Music theory for nerds (2016)

On the question of &quot;why 12 notes&quot;, I <i>STRONGLY</i> recommend this video by David Bennett:<p><a href="https:&#x2F;&#x2F;www.youtube.com&#x2F;watch?v=lvmzgVtZtUQ" rel="nofollow">https:&#x2F;&#x2F;www.youtube.com&#x2F;watch?v=lvmzgVtZtUQ</a><p>There&#x27;s a really important step missing in the article which has been aluded to by a couple of comments already. It&#x27;s the step between:<p><pre><code> 1. a series of notes defined by the harmonic series, that is, exact integer multiples (or ratios, if you prefer) of a base frequency, a physical phenomena found in nature </code></pre> and<p><pre><code> 2. the well-tempered scales used by most western music, which adjusts the ratios to allow certain compositional &quot;tricks&quot; to be used without sounding dissonant. </code></pre> The set of notes defined by (1) are typically referred to as &quot;Just Intonation&quot;, and was the basis for most western music (and some non-western music) until somewhere between about 1400 and 1600 (lots of room for discussion&#x2F;debate there).<p>The &quot;problem&quot; with just intonation, if indeed it is a problem, is that if you define two series of notes (call them a scale, if you like) using these integer ratios but starting from two different notes, you will trivially find cases where &quot;the next higher E&quot; is a different frequency. So starting from (say) A and moving by integer ratios gets you to a different (say) F than if you start from (say) B.<p>This means that if you are writing in a just intonation scale with (say) A as the root, you have a set of notes that are not actually the same frequencies as if your scale started on (say) B.<p>By itself, this is not a problem at all - there is all kinds of lovely music written through history that works just fine with notes and scales defined this way. You just stay in the same scale throughout, and there are no issues. There are even some scale changes you can make that still work, you just have to know what they are (and they depend on the root note and the set of integer ratios you&#x27;re using, so it gets complicated).<p>But ... somewhere in the period mentioned above, a subset of western musical culture started to want to experiment with &quot;modulation&quot; - changing from one scale to another in the midst of piece. On continuous pitch instruments (e.g. violin, voice), this is entirely possible to do, since they can play any frequency in their range at all. However, it does require significant skill on the part of the performer, since the pitch of a (say) &quot;F&quot; will differ depending on the scale currently in use in the piece.<p>The breaking point, such as it was, came with the development of fixed pitch instruments (keyboards). These can only play the notes they are currently tuned to, and so if the (say) F in two scales is a different frequency, you cannot play in both scales without retuning - an obvious impossibility in the middle of a piece.<p>So, &quot;well-tempered&quot; tuning was developed - the ratios described in TFA. These are tweaked by relatively small amounts so that the notes are close to where Just Intonation (integer ratios) would have placed them, but not precisely the same. These small shifts mean that the (say) &quot;F&quot; is the same frequency whether your scale began on (say) A or B. You can now play a fixed-tuning instrument like a harpsichord or a piano, change from one scale to another, and everything remains &quot;in tune&quot;.<p>Of course, to ears used to Just Intonation, &quot;well-tempered&quot; tuning sounds out of tune. But in the west, most people (even within our musical academies) have grown up used to the sound of well-tempered tuning, and it is Just Intonation that sounds &quot;off&quot;.

I really don&#x27;t understand all the hate in the comments for this article. This seems like a really fine introduction to a lot of the concepts. I had to scour the web a long time before understanding that the scales were constructed because of the simple&#x2F;reduced ratio of root note to the others, which is clearly spelled out in this article.<p>Music theory is, in my experience, typically taught as a list of facts to remember. Deriving it from first principles, insofar as it can be, is not common. This article is attempting to start that process.<p>Some commentators are saying that the article doesn&#x27;t address why we have 12 notes but I wonder if they even know why (one answer is because it&#x27;s a good compromise between number of notes and that have simple&#x2F;reduce ratios to each other [0]). I&#x27;m also skeptical of music theorists that can&#x27;t even attempt an answer to basic questions about note length frequency, why some chords are &quot;sad&quot; or &quot;happy&quot; and other basic questions. It&#x27;s difficult because music theory is a hodge podge of theory that&#x27;s attempting to describe what&#x27;s effectively an evolved language (with different music evolution for different regions), but there are some basic tenets that probably apply.<p>I don&#x27;t claim to have deep knowledge but there are a few key facts about music and music theory that are really obscure unless you know where to look. Any attempt at coming to music theory from a more rigorous foundation should be encouraged.<p>[0] &quot;Measures of Consonances in a Goodness-of-fit Model for Equal-tempered Scales&quot; by Aline Honigh (<a href="https:&#x2F;&#x2F;github.com&#x2F;abetusk&#x2F;papers&#x2F;blob&#x2F;release&#x2F;Music&#x2F;measures-consonances_honingh.pdf" rel="nofollow">https:&#x2F;&#x2F;github.com&#x2F;abetusk&#x2F;papers&#x2F;blob&#x2F;release&#x2F;Music&#x2F;measure...</a>)<p>EDIT: corrected spelling (thanks for the comments pointing it out)

Music was taught as part of the quadrivium (alongside arithmetic, geometry, and astronomy), so it has a long history of being thought about in mathematical terms. There&#x27;s nothing wrong with looking at music that way. What is so interesting about music is its ability to turn mathematical beauty into aesthetic, artistic beauty.<p>Sometimes the math is just there in the background: &quot;This chord progression is based on certain physical facts related to the harmonic series. That makes it sound good, and I like it&quot;. Sometimes it is brought to the forefront: &quot;All of this music appears to be generated from a short pattern of notes which is recognizable even as it is subjected to rotations, elongations, truncations, etc. That is beautiful.&quot;<p>Certain types of music play up the lovely-pattern-ness of music, and some don&#x27;t. Neither is better than the other (denigrate the art of Justin Bieber who dare), but you can probably guess which I prefer :)<p>Consciousness, pattern-recognition, spatial rotation of abstract objects, natural mathematical beauty, poetry, drama, acting&#x2F;performance, nonverbal collaboration, shared rhythm: music is a nexus for all of these things. Music is nothing short of awesome, in the truest sense of that word.

I guess I fell into the same trap as a lot of others reading this and commenting it and having the programmer side of my brain see one thing and the musician side see something else.<p>Computer programs must follow the rules... the rules get made up first and the programs get built to conform to the rules.<p>Music theory isn&#x27;t like that. The music comes first. The theory gets made up to try and describe &amp; understand what is going on with the music. It&#x27;s not at all like a program where the rules &amp; theory come first and you build something that fits into that.<p>If you write a piece of music that doesn&#x27;t conform to some kind of arbitrary rule in music theory that&#x27;s fine. If your music is successful and important enough someone will modify music theory to explain why your music was successful and important.<p>There is a cycle where you learn some music theory to help learn music but the theory just helps you understand what&#x27;s going on, it doesn&#x27;t force you to do anything a certain way. You can&#x27;t come at music and think about it like you&#x27;re building a computer program.

I love this style of clear, unambiguous writing starting with no knowledge assumptions.<p>[but] The author is missing or skipping key pieces of information about why we use the notes we use, how they were derived, and why they sound the way they do to us.<p>Edit. There was zero sarcasm in my comment. I really love this style of writing.<p>The missing concept here is the harmonic series. A pitched note is not generally a sine wave, it is properly modeled as a series of sines waves combined. These sine waves go up in mathematical multiples.<p>So say we have a note at 100hz. Simplistically it is actually 100, 200, 300, 400 etc all at once. So if 100 were A, 200 is A, 400 is A, 300 is … E, so 150 is E. Etc.<p>None of this is arbitrary because the When the waves combine the result is more ordered or more chaotic, depending on the ratios of the waves, which our ears hear as consonant or dissonant.<p>Best I can do from a phone.

As an amateur musician (and amateur nerd) I found this article to be insightful in many ways. The author’s opening disclaimer renders most of the HN criticism moot.

The article says:<p><pre><code> 0 1.000 = 1:1 (unison) 1 1.059 (semitone; minor second) 2 1.122 ≈ 1.125 = 9:8 (whole tone; major second) 3 1.189 (minor third) 4 1.260 ≈ 1.250 = 5:4 (major third) 5 1.335 ≈ 1.333 = 4:3 (perfect fourth) 6 1.414 7 1.498 ≈ 1.500 = 3:2 (perfect fifth) 8 1.587 (minor sixth) 9 1.682 ≈ 1.667 = 5:3 (major sixth) 10 1.782 (minor seventh) 11 1.888 ≈ 1.889 = 17:9 (major seventh) 12 2 = 2:1 (octave) </code></pre> This table has many omissions and some errors. The way people usually play these intervals on just intonation instruments, minor second should be 16:15, major second can be 9:8 or 10:9 depending on context, minor third is 6:5, minor sixth is 8:5, minor seventh can be 9:5 or 16:9 (or 7:4 in barbershop), major seventh is 15:8.

He left off right where it starts to get really interesting.<p>Most of what he covered is stuff you learn the first day playing and never really need to worry about again.<p>The interactions of how different notes &amp; chords in a scale draw our ears, how 7ths and other intervals work&#x2F;sound. Modes, chord voicings, inversions, modulations.. all that stuff gets incredibly fascinating.

Ultimately a lot of things around music are cultural and&#x2F;or arbitrary. It&#x27;s like languages in that sense. Personally I don&#x27;t find this sort of analytical study to be that helpful for actually playing or improvising music, but it is neat information. You don&#x27;t have to clinically study grammar and language theory in order to be a great writer; it&#x27;s the same with music.<p>Something that&#x27;s very important to note is that music theory and playing music are very different. Music theory allows us to understand what someone played, and lets us communicate to other musicians in a compact manner. Theory by no means imposes rules, unless you are specifically wanting to write music exactly like Bach or something.<p>As a logical-minded person myself, it&#x27;s tempting to really dive into theory and the root of things, but I&#x27;ve found that it&#x27;s a trap most of the time. For getting better at songwriting or composing, I&#x27;ve found the best thing to do is transcribe the songs you like (or portions of them) and see what they do, then incorporate those ideas into your own playing.<p>I also get the frustration around enharmonics (e.g. C# vs Db), but honestly it&#x27;s not an issue in practice. I had no opinion until I started typesetting songs myself, and it just looks cleaner to sometimes have Cb or B# depending on the context. The circle of 5ths also neatly puts scales in the order of how many sharps or flats they have. Also, C# is not a commonly used key anyways.<p>From a theoretical perspective C major and A minor share the same notes, but when I improvise I do find the distinction meaningful. Having multiple ways to think about things is helpful. There are other things too: C minor 7 is the same as Eb major 6. Or if you play a rootless voicing, the harmony could be ambiguous. Etc.<p>Yes, our music notation system has some issues. But this article is overanalyzing it from an armchair. It&#x27;s mostly fine in practice.

&gt; More “fake” notes exist than E♯, too; I hear rumor of such nonsense as G𝄪, “G double sharp”, which I would rather call “A”.<p>It starts making more sense when you&#x27;re working with harmony and chords. The wikipedia example section on enharmonic equivalents has a nice example of a Shubert sonata that makes use of this [1].<p>[1]<a href="https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Enharmonic" rel="nofollow">https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Enharmonic</a>

&gt; That problem is that sheet music is terrible.<p>Something that people with casual exposure to musical notation probably don&#x27;t know is that before the standard 5-staff notation became ubiquitus there were quite a few alternatives, maybe the most interesting from a mathematical perspective being the Byzantine notation<p>Broadly speaking, while the modern system focuses on pitch <i>values</i> and an elaborate (modulated) map from frequency space to physical space (paper), older systems used in the Byzantium used &quot;deltas&quot; or the first differences of pitch values.<p>So you start with a base note (lets say C) and then you go +1, +1, -2 to indicate pitch changes (in semitones). This is quite well adapted to monophonic chant. This notation was never developed to cope with the complexity of modern music but its not immediately obvious that it can&#x27;t be done<p>There is no easily accessible exposition of this musical notatin style, this cheatsheet gives a flavor <a href="http:&#x2F;&#x2F;www.byzantinechant.org&#x2F;notation&#x2F;Table%20of%20Byzantine%20Notation%20Symbols.pdf" rel="nofollow">http:&#x2F;&#x2F;www.byzantinechant.org&#x2F;notation&#x2F;Table%20of%20Byzantin...</a>

When I listen to the three sinewaves, two of which are supposed to feel more similar because they a factor two apart, I don&#x27;t actually get any such similarity effect. They all seem just about equally similar to each other, even when played together. Is everyone here picking this sameness up? I&#x27;m curious what kind of mechanic underlies those feelings of sameness and what it feels like.

&gt; I don’t know why twelve in particular has this effect, or if other roots do as well<p>Here&#x27;s a 31-equal temperament song. Microtonal music has a reputation for sounding bad, but it doesn&#x27;t have to.<p><a href="https:&#x2F;&#x2F;www.youtube.com&#x2F;watch?v=u-PEhbSOh74&amp;list=PLC6ZSKWKnVz3lBogeQ66N8W3x_GwUhTFw&amp;index=17" rel="nofollow">https:&#x2F;&#x2F;www.youtube.com&#x2F;watch?v=u-PEhbSOh74&amp;list=PLC6ZSKWKnV...</a><p>&gt; ... but it’s probably why Western music settled on twelve.<p>The curious part is that Western music settled on twelve a long time before equal temperament. They used to have 12 notes but use for example quarter-comma meantone. Great mozart meantone piano piece (mozart never used equal temperament):<p><a href="https:&#x2F;&#x2F;youtu.be&#x2F;lzsEdK48CDY?t=700" rel="nofollow">https:&#x2F;&#x2F;youtu.be&#x2F;lzsEdK48CDY?t=700</a>

Hold on. a C# and a Db are not the same note. They are a comma apart.<p><a href="https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Comma_(music)" rel="nofollow">https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Comma_(music)</a>

Math and music are similar in so many aspects. Are the people good at one also good at other? I mean, do mathematicians understand musical theory better than those that have no background in mah?

I wish there was a theory for why a certain set of sounds is music and why it isn&#x27;t music. I&#x27;m positive I can follow all the rules according to standard &quot;music theory&quot; and produce sounds that aren&#x27;t classified as music.

I really like 12tone&#x27;s breakdown of pop songs. Here&#x27;s an example with The Immigrant Song: <a href="https:&#x2F;&#x2F;www.youtube.com&#x2F;watch?v=gwsfR5sJPbM" rel="nofollow">https:&#x2F;&#x2F;www.youtube.com&#x2F;watch?v=gwsfR5sJPbM</a>

Discussed at the time:<p><i>Music theory for nerds</i> - <a href="https:&#x2F;&#x2F;news.ycombinator.com&#x2F;item?id=12528144" rel="nofollow">https:&#x2F;&#x2F;news.ycombinator.com&#x2F;item?id=12528144</a> - Sept 2016 (378 comments)

Love it! Learning in public FTW. (I&#x27;ve been making music for over 40 years, and some of this gave me a fresh appreciation for how convoluted and borderline impenetrable music theory can be.)

My quick notes on music theory: <a href="https:&#x2F;&#x2F;calebmadrigal.com&#x2F;music-theory-notes&#x2F;" rel="nofollow">https:&#x2F;&#x2F;calebmadrigal.com&#x2F;music-theory-notes&#x2F;</a>

I’m working on an IDE for music composition <a href="https:&#x2F;&#x2F;ngrid.io" rel="nofollow">https:&#x2F;&#x2F;ngrid.io</a>

Once you&#x27;ve understood the basics, get the Gradus Ad Parnassum by Johann Joseph Fux. A word to the wise is sufficient.

I read eev.ee&#x27;s tutorial on platformer physics when doing a little gamejam.<p>Worked great! What a talented person!

I have always struggled to learn music theory. If anyone has a good source please let me know.

Sorry, a lot of this is wrong. A major 7th is not 17:9 but 15:8. Our ear tries to represent all these rich resonances in as simple primes as possible (Think, combos of 2&#x27;s 3&#x27;s and 5&#x27;s - and <i>occasionally</i> 7&#x27;s in blue notes).<p>A minor 6th isn&#x27;t just a note in between, but a simple 4:5, The <i>minor</i> sound that we associate with sadness is simply the reciprocal nature of the ratio - the more complicated prime is in the denominator (as opposed to the overtonal nature of the M7 above)<p>Short answer: read Harmonic Experience by W.A. Mathieu

&quot;This has got to be some of the worst jargon and notation for anything, ever.&quot;<p>Ummm, do you math? ;)<p>My point of view of music theory is: sound is objective. music is subjective. where the two meet is music theory.

Much of the confusion and misunderstanding is because when this subject is taught and explained, the teacher usually jumps from sine waves to music, where a musical note is thought of as a sine wave.<p>But it is better to think of a musical note as a set of sine waves. Each sine wave in the set has a frequency which is an integer multiple of the frequency of the longest sine wave. These frequencies are known as harmonics.<p>If we plot these frequencies on a logarithmic scale, they&#x27;ll be evenly spaced.<p><a href="https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;File:Spectrogram_of_violin.png" rel="nofollow">https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;File:Spectrogram_of_violin.png</a> <a href="https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;File:Human_voice_spectrogram.jpg" rel="nofollow">https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;File:Human_voice_spectrogram.j...</a><p>Imagine a note as a comb, where each tooth represents the frequency of one of the sine waves. Now imagine a second comb where the teeth are spaced farther apart. Now imagine that you put one comb on top of the other so that the teeth line up. You&#x27;ll see a nice pattern. This is the harmony of the combs.<p><pre><code> C4 | | | | | | | | G4 | | | | | | | | C5 | | | | | | | | E4 | | | | | | | | </code></pre> But these combs aren&#x27;t only in an abstract frequency space. The cochlea inside your ear maps each frequency to a specific physical location where your ear hairs are tickled, forming a comb-like pattern.<p><a href="https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;File:Journey_of_Sound_to_the_Brain.ogg" rel="nofollow">https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;File:Journey_of_Sound_to_the_B...</a><p>Consider this chord composed of the notes G4, C5, and E5, which some of you may know as C 2nd inversion or G⁶₄.<p><pre><code> chord | | | | | | | | | | | | | | | | | | | | </code></pre> This chord will tickle your ears in many of the same places as the note C3.<p><pre><code> C3 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | </code></pre> One way we can tell human voices apart is by the relative intensities of the harmonics. And the way we tell different vowel sounds apart is by the relative intensities of the harmonics. So our brains have lots of wiring for interpreting relative intensities of harmonics.<p><a href="https:&#x2F;&#x2F;auditoryneuroscience.com&#x2F;sites&#x2F;default&#x2F;files&#x2F;Fig1-16color.jpg" rel="nofollow">https:&#x2F;&#x2F;auditoryneuroscience.com&#x2F;sites&#x2F;default&#x2F;files&#x2F;Fig1-16...</a> (source: <a href="https:&#x2F;&#x2F;auditoryneuroscience.com&#x2F;vocalizations-speech&#x2F;formants-harmonics" rel="nofollow">https:&#x2F;&#x2F;auditoryneuroscience.com&#x2F;vocalizations-speech&#x2F;forman...</a>)<p>So when we hear this chord, our brains are likely using some of the same wiring they would use for a human voice singing the note C3.

Seems not knowing the struggle with the note system. The struggle eg ## which is not really the note above because the # and b is not perfect. Bath well pieces probably not in his understanding. Not really a music theory.

Why did we settle on 12 notes?<p>Rule 1 (also stated in the article): Doubling frequency gives us practically the same note. It is so pleasant together with the original note, that it is boring.<p>Lets make a set of notes that sound nice together, and are more or less equally distributed between a note and its frequency double (octave).<p>The next most pleasant note, is a note with 3 times the frequency, then 4, etc...<p>(Why is this pleasant? Because its a close natural harmonic. When you vibrate a string with frequency 1, it will start vibrating with a frequency of 2 as well, but softer. This doubling in frequency is because the string will behave both as a string with length L, and as two separate strings with length L&#x2F;2. Because the length is halve, the frequency is doubled.)<p>Now, 3 times the frequency, is the same as 1.5 times the frequency. If we do the same from that base note, we get 1.5 * 1.5 = 2.25 times the frequency. You could decide that that 2.25 is close to 2, call it a day, and settle on a two note scale.<p>Or, you go further. 2.25&#x2F;2 = 1.125, so now we have the notes 1, 1.125, 1.5 and 2. Not really evenly spaced yet. 1.125*1.5=1.6875...: [1, 1.125, 1.265, 1.5, 1.687, 1.89]. Oh, 1.89 is almost 2. Let&#x27;s call it a day and settle on a five note scale. (there are plenty of cultures with a five not scale folk music tradition).<p>Note that the error gets smaller, it was 2.25&#x2F;2 for our 2 note scale, it&#x27;s 1.89&#x2F;2 for the 5 note scale. Now it happens that the next smallest error is with a 12 note scale. But there&#x27;s still an error, you don&#x27;t arrive at exactly 2. The intervals between your notes are not even, and your last interval is all retarded.<p>That&#x27;s why people settled on &quot;equal temperament&quot;, where all the intervals between the notes are made equal, and the number 12 looks seemingly arbitrary.<p>I did the math, see table below. The first column are the frequency of the notes, with base frequency 1, where I multiplied by 1.5, and divided by 2 to keep the notes between frequency 1 and 2. The 4th column gives you the &quot;error&quot;, how close is the last note of the scale, to the octave of the base note. The closer it is, the better, it means we have an equally divisible range of notes that all have nice (with ratio 1.5) intervals with each other.<p>The arrows indicate where we could stop adding notes to our scale, because we arrived at a local optimum: a five not scale is small but has quite an error. The 12 note scale is quite elaborate, but reduces the error a lot. The next best scale would be a 41 note scale, with only a slightly diminished error compared to the 12 note scale.<p><pre><code> Freq. n&#x2F;1 2&#x2F;n min(n&#x2F;1;2&#x2F;n) 1.500 1.50 1.33 1.333 1.125 1.13 1.78 1.125 1.688 1.69 1.19 1.185 1.266 1.27 1.58 1.266 1.898 1.90 1.05 1.053 &lt;-- (5 notes) 1.424 1.42 1.40 1.405 1.068 1.07 1.87 1.068 1.602 1.60 1.25 1.249 1.201 1.20 1.66 1.201 1.802 1.80 1.11 1.110 1.352 1.35 1.48 1.352 1.014 1.01 1.97 1.014 &lt;-- (12 notes) 1.520 1.52 1.32 1.315 1.140 1.14 1.75 1.140 1.711 1.71 1.17 1.169 1.283 1.28 1.56 1.283 1.924 1.92 1.04 1.039 1.443 1.44 1.39 1.386 1.082 1.08 1.85 1.082 1.624 1.62 1.23 1.232 1.218 1.22 1.64 1.218 1.827 1.83 1.09 1.095 1.370 1.37 1.46 1.370 1.027 1.03 1.95 1.027 1.541 1.54 1.30 1.298 1.156 1.16 1.73 1.156 1.734 1.73 1.15 1.153 1.300 1.30 1.54 1.300 1.951 1.95 1.03 1.025 1.463 1.46 1.37 1.367 1.097 1.10 1.82 1.097 1.646 1.65 1.22 1.215 1.234 1.23 1.62 1.234 1.852 1.85 1.08 1.080 1.389 1.39 1.44 1.389 1.041 1.04 1.92 1.041 1.562 1.56 1.28 1.280 1.172 1.17 1.71 1.172 1.758 1.76 1.14 1.138 1.318 1.32 1.52 1.318 1.977 1.98 1.01 1.012 &lt;-- (41 notes)</code></pre>

This has about as much to do with &quot;music&quot; as analyzing grammar when talking about poetry.<p>Musical notation, and lots of &quot;rules&quot; that come with it, came after the fact. Just like with natural language.<p>And just like staring at grammar rules is a terrible way to learn to speak a new language, analyzing music &quot;theory&quot; at this level does not teach you anything about actual music.

&gt;Not counting the octave, there are seven fairly nice fractions here. Hmm. Seven. What a conspicuous number.<p>There&#x27;s seven nice fraction because of how they were rounded.<p>But there&#x27;s no explanation of why he rounds 1.682 to 5&#x2F;3 but doesn&#x27;t round 1.587 to the even closer 8&#x2F;5.<p>It&#x27;s because scales are a social construction, not a result of vague math.

&gt;<i>If we say 440 Hz produces a note called A, then 880 Hz, 220 Hz, 1760 Hz, 110 Hz, and so forth will also produce a note called A. An important consequence of this is that all distinct notes we could possibly come up with must exist somewhere between 440 Hz and 880 Hz. Any other pitch could be doubled or halved until it lies in that range, and thus would produce a note in that range.</i><p>That&#x27;s inaccurate and backwards. 338 Hz is not in that range, and it can&#x27;t be doubled or halved to fit in that, but it&#x27;s a perfectly fine not. The range is more about tuning (which is a choice) than about 440-880Hz being something special.<p>&gt;<i>All of these things messily overlap and create multiple conflicting names for the same things, because they’re attempts to describe human intention rather than an objective waveform.</i><p>Nope, they don&#x27;t overlap any more messily than 1+2 == 2+1, or in how 0.99999... == 1.