not for prosecuting abortions as above, but for inferring relationships and life circumstances.

The sheer amount of surveillance available to state actors is dizzying. But you’ll be fine, right?

Oh wow, no, we need regulation, especially to protect kids!

It’s absolutely true that individual action is not going to put a stop to this, and larger, collectivised action is necessary. The GDPR in the EU is a great step, even if it took them a very long time to act and fines were small relative to Meta’s income.

The situations in the US and the UK, however, are a long way off from the EU. The UK is too small and isolated to act with any real teeth and the US is currently pro-abuse.

You may be thinking of some proposed legislation purported to benefit the online safety of kids. But those proposed laws were written by the companies they’re meant to legislate. They’re written in such a way that no social media site could possibly comply with them unless they have the resources of Meta. The version proposed in the UK would have outlawed Wikipedia. They’re meant to extend monopolies, not to protect kids.

Indeed, the language they use for marketing these ideas is the same language used by the governor of Florida while enacting rules against trans youth. “Letting kids be kids” means keeping them away from knowledge about trans lives, gay lives, protection against STIs, or any kind of sex or gender education. Enshrouding children with enforced innocence is compulsory cisgender heterosexuality. It is “anti woke” ignorance in which discrimination is tolerated by its antidotes are not.

The version of this just passed in Australia requires age verification for the entire country for many normal activities. This hampers anonymity, puts people subject to abuse at greater risk, and deprives kids of vital information.

We must stay and fight!

Karl Marx thought that the revolution would come when the workers seized the tools of production. Even now, people are clinging to Twitter – a site owned and controlled by somebody who Seig Heils – vowing to hold their ground.

But this isn’t like holding on to a piece of land against an advancing army. The oligarchs not only own the land, they own the physics. It’s like fighting G-d. They control who, if anyone, sees your posts and everything that you see. That is not at all like real life or even like ancient myth. G-d sometimes plunged people into darkness or plagued them with flies or even opened chasms beneath them, but everyone present was in the same reality, seeing the same things. That’s not true on a virtual platform.

Fox News used to run a show called Hannity and Colmes. Hannity was a tough, jockular bloke who was right wing. Colmes was a tame, soft-spoken liberal. They faced off each other to debate, except they didn’t. The terms, the framing, the guests, and everything about the show was meant to give right wing audiences an illusion of debate, but it was never a fair match. Even the settings of the microphones was such that the right wing voices were objectively louder than the liberals. If Colmes had been actually effective at countering right wing narratives and framing, he would have been fired.

Nobody is actually “staying and fighting” on Twitter. They’re just the loyal opposition. They’re a figleaf of balance where none exists. They’re being used and exploited and unlike Colmes, aren’t even getting anything in return. He got paid. Liberals on twitter are giving resources and cover to Musk.

Some of the corporate platforms may “feel” more balanced, but their owners are loyal to Trump and are quickly lining up. In the West, the workers did not ever end up seizing the factories they worked in. The means of production stayed in the hands of capital. But at least Marxists had a credible plan for how victory might have come about. There is no credible path wherein Facebook users seize Meta. There’s just not.

You Get a better world by building it.

Leftists talk about the “power of the people”. The people are building alternative platforms which are not under oligarch control and are structurally resistant to capture. That’s the way forward. Join a movement that will immediately (although individually) solve several of the problems listed above and which provides a route for the future.

A better world is possible and is much less far away than it seems.

But it does mean stopping giving all your data to Meta, Google, Apple, Amazon, and X.

Join the movement. Go to fedi.

But what about Bluesky?

This platform – invented by the guy who started Twitter, then refused to moderate it properly and eventually forced the sale to Musk after Musk wanted to back out- sorry, where was I?

Bluesky has a single point of failure. They could decide to sell it to Musk too. Or to Meta. Or to Google. Or to Amazon, or to whoever. It’s backed by capital, so it’s for sale.

To prevent this, there’s a group of people trying to build a “spare” server so Bluesky users can easily hop between them when the main Bluesky stops being in the fun early phases of trying to build up a user base and moves on phase 2 where it tryies to wring data and profit out of everyone. The people making the spare estimate that their server will cost $30 million USD and take three years to set up. That’s apparently not counting running costs after those three years are up. Which may be a moot point, as currently, this is also not technically possible. It requires the capital-backed team at Bluesky to build the tools to make this possible.

And if the hypothetical $30 million dollar spare server also gets caught up in US politics, what then? Do they need to raise another $30 million?

By contrast, a small mastodon (or other fediverse) server can go up in an afternoon and be run for a few dollars a month. Nobody quite knows how many are running right now- it’s in the thousands. Some only have one user. Some have hundreds.

The social media project with thousands of people working independently as part of a loose movement is harder to capture than a centralised, expensive project. It’s not just run by one or two teams. It’s not in one or two countries.

I know your friends are on Bluesky. . . . Like they are on facebook and were on twitter and were on myspace. . ..

Part of the reason people like the fediverse is that it’s really very easy to move between instances. If queer.party turns out too be too unserious, I can go to scholar.social and keep my contacts. I would argue that this is better than herding ourselves into yet another walled garden that cages us. To another billionaire who turns fascist.

You can be on many websites at a time and you don’t have to choose – it’s possible to have both kinds of account. I’m just kind of tired of being caught in traps again and again and again.

Get Away from Meta and X Now

We’ve all seen the pictures from Trump’s inauguration – Elon Musk, CEO of Twitter and Mark Zuckerberg, CEO of Facebook, WhatsApp and Instagram; stood in the front row. And of Musk giving a Nazi salute. The CEO of TikTok was not in those pictures, but he was also there. The collaboration has already started. Facebook has already helped arrest and prosecute people for abortion.

Ditch WhatsApp

Get Signal instead. I’m on there. Find me via my phone number or send me an email.

Ditch Insta

The hot new platform is PixelFed. You pick an “instance” to join and then can follow any other PixelFed user on any instance. I’m trying out Pix.lgbt, which is run by a trans woman.

Pixelfed is part of the fediverse. (keep reading for more)

Ditch Twitter (or Threads)

As in, actually delete your X account. Don’t just stop using it. You need to deactivate the account, wait 30 days and then demand that the data actually be deleted.

The popular replacement for twitter is Mastodon. But don’t just pick an instance at random, because they all have different moderation policies. For queers and allies, a good one is https://lgbt.io. If you’re a musician, you might enjoy https://sonomu.club/. If you’re Jewish (and get on well with liberal zionists), https://babka.social/. If you’re not sure, just do lgbt.io. It’s extremely easy to move your account between servers, so if you decide later that you should have been on a different server, you can move. However, your early experiences are going to make you feel happy or not, so do get a recommendation for an instance. The flagship one has moderation problems and puts many people off.

Mastodon is part of the fediverse. (keep reading for more)

Ditch Facebook

You could just join Mastodon, but for people who like nicely threaded replies and discussions where you can see who has replied to what, people like Misskey (or the million sub-variants of Misskey). I’m trying out https://blahaj.zone/ which is very queer friendly.

Misskey is part of the fediverse. (keep reading for more)

The fedi-what?

Mastodon, Pixelfed, Misskey and several other platforms all interoperate. If you join any Matsodon instance, you can follow anyone on any of them, on any instance. The differences between them are the interface, the moderation, and the community that is local to each instance. (Pixelfed, reasonably, only shows posts that contain pictures, so isn’t a good way to follow Mastodon users.)

In practice, this means that wherever you join, you can follow me: @celesteh@lgbt.io. (You can also follow this blog: @celesteh )

This is important because it means that this network cannot just be purchased by Elon Musk, even if he buys this blog, he can’t buy every single Mastodon server. This network can never be fully captured by oligarchs.

Getting started

I just wrote a thing about how to sign up at another blog yesterday, so go read that.

Once you’re signed up, upload a profile image, fill out your bio and write a little #introduction post.

Then follow me. Send me a message if we know each other online or in real life.

Tell your friends where you’ve gone. And why. If you want to post this message to X or any Meta property, you’re going to need to be cryptic, because they will not let you link to their competitors in a post.

Octatonic Scales in SuperCollider

You can generate your own Octatonic scale in an arbitrary Equal Temperament using the following code.

Change octaveRatio to the ratio you’d like and steps to the number of steps. The Scale is saved to the global variable o;

(

var octaveRatio = 2, steps = 12;
var ratio, tuning_arr, tuning, octatonic_arr, octatonicScale, index;

ratio = octaveRatio.pow(steps.reciprocal);

tuning_arr = steps.collect({|i| ratio.pow(i).ratiomidi });
tuning = Tuning(tuning_arr, octaveRatio);

index = 0;
octatonic_arr =[];

{index < steps }.while({
	octatonic_arr = octatonic_arr.add(index);
	index = index+2;
	(index <= steps).if({
		octatonic_arr = octatonic_arr.add(index);
	});
	index = index + 1;
});


octatonicScale = Scale(octatonic_arr, tuning: tuning);

o = octatonicScale;
)

You can then use this in a Pbind by using \scale. For example:

(
Pbind(
	\scale, o,
	\degree, Prand((0..7), 7)
).play
)

Try out different Equal Temperaments

Note: Code for this post is available on github here.

Tuning scales is about ratios. We multiply the root frequency by a given ratio to get a note in the scale. In Equal Temperament, all ratios are equal, the 12th root of 2. Which is 2^1⁄12. We multiply a frequency by that to get the next frequency in the scale. When we’ve gone through all 12, we get the octave. (2^1⁄12)¹² = 2.

Let’s say we want the 3rd note in the chromatic scale. We have the root and multiply by the ratio for the second and then for the third. For the fourth, we do it three times. For the fifth, four times. Therefore, for any chromatic scale step 𝘯, we multiply the root by 2^{(𝘯-1)⁄12}

But, especially when we’re using computers, we can try out putting the notes in different places! What if we have 10 steps per octave? Then our ratio is Which is 2^1⁄10. The composer William Sethares has written music using 10 tone equal temperament and in other unusual tunings, which you can listen to on his web page.

We can even forego octaves entirely. The Bohlen-Pierce scale is based on divisions of 3, rather than 2. When people use equal temperament with that scale, they typically have 13 steps in the octave, which makes their ratio 3^1⁄13. The composer Elaine Walker is one of many who has written music using Bolhen Pierce and you can find examples on her website.

We can also try out different tunings ourselves! Below, you can try out different Equally Tempered scales. Change the steps value for the number of divisions you want. If you want to try out Bohlen-Pierce, change the octave ratio to 3. Or try whatever tickles your fancy.

Your tuning ratio is 2^1⁄12, which is equal to 1.0594630943592953

12tet’s ratio of 2^1⁄12 is equal to 1.0594630943592953

It can sometimes be difficult to hear the differences in pitches just going up and down a chromatic scale. Modes like major and minor are very strongly tied to a 12 note chromatic scale and it doesn't make sense to try to, say, play a 10 note major scale. However, the octatonic scale is a mode that can potentially work for any tuning. It alternates whole and half steps. Perhaps listening to the octatonic versions of your scale and 12tet will demonstrate the differences more clearly.

Or we can try a phrase by Debussy:

Plugins for music, equations, etc

In the hope of making my text here more accessible, I’ve installed a few new plugins. Rather than take screenshots of a notation program, to show notes, I’ve installed Music Sheet Viewer. This is supposed to support Plaine and Easie Code, which is meant to be a dead easy way to input a few lines of notes. However, I couldn’t get that to work, so I input the Plaine and Easy Code into a free online converter, which turns it to MusicXML. It supports this format without a hitch. I’m sure I’m doing something wrong and will be able to simplify this soon.

For maths formulas, I tried very many plugins. The one that finally worked was QuickLaTex. As the name implies, it uses LaTex syntax for layouts. I’m under the impression that this increases accessibility for screen reader viewers, although perhaps not as much as MathsML. I tried many plugins nad this is the only one I could get to work. Of course, MathML is in Jetpack, but so is a bunch of SEO garbage that I’d rather live without.

Finally, I’ve enabled the ability to upload SVGs. I used WPCode, which is a code snippet library. It added a function for SVGs. This was better than trying to do this by hand, especially as it worked the first time without breaking my site.

My next step is to write or deploy a little javascript toy to let people try out different equal temperaments.

Science of Sound Week 2

Frequency

Previously, we talked about wave length and frequency. We measure frequency in Herz, abbreviated as Hz. A 1Hz sine wave goes through a complete cycle one time per second. A 440Hz sound wave goes through a complete cycle 440 times per second. The frequency is the reciprocal of the duration. A single cycle of a 440 Hz sine wave is $\frac {1} {440}$ th of a second.

We also talked about the speed of sound, which is 340 m/s at 20 degrees celsius. If we have a 1 Hz wave, travelling at 340m/s, it takes one full second to get through the complete cycle. Which means that the front of the sound wave is 340 metres away from the back. The wavelength is 340 metres.

A 2 Hz sine wave also travels 340m/s. The time it takes to get through each cycle is half a second. In half a second, the front has travelled 170 metres, which is to say that’s the wave length.

A 10 Hz sine wave lasts $\frac{1}{10}$ th of a second, so the wave length is $\frac{340}{10}$ , which is to say 34 metres.

A 100 Hz sine wave is $\frac{340}{100} = 3.4$ metres. The octave higher, 200 Hz, is 1.7 metres. The wave length is the speed of sound ( $c$ in the formula) divided by the frequency. $\lambda = \frac{c}{f}$

Tuning

We mentioned 440 Hz in the first paragraph. If that sounds familiar, it’s because it’s also the frequency of most tuning forks. It’s the defined frequency for A.

MusicSheetViewerPlugin 4.1

We also know that if we double the frequency to 880, that’s also and A. Or if we halve it to 220.

MusicSheetViewerPlugin 4.1

110 Hz, 55 Hz and 27.5 Hz are also As. As we get lower the frequencies get closer together and as we get higher they’re farther apart. 7040 Hz and 14080 Hz are also As.

Figure 3: Diatonic scale notes graphed by frequency. The curve of the notes is exponential. By SharkD, Public domain, via Wikimedia Commons.

We know that all As are 440 multiplied or divided by a power of 2. We also know that doubling any frequency gives us an octave of that frequency. We can generalise from this to come up with a formula for a one note scale based on the octave. Where $f$ is frequency, $f \times x = 2f$ . It’s obvious here that x is 2.

What if we want a two note scale that uses Equal Temperament? This is a system where all the notes are equally distant from each other perceptually. We know that this has to be based on multiplication. We want an equal ratio between all the notes. Therefore to get from the bottom note to the next one, we need to multiply by some number x. And then to get from the middle note to the octave, we multiply by x again. $f \times x \times x = 2f$ We can simplify those two xs. $\therefore f \times x^2 = 2f$ And divide both sides by f. $\therefore x^2 = 2$ Solving for x: $\therefore x = \sqrt{2}$ . Our two note scale is 440, 622.25, 880. This is because $440 \times \sqrt{2} = 622.25$ and $622.25 \times \sqrt{2} = 880$

What about a three note scale? $f \times x \times x \times x = 2f$ Which means $\therefore x^3 = 2$ and so $\therefore x = \sqrt[3]{2}$ To work out this scale, $440 \times \sqrt[3]{2} = 554.37$ , $554.37 \times \sqrt[3]{2} = 698.46$ , and $698.46 \times \sqrt[3]{2} = 880$ .

If we want a 4 note scale, we can use $\sqrt[4]{2}$ or for a five note scale $\sqrt[5]{2}$ . But for a piano, we want 12 notes, including all the white and black keys.

Figure 4: A one octave keyboard by Lauri Kaila CC BY 3.0 https://creativecommons.org/licenses/by/3.0, via Wikimedia Commons

Therefore, the tuning used by the piano, called “12 Tone Equal Temperament” (or 12tet) uses $\sqrt[12]{2}$ .

We know that the frequencies are exponential, but perceptually, the difference between a C and and A is the same in any octave. Our scales and keyboards and the musical concept of pitch is linear. Every octave may double in frequency, but it’s always only 12 semitones.

Figure 5: “Logarithmic plot of frequency in hertz versus pitch of a chromatic scale starting on middle C.” via https://en.wikipedia.org/wiki/Musical_note. Image by Jono4174, public domain via Wikimedia Commons.

You now know enough to work out the frequency for every single note on the piano. (Or, you can look it up on wikipedia.) You can also work out the wavelength for every frequency on the keyboard. If the lowest note is A₀, the frequency is 27.5 Hz, so the wavelength $\lambda = \frac {340}{27.5} =$ 12.4 metres. And the highest note, C₈ is 4186 Hz, so $\lambda = \frac {340}{4186} =$ 0.081 metres. What a range! And that’s not even the highest note we can hear!

Going Further

Not all scales are based on octaves! The Bohlen-Pierce scale is based on multiplying frequencies by 3. How could you compute an equally tempered scale for Bohlen Pierce? If you wanted the scale steps to be roughly the same size as 12tet, how many scale steps would you use?