Basic introduction to xenharmonic music

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Just a rough draft. [[Mike Battaglia]] and [[Keenan Pepper]] are writing it but suggested improvements are more than welcome. Will be TeXed up later into a document intended for very wide distribution.
Things to cover:
* What 12edo actually is (must drive home the point that there are 12 notes per octave, not, say, 8, as some people would say if you asked them)
* The circle/chain of fifths, "G# and Ab can be different", with historical info
* Different circle-of-fifths EDOs and their different enharmonic equivalences (with 19edo as the obvious example to focus on, but also 17edo)
* [[#htmldiff1]]Circles of other intervals, such as minor thirds? 
----
* Frequency, ratios, cents, Hz, the harmonic series (just briefly mentioning logarithms and saying what the practical implications are - this is not for mathematicians!)
* Approximate relationships between 12edo and JI - the fact that each interval represents *many* JI ratios (important because a lot of sources give exactly one JI ratio per 12edo interval which is very misleading)
* JI lattices (VERY BRIEFLY)
* Commas, the syntonic comma, puns, comma pumps
* How meantone temperament works
* MOS series (never mentioning continued fractions except maybe in a footnote)
* Finally, non-meantone rank-2 temperaments (this should be the pinnacle of the document, for which everything else is a required prerequisite) Porcupine is a good example that everybody loves. Could perhaps also introduce pajara as a non-octave-period rank-2 temperament.


=Mike's suggested ordering= 
**Unit 1** - picking up where the AP music theory test leaves off (and getting everyone else up to speed)
# What 12edo really is (must drive home the point that there are 12 notes per octave, not, say, 8, as some people would say if you asked them. just very basic stuff, no acoustics)
# What Western notation really is (note that there's both C# and Db for some reason, and that they both map to the same note)
# Explain that historically, C# and Db were different things, and that our notation is a remnant of the historic system (most music students are up to speed at this point)
# Introduce the idea that instead of this leading to a "12 tone unequal temperament," it actually leads to an "infinite" tuning system that goes on forever (e.g. meantone isn't just a 12 tone unequal tuning - there exist more than 12 notes in meantone. this is the first 'aha' moment that most music students will have)
# Introduce the idea of the circle of fifths "opening up" into the full neverending chain, and 12 being a particular way to "close" the chain, turning C# and Db into the same note
# Introduce the idea that 19 is another way to close the chain, and that this introduces "new enharmonic equivalences." (epic musical examples needed)
# Touch on other ways to close the chain - 17, 7, 5, 26, 31, etc. Possibly introduce the word "temperament" to explain this idea of dimensionality reduction
# Introduce the idea that there's even more to music than just the circle of fifths, and that "meantone" itself is a temperament of something more fundamental (Maybe remind people about the diminished scale and augmented and so on, I dunno. BE SURE TO NOT LEAD PEOPLE INTO THE RATIOS = CATEGORIES FALLACY)
# Play tantalizing porcupine/blackwood/machine clips and so on
# Introduce the idea that to understand what's going on, we'll dig even deeper than music and start looking at the building blocks of sound itself (except not really, lulz)

**Unit 2** - frequencies, ratios, etc
# Talk about frequencies, ratios, cents, Hz, the harmonic series (just briefly mentioning logarithms and saying what the practical implications are - this is not for mathematicians!)
# Approximate relationships between 12edo and JI - the fact that each interval represents *many* JI ratios (important because a lot of sources give exactly one JI ratio per 12edo interval which is very misleading)
# JI lattices (VERY BRIEFLY) (epic animation of the circle of fifths opening into a chain and then the chain itself opening into the 5-limit lattice!)
# Commas are enharmonic equivalences on steroids
# The syntonic comma makes the world go round (probably a bad idea to write <12 19 28|-4 4 -1>=0 here, haha...)
# Finally introduce new rank-2 temperaments, like blackwood and porcupine (epic musical examples needed here!)
# BRIEFLY introduce the concept of those rank-2 temperaments closing into other equal temperaments (e.g. as an approach to answer questions like "how do I use 15-EDO?")
# List some awesome temperaments and related scales using them (equip these scales with #/b accidentals plz)
# Awesome musical examples!

**Unit 3** - how to actually use temperaments
# Temperaments are broad objects involving intonation. WTF do I do with them?
# Puns! Comma pumps!
# Talk about MOS series as a generalization of pentatonic/diatonic/chromatic scales
# Talk about MODMOS's
# Higher-limit JI!
# Tuning accuracy for crunchy stuff, lower accuracy for color!
# (INVENT MORE COOL STUFF IN REAL LIFE AND WRITE IT HERE)

Feel free to change

Original HTML content:

<html><head><title>Basic introduction to xenharmonic music</title></head><body>Just a rough draft. <a class="wiki_link" href="/Mike%20Battaglia">Mike Battaglia</a> and <a class="wiki_link" href="/Keenan%20Pepper">Keenan Pepper</a> are writing it but suggested improvements are more than welcome. Will be TeXed up later into a document intended for very wide distribution.<br />
Things to cover:<br />
<ul><li>What 12edo actually is (must drive home the point that there are 12 notes per octave, not, say, 8, as some people would say if you asked them)</li><li>The circle/chain of fifths, &quot;G# and Ab can be different&quot;, with historical info</li><li>Different circle-of-fifths EDOs and their different enharmonic equivalences (with 19edo as the obvious example to focus on, but also 17edo)</li><li><!-- ws:start:WikiTextAnchorRule:2:&lt;img src=&quot;/i/anchor.gif&quot; class=&quot;WikiAnchor&quot; alt=&quot;Anchor&quot; id=&quot;wikitext@@anchor@@htmldiff1&quot; title=&quot;Anchor: htmldiff1&quot;/&gt; --><a name="htmldiff1"></a><!-- ws:end:WikiTextAnchorRule:2 -->Circles of other intervals, such as minor thirds?</li></ul><hr />
<ul><li>Frequency, ratios, cents, Hz, the harmonic series (just briefly mentioning logarithms and saying what the practical implications are - this is not for mathematicians!)</li><li>Approximate relationships between 12edo and JI - the fact that each interval represents *many* JI ratios (important because a lot of sources give exactly one JI ratio per 12edo interval which is very misleading)</li><li>JI lattices (VERY BRIEFLY)</li><li>Commas, the syntonic comma, puns, comma pumps</li><li>How meantone temperament works</li><li>MOS series (never mentioning continued fractions except maybe in a footnote)</li><li>Finally, non-meantone rank-2 temperaments (this should be the pinnacle of the document, for which everything else is a required prerequisite) Porcupine is a good example that everybody loves. Could perhaps also introduce pajara as a non-octave-period rank-2 temperament.</li></ul><br />
<br />
<!-- ws:start:WikiTextHeadingRule:0:&lt;h1&gt; --><h1 id="toc0"><a name="Mike's suggested ordering"></a><!-- ws:end:WikiTextHeadingRule:0 -->Mike's suggested ordering</h1>
 <strong>Unit 1</strong> - picking up where the AP music theory test leaves off (and getting everyone else up to speed)<br />
<ol><li>What 12edo really is (must drive home the point that there are 12 notes per octave, not, say, 8, as some people would say if you asked them. just very basic stuff, no acoustics)</li><li>What Western notation really is (note that there's both C# and Db for some reason, and that they both map to the same note)</li><li>Explain that historically, C# and Db were different things, and that our notation is a remnant of the historic system (most music students are up to speed at this point)</li><li>Introduce the idea that instead of this leading to a &quot;12 tone unequal temperament,&quot; it actually leads to an &quot;infinite&quot; tuning system that goes on forever (e.g. meantone isn't just a 12 tone unequal tuning - there exist more than 12 notes in meantone. this is the first 'aha' moment that most music students will have)</li><li>Introduce the idea of the circle of fifths &quot;opening up&quot; into the full neverending chain, and 12 being a particular way to &quot;close&quot; the chain, turning C# and Db into the same note</li><li>Introduce the idea that 19 is another way to close the chain, and that this introduces &quot;new enharmonic equivalences.&quot; (epic musical examples needed)</li><li>Touch on other ways to close the chain - 17, 7, 5, 26, 31, etc. Possibly introduce the word &quot;temperament&quot; to explain this idea of dimensionality reduction</li><li>Introduce the idea that there's even more to music than just the circle of fifths, and that &quot;meantone&quot; itself is a temperament of something more fundamental (Maybe remind people about the diminished scale and augmented and so on, I dunno. BE SURE TO NOT LEAD PEOPLE INTO THE RATIOS = CATEGORIES FALLACY)</li><li>Play tantalizing porcupine/blackwood/machine clips and so on</li><li>Introduce the idea that to understand what's going on, we'll dig even deeper than music and start looking at the building blocks of sound itself (except not really, lulz)</li></ol><br />
<strong>Unit 2</strong> - frequencies, ratios, etc<br />
<ol><li>Talk about frequencies, ratios, cents, Hz, the harmonic series (just briefly mentioning logarithms and saying what the practical implications are - this is not for mathematicians!)</li><li>Approximate relationships between 12edo and JI - the fact that each interval represents *many* JI ratios (important because a lot of sources give exactly one JI ratio per 12edo interval which is very misleading)</li><li>JI lattices (VERY BRIEFLY) (epic animation of the circle of fifths opening into a chain and then the chain itself opening into the 5-limit lattice!)</li><li>Commas are enharmonic equivalences on steroids</li><li>The syntonic comma makes the world go round (probably a bad idea to write &lt;12 19 28|-4 4 -1&gt;=0 here, haha...)</li><li>Finally introduce new rank-2 temperaments, like blackwood and porcupine (epic musical examples needed here!)</li><li>BRIEFLY introduce the concept of those rank-2 temperaments closing into other equal temperaments (e.g. as an approach to answer questions like &quot;how do I use 15-EDO?&quot;)</li><li>List some awesome temperaments and related scales using them (equip these scales with #/b accidentals plz)</li><li>Awesome musical examples!</li></ol><br />
<strong>Unit 3</strong> - how to actually use temperaments<br />
<ol><li>Temperaments are broad objects involving intonation. WTF do I do with them?</li><li>Puns! Comma pumps!</li><li>Talk about MOS series as a generalization of pentatonic/diatonic/chromatic scales</li><li>Talk about MODMOS's</li><li>Higher-limit JI!</li><li>Tuning accuracy for crunchy stuff, lower accuracy for color!</li><li>(INVENT MORE COOL STUFF IN REAL LIFE AND WRITE IT HERE)</li></ol><br />
Feel free to change</body></html>