Nick Vuci's Fundamentals of Xen: Difference between revisions

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=== Harmonic Series ===

A fundamental aspect of JI is the harmonic series. The harmonic series is the series that happens when you keep adding the first frequency over and over, ad infinitum. Which is 1:2:3:4:5:6:7… If we take a segment of it, we get different intervals and chords. For example, 4:5:6 is the major triad. ''Try it out in'' [https://scaleworkshop.plainsound.org Scale Workshop] ''by creating a new scale from the ‘harmonic series segment’''!

[[File:4-5-6 segment.mp4|none|thumb|4:5:6 segment (major triad)]]

''Clarifaction: something that confused me when I was learning about Just Intonation is the conflation I made between JI intervals and divisions of string vibrations. That’s because it’s two ways of looking at the same thing. See the physics forumla wavelength is proportionate to the speed of sound (or light) over frequency: lambda equals v over f.''

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[[File:Subharmonic series 7-1.png|thumb|Subharmonic Series segment 7-1|none]]

Now remember how I said 4:5:6 is the major triad? Well, the subharmonic version is the minor chord: 8/6, 8/5, 8/4 or 1/(6:5:4). This means 4:5:6 and 1/(6:5:4) are Mathematical inversions for each other, or the ''otonal'' and the ''utonal'' versions (for overtone and undertone).

[[File:Harmonic Series segment 1-(6-5-4) (minor triad).mp4|none|thumb|Harmonic Series segment 1/(6:5:4) (minor triad)]]

=== Tonality Diamond ===

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You can generate any temperament by stacking an interval over and over. Whatever that interval is, that’s the generator of the temperament. A generator is a universal concept in Xenharmonics. The JI lattice the infinite 5-limit diamond, is made using 3/2 and 5/4 as generators. 12edo? You can make using 100 cents as a generator. Generators are intervals you stack on themselves to make some structure.

<blockquote>~~explain rank-2 temperaments Nick Vuci — 2:44 AM No, but I get why you are confused.~~ Basically think of it like this Imagine JI 5-limit So what do you have? You have 2.3.5 JI Which is rank-3 fm’latghor — 2:45 AM 3 generators on the tonality diamond Nick Vuci — 2:46 AM Like you need 3 generators: 2.3.5 And you get 4:5:6 triads Now why happens with meantone? You temper out a comma and collapse a rank. In this case you temper so that a single generator gives you an approximation of both 3 and 5 So meantone is a 2.3.5 (5-limit) temperament but it’s only rank-2 since you can make it by using an octave and a meantone generator (where a tempered 3 also gives you a 5) Then you can reduce any temperament to an equal tuning, where again you reduce rank so that a single generator gives you the 2, the 3, and the 5 So all equal tunings are rank-1 Lots of this stuff is kinda abstract

<blockquote>Basically think of it like this Imagine JI 5-limit So what do you have? You have 2.3.5 JI Which is rank-3

fm’latghor — 2:45 AM 3 generators on the tonality diamond

Nick Vuci — 2:46 AM Like you need 3 generators: 2.3.5 And you get 4:5:6 triads Now why happens with meantone? You temper out a comma and collapse a rank. In this case you temper so that a single generator gives you an approximation of both 3 and 5 So meantone is a 2.3.5 (5-limit) temperament but it’s only rank-2 since you can make it by using an octave and a meantone generator (where a tempered 3 also gives you a 5) Then you can reduce any temperament to an equal tuning, where again you reduce rank so that a single generator gives you the 2, the 3, and the 5 So all equal tunings are rank-1 Lots of this stuff is kinda abstract

</blockquote>

== MOS ==

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 === Harmonic Series ===
 A fundamental aspect of JI is the harmonic series. The harmonic series is the series that happens when you keep adding the first frequency over and over, ad infinitum. Which is 1:2:3:4:5:6:7… If we take a segment of it, we get different intervals and chords. For example, 4:5:6 is the major triad. ''Try it out in'' [https://scaleworkshop.plainsound.org Scale Workshop] ''by creating a new scale from the ‘harmonic series segment’''!
+[[File:4-5-6 segment.mp4|none|thumb|4:5:6 segment (major triad)]]
 ''Clarifaction: something that confused me when I was learning about Just Intonation is the conflation I made between JI intervals and divisions of string vibrations. That’s because it’s two ways of looking at the same thing. See the physics forumla wavelength is proportionate to the speed of sound (or light) over frequency: lambda equals v over f.''
@@ Line 33: / Line 34: @@
 [[File:Subharmonic series 7-1.png|thumb|Subharmonic Series segment 7-1|none]]
 Now remember how I said 4:5:6 is the major triad? Well, the subharmonic version is the minor chord: 8/6, 8/5, 8/4 or 1/(6:5:4). This means 4:5:6 and 1/(6:5:4) are Mathematical inversions for each other, or the ''otonal'' and the ''utonal'' versions (for overtone and undertone).
+[[File:Harmonic Series segment 1-(6-5-4) (minor triad).mp4|none|thumb|Harmonic Series segment 1/(6:5:4) (minor triad)]]
 === Tonality Diamond ===
@@ Line 102: / Line 104: @@
 You can generate any temperament by stacking an interval over and over. Whatever that interval is, that’s the generator of the temperament. A generator is a universal concept in Xenharmonics. The JI lattice the infinite 5-limit diamond, is made using 3/2 and 5/4 as generators. 12edo? You can make using 100 cents as a generator. Generators are intervals you stack on themselves to make some structure.
-<blockquote>explain rank-2 temperaments Nick Vuci — 2:44 AM No, but I get why you are confused. Basically think of it like this Imagine JI 5-limit So what do you have? You have 2.3.5 JI Which is rank-3 fm’latghor — 2:45 AM 3 generators on the tonality diamond Nick Vuci — 2:46 AM Like you need 3 generators: 2.3.5 And you get 4:5:6 triads Now why happens with meantone? You temper out a comma and collapse a rank. In this case you temper so that a single generator gives you an approximation of both 3 and 5 So meantone is a 2.3.5 (5-limit) temperament but it’s only rank-2 since you can make it by using an octave and a meantone generator (where a tempered 3 also gives you a 5) Then you can reduce any temperament to an equal tuning, where again you reduce rank so that a single generator gives you the 2, the 3, and the 5 So all equal tunings are rank-1 Lots of this stuff is kinda abstract
+<blockquote>Basically think of it like this Imagine JI 5-limit So what do you have? You have 2.3.5 JI Which is rank-3
+fm’latghor — 2:45 AM 3 generators on the tonality diamond
+Nick Vuci — 2:46 AM Like you need 3 generators: 2.3.5 And you get 4:5:6 triads Now why happens with meantone? You temper out a comma and collapse a rank. In this case you temper so that a single generator gives you an approximation of both 3 and 5 So meantone is a 2.3.5 (5-limit) temperament but it’s only rank-2 since you can make it by using an octave and a meantone generator (where a tempered 3 also gives you a 5) Then you can reduce any temperament to an equal tuning, where again you reduce rank so that a single generator gives you the 2, the 3, and the 5 So all equal tunings are rank-1 Lots of this stuff is kinda abstract
 </blockquote>
 == MOS ==