writing challenge 96/100
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</article>
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<article>
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<h2>Latest gomepost</h2>
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<a class='journal-link' href='journal/candy.html'>
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<h3>Trying new candy</h3>
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<div class='line stop-2'></div>
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<time datetime='Wed, 12 Apr 2023 23:00:00 CDT'>12 Apr 2023, 11:00 PM</time>
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<p>I only ever buy international candy</p>
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<a class='journal-link' href='journal/just-intonation-5.html'>
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<h3>Just intonation: 5-limit tuning</h3>
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<div class='line stop-1'></div>
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<time datetime='Thu, 13 Apr 2023 23:30:00 CDT'>13 Apr 2023, 11:30 PM</time>
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<p>How do you build intervals out of primes?</p>
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</a>
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</article>
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<article id='listening'>
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<main>
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<h1>Journal</h1>
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<a href='rss/feed.xml' class='rss-link'>RSS feed</a>
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<a class='journal-link' href='just-intonation-5.html'>
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<h3>Just intonation: 5-limit tuning</h3>
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<div class='line stop-1'></div>
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<time datetime='Thu, 13 Apr 2023 23:30:00 CDT'>13 Apr 2023, 11:30 PM</time>
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<p>How do you build intervals out of primes?</p>
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</a>
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<a class='journal-link' href='candy.html'>
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<h3>Trying new candy</h3>
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<div class='line stop-2'></div>
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Are you interested in writing it?
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Does the new approach I’m trying work for you?
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Let me know your thoughts at my Ctrl-C email: <code>gome<span style='user-select: none;'> ​</span>@<span style='user-select: none;'> ​</span>ctrl-c.club</code>.
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</p><p class='footnote'>
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</p>
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<h4>Next post in series</h4>
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<a class='journal-link' href='just-intonation-5.html'>
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<h3>Just intonation: 5-limit tuning</h3>
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<div class='line stop-1'></div>
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<time datetime='Thu, 13 Apr 2023 23:30:00 CDT'>13 Apr 2023, 11:30 PM</time>
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<p>How do you build intervals out of primes?</p>
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</a>
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<p class='footnote'>
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* You can also get all the intervals just <a href='https://en.wikipedia.org/wiki/Pythagorean_tuning'>with combinations of ×2 and ×3</a>, but I think it’s kind of boring compared to 5-limit. Still worth reading about for further learning.
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</p>
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<!DOCTYPE html>
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<html lang='en'>
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<head>
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<title>gome — just intonation pt. 5</title>
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<meta charset='utf-8'/>
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<meta name='theme-color' content='#efe5d7'>
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<link rel='icon' type='image/x-icon' href='../favicon.ico'>
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<link rel='preconnect' href='https://fonts.googleapis.com'>
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<link rel='preconnect' href='https://fonts.gstatic.com' crossorigin>
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<link href='https://fonts.googleapis.com/css2?family=Lora:ital@0;1&display=swap' rel='stylesheet'>
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<link rel='stylesheet' type='text/css' href='../css/style.css'/>
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</head>
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<body>
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<header id='header'>
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<nav>
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<a href='..'>back to gomepage</a>—<a href='.'>journal</a>
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</nav>
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</header>
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<main>
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<article>
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<h1 id='title'>Just intonation: 5-limit tuning</h1>
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<time datetime='Thu, 13 Apr 2023 23:30:00 CDT'>13 Apr 2023, 11:30 PM</time>
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<p>
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Writing about just intonation is so hard, and I’m pretty sure most of my audience doesn’t care about it, and yet I must keep doing it!
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In <a href='just-intonation-4.html' target='_blank'>the last just intonation post</a>, I shared a table of intervals for the major scale in just intonation.
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Today we’ll look at the just intonation equivalents of all twelve tones in a similar fashion.
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Before the table, here’s some groundwork.
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</p><h3>Interval Quality</h3><p>
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It’s helpful to talk about two intervals that are the same under octave equivalence as the same interval.
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I’ll use the term <b>quality</b> to describe the property of an interval that doesn’t change depending on the specific octave it’s in.
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</p><p>
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In 12-tone equal temperament, you can find an interval’s quality by taking its size in semitones <a href='https://en.wikipedia.org/wiki/Modulo'>modulo 12</a>.
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For example, a major tenth, (16 semitones), has the same quality as a major third, (4 semitones), because 16 mod 12 and 4 mod 12 both equal 4.
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In just intonation, you can take the number representing the interval and remove all factors of two from its numerator and denominator (i.e., by multiplying by some number 2<sup><i>n</i></sup>, which is equivalent to transposition by <i>n</i> octaves).
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</p><p>
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The table below will include a column representing the quality of the interval derived in this way.
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</p>
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<figure>
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<img src='img/quadrangularis_reversum.webp' width='500' height='358' />
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<figcaption>Photo credit: <a href='https://commons.wikimedia.org/wiki/File:Gourd_Tree_%26_Cone_Gongs.jpg#Licensing' target='_blank'>HorsePunchKid</a></figcaption>
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</figure>
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<h3>Multiple intervals for a single name</h3><p>
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Traditional interval names such as “major second” or “augmented fourth” don’t always map cleanly to specific intervals.
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The name “major second” only maps to a single 12-tone interval, i.e., 2 semitones,
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but in just intonation, both ×9/8 and ×10/9 are major seconds, with the former just slightly sharper than the latter.
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Where I think it’s relevant, I’ll include both variants.
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</p><h3>Constructing the intervals</h3><p>
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If we didn’t have a table that already told us, how would we go about constructing the intervals we need?
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Our basic building blocks for this exercise are the primes up to 5: ×2 the octave, ×3 the large fifth, and ×5 the large third.
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We need to construct at least twelve interval qualities, to match the twelve tones of equal temperament.
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Since we are looking at qualities, we can keep ×2’s out of the picture and talk about everything in terms of qualities, treating octaves as equivalent.
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In the table, these qualities will be normalized into the range between ×1 and ×2, so, within a single octave.
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</p><p>
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So with just a major third and a perfect fifth, how would we construct the rest of our intervals?
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Well, let’s start by just combining them.
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A fifth plus a major third makes a major seventh.
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In equal temperament, you can get the interval by adding semitones: 7 + 4 = 11.
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In just intonation, the equivalent calculation, 3 × 5 = 15, yields our just major seventh, ×15.
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Now that we have a major seventh, we can invert it to get a minor second: ×1/15.
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</p><p>
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What if we subtract a major third from a fifth? That gives us a minor third quality, ×3/5, and inverting that gets us a minor sixth, ×5/3.
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How would we make a major second? Well, we can stack two fifths for ×9, or we can go down two fifths and up a major third, yielding ×5/9.
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</p><p>
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All the intervals in the table below can be constructed in the same manner.
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If you want to verify any of them for yourself, you can reference the prime factors column.
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Starting from middle C on a keyboard, move an octave for every 2 you see, move by a perfect fifth for every 3 you see, and move by a major third for every 5 you see.
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If the number is in the numerator, move up by the corresponding interval; if it’s in the denominator, move down.
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This will land you at the correct interval relative to middle C.
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The same process can be followed with the quality column, but it won’t land you in the correct octave, since interval quality ignores octaves.
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</p><table style='border-collapse: collapse;'>
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<tr><th>Musical interval name</th><th style='width: min-content;'>12-tone equivalent</th><th>Interval</th><th>Prime factors</th><th>Quality</th></tr>
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<tr><td>unison</td> <td>0</td> <td>×1</td> <td>∅</td> <td>1 / 1</td></tr>
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<tr><td>minor second</td> <td>1</td> <td>×16/15</td> <td>2×2×2×2 / 3×5</td> <td>1 / 3×5</td></tr>
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<tr><td>major second</td> <td>2</td> <td>×10/9</td> <td>2×5 / 3×3</td> <td>5 / 3×3</td></tr>
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<tr><td>major second</td> <td>2</td> <td>×9/8</td> <td>3×3 / 2×2×2</td> <td>3×3 / 1</td></tr>
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<tr><td>minor third</td> <td>3</td> <td>×6/5</td> <td>2×3 / 5</td> <td>3 / 5</td></tr>
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<tr><td>minor third</td> <td>3</td> <td>×32/27</td> <td>2<sup>5</sup> / 3×3×3</td> <td>1 / 3×3×3</td></tr>
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<tr><td>major third</td> <td>4</td> <td>×5/4</td> <td>5 / 2×2</td> <td>5 / 1</td></tr>
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<tr><td>perfect fourth</td> <td>5</td> <td>×4/3</td> <td>2×2 / 3</td> <td>1 / 3</td></tr>
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<tr><td>augmented fourth</td> <td>6</td> <td>×25/18</td> <td>5×5 / 2×3×3</td> <td>5×5 / 3×3</td></tr>
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<tr><td>augmented fourth</td> <td>6</td> <td>×45/32</td> <td>3×3×5 / 2<sup>5</sup></td> <td>3×3×5 / 1</td></tr>
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<tr style='border-top: 2px solid #323a42;'><td>diminished fifth</td> <td>6</td> <td>×64/45</td> <td>2<sup>6</sup> / 3×3×5</td> <td>1 / 3×3×5</td></tr>
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<tr><td>diminished fifth</td> <td>6</td> <td>×36/25</td> <td>2×2×3×3 / 5×5</td> <td>3×3 / 5×5</td></tr>
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<tr><td>perfect fifth</td> <td>7</td> <td>×3/2</td> <td>3 / 2</td> <td>3 / 1</td></tr>
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<tr><td>minor sixth</td> <td>8</td> <td>×8/5</td> <td>2×2×2 / 5</td> <td>1 / 5</td></tr>
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<tr><td>major sixth</td> <td>9</td> <td>×5/3</td> <td>5 / 3</td> <td>5 / 3</td></tr>
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<tr><td>major sixth</td> <td>9</td> <td>×27/16</td> <td>3×3×3 / 2×2×2×2</td> <td>3×3×3 / 1</td></tr>
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<tr><td>minor seventh</td> <td>10</td> <td>×16/9</td> <td>2×2×2×2 / 3×3</td> <td>1 / 3×3</td></tr>
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<tr><td>minor seventh</td> <td>10</td> <td>×9/5</td> <td>3×3 / 5</td> <td>3×3 / 5</td></tr>
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<tr><td>major seventh</td> <td>11</td> <td>×15/8</td> <td>3×5 / 2×2×2</td> <td>3×5 / 1</td></tr>
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<tr><td>octave</td> <td>12</td> <td>×2</td> <td>2 / 1</td> <td>2 / 1</td></tr>
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</table><p>
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I don’t really have time to write much else, but it’s useful to look at pairs of inverted intervals (those across the bold center line from each other) like ×4/3 and and ×3/2.
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Looking at their qualities, they are always inverses of each other, and looking at their prime factors, they’re always off by just a factor of ×2.
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Why might that be?
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</p><p>
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Are you learning anything from the just intonation series?
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Have you listened to any just intonation music?
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Did you notice a mistake in the big chart?
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Let me know your thoughts at my Ctrl-C email: <code>gome<span style='user-select: none;'> ​</span>@<span style='user-select: none;'> ​</span>ctrl-c.club</code>.
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</p>
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</article>
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</main>
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</body>
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</html>
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<width>144</width>
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<item>
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<title>Just intonation: 5-limit tuning</title>
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<author>gome</author>
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<pubDate>Thu, 13 Apr 2023 23:30:00 CDT</pubDate>
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<link>https://ctrl-c.club/~gome/journal/just-intonation-5.html</link>
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<guid>https://ctrl-c.club/~gome/journal/just-intonation-5.html</guid>
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<description>How do you build intervals out of primes?</description>
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<enclosure url="https://ctrl-c.club/~gome/journal/img/quadrangularis_reversum.webp" length="60140" type="image/webp" />
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</item>
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<item>
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<title>Trying new candy</title>
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<author>gome</author>
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<width>144</width>
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</image>
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<item>
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<title>Just intonation: 5-limit tuning</title>
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<author>gome</author>
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<pubDate>Thu, 13 Apr 2023 23:30:00 CDT</pubDate>
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<link>https://ctrl-c.club/~gome/journal/just-intonation-5.html</link>
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<guid>https://ctrl-c.club/~gome/journal/just-intonation-5.html</guid>
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<description>How do you build intervals out of primes?</description>
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<enclosure url="https://ctrl-c.club/~gome/journal/img/quadrangularis_reversum.webp" length="60140" type="image/webp" />
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</item>
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<item>
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<title>Trying new candy</title>
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<author>gome</author>
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