Nick Vuci's Fundamentals of Xen - Revision history

FloraC: + categories

2026-02-26T11:27:19Z

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	[[File:Tonnetz Lattice.png\|thumb\|Tonnetz Lattice\|none]]		[[File:Tonnetz Lattice.png\|thumb\|Tonnetz Lattice\|none]]
	A nice way to lay out chord progressions is through a tonnetz lattice. Tonnetz originates from Neo-Riemannian Music Theory and the Mathematical toys of figures like Euler. [https://generalizedtonnetz.nickvuci.com/ Play with Vuci’s Generalized Tonnetz here]. 1. Input the EDO, here we do 12edo 2. Input the intervals for a minor triad, X: 7, Y: 3 3. For the diatonic scale, input the relative scale degrees in the drop down as “2,2,2,1,2,2,1” (which is MOS 5L 2s 2:1) 4. Highlight “0” if you want to 5. Input the minor triad for the red chord overlay: 0,3,7 6. Input the major triad for the blue chord overlay: 0,4,7 7. Then click on triangles and try out that progression in your DAW.		A nice way to lay out chord progressions is through a tonnetz lattice. Tonnetz originates from Neo-Riemannian Music Theory and the Mathematical toys of figures like Euler. [https://generalizedtonnetz.nickvuci.com/ Play with Vuci’s Generalized Tonnetz here]. 1. Input the EDO, here we do 12edo 2. Input the intervals for a minor triad, X: 7, Y: 3 3. For the diatonic scale, input the relative scale degrees in the drop down as “2,2,2,1,2,2,1” (which is MOS 5L 2s 2:1) 4. Highlight “0” if you want to 5. Input the minor triad for the red chord overlay: 0,3,7 6. Input the major triad for the blue chord overlay: 0,4,7 7. Then click on triangles and try out that progression in your DAW.

			[[Category:Essays]]
			[[Category:Guides]]

Fmlatghor at 21:37, 25 February 2026

2026-02-25T21:37:58Z

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	=== Lattices ===		=== Lattices ===
	Lattices are another way to visualize JI: [[File:Wilsonic lattice.mp4\|thumb\|Wilsonic ~~lattice~~\|none]]		Lattices are another way to visualize JI: [[File:Wilsonic lattice.mp4\|thumb\|Wilsonic Lattice\|none]]
	=== Conclusion ===		=== Conclusion ===
	In essence, JI is when intervals can be defined as frequencies that align and form whole number ratios. The tonality diamond is the conceptual layout of an odd-limit.		In essence, JI is when intervals can be defined as frequencies that align and form whole number ratios. The tonality diamond is the conceptual layout of an odd-limit.
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	== Tonnetz ==		== Tonnetz ==
	[[File:Tonnetz Lattice.png\|thumb\|Tonnetz Lattice]]		[[File:Tonnetz Lattice.png\|thumb\|Tonnetz Lattice\|none]]
	A nice way to lay out chord progressions is through a tonnetz lattice. Tonnetz originates from Neo-Riemannian Music Theory and the Mathematical toys of figures like Euler. [https://generalizedtonnetz.nickvuci.com/ Play with Vuci’s Generalized Tonnetz here]. 1. Input the EDO, here we do 12edo 2. Input the intervals for a minor triad, X: 7, Y: 3 3. For the diatonic scale, input the relative scale degrees in the drop down as “2,2,2,1,2,2,1” (which is MOS 5L 2s 2:1) 4. Highlight “0” if you want to 5. Input the minor triad for the red chord overlay: 0,3,7 6. Input the major triad for the blue chord overlay: 0,4,7 7. Then click on triangles and try out that progression in your DAW.		A nice way to lay out chord progressions is through a tonnetz lattice. Tonnetz originates from Neo-Riemannian Music Theory and the Mathematical toys of figures like Euler. [https://generalizedtonnetz.nickvuci.com/ Play with Vuci’s Generalized Tonnetz here]. 1. Input the EDO, here we do 12edo 2. Input the intervals for a minor triad, X: 7, Y: 3 3. For the diatonic scale, input the relative scale degrees in the drop down as “2,2,2,1,2,2,1” (which is MOS 5L 2s 2:1) 4. Highlight “0” if you want to 5. Input the minor triad for the red chord overlay: 0,3,7 6. Input the major triad for the blue chord overlay: 0,4,7 7. Then click on triangles and try out that progression in your DAW.

Fmlatghor at 21:37, 25 February 2026

2026-02-25T21:37:11Z

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	* Temperaments		* Temperaments

	With EDOs, MOS, and temperaments, it’s important to know that although these concepts are ~~all~~ presented within ~~the same~~ context, they ~~can~~ exist ~~separately or all within the same system~~.		With EDOs, MOS, and temperaments, it’s important to know that although these concepts are presented within a unified context, they exist as modular components.

	== Just Intonation (JI) ==		== Just Intonation (JI) ==

Fmlatghor at 21:35, 25 February 2026

2026-02-25T21:35:14Z

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	The four pillars of Xen theory are: - Just Intonation (JI) - Equal Divisions of the Octave (EDO) - Moments of Symmetry (MOS) - Temperaments		The four pillars of Xen theory are:

			* Just Intonation (JI)
			* Equal Divisions of the Octave (EDO)
			* Moments of Symmetry (MOS)
			* Temperaments

	With EDOs, MOS, and temperaments, it’s important to know that although these concepts are all presented within the same context, they can exist separately or all within the same system.		With EDOs, MOS, and temperaments, it’s important to know that although these concepts are all presented within the same context, they can exist separately or all within the same system.

Fmlatghor at 21:34, 25 February 2026

2026-02-25T21:34:09Z

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	=== Harmonic Series ===		=== 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’''!		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.''		''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]]		[[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).		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 ===		=== 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.		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>		</blockquote>
	== MOS ==		== MOS ==

Fmlatghor at 21:23, 25 February 2026

2026-02-25T21:23:30Z

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	In essence, JI is when intervals can be defined as frequencies that align and form whole number ratios. The tonality diamond is the conceptual layout of an odd-limit.		In essence, JI is when intervals can be defined as frequencies that align and form whole number ratios. The tonality diamond is the conceptual layout of an odd-limit.

	Some extra things on Just Intonation is that there are other approaches to Just-Intonation besides the basic tonality diamonds. You have [https://en.xen.wiki/w/Primodality primodality] that exploits prime modes of the harmonic series and incorporates commas, which are ~~usually~~ seen as a musical ~~inconvience or~~ blight, and then you have Hexany by Erv Wilson which is a way to make JI without a tonal-center using Combination Product Sets. The [https://www.wilsonic.co/ Wilsonic software] makes all these things accessible.		Some extra things on Just Intonation is that there are other approaches to Just-Intonation besides the basic tonality diamonds. You have [https://en.xen.wiki/w/Primodality primodality] that exploits prime modes of the harmonic series and incorporates commas, which are conventionally seen as a musical inconvenience / blight, and then you have Hexany by Erv Wilson which is a way to make JI without a tonal-center using Combination Product Sets. The [https://www.wilsonic.co/ Wilsonic software] makes all these things accessible. What's important to note about high-complexity JI music psycho-acoustically is that extremely high-limit JI isn't going to be perceived as high limit JI, rather it is that super complex JI intervals end up being reinterpreted as approximations of simpler JI intervals. Point of diminishing returns.

	== Equal Divisions of the Octave (EDOs) ==		== Equal Divisions of the Octave (EDOs) ==

Fmlatghor at 21:17, 25 February 2026

2026-02-25T21:17:20Z

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	Moments of Symmetry (MOS) are special scales based on a simple principle. Take some generator and reduce in some period (usually the octave). Stack the generator over and over. When you get some collection of notes by doing this that have only two step sizes, you have a moment of symmetry. Another way to think of it is to consider you have some number of large (L) and small (s) steps, say 5L and 2s. Make it so that the 2s are as evenly distributed amongst the 5L as possible. This looks like LLsLLLs or sLLsLLL. This is called maximal evenness.		Moments of Symmetry (MOS) are special scales based on a simple principle. Take some generator and reduce in some period (usually the octave). Stack the generator over and over. When you get some collection of notes by doing this that have only two step sizes, you have a moment of symmetry. Another way to think of it is to consider you have some number of large (L) and small (s) steps, say 5L and 2s. Make it so that the 2s are as evenly distributed amongst the 5L as possible. This looks like LLsLLLs or sLLsLLL. This is called maximal evenness.

	=== Step ~~size ratio:~~ ===		=== Step-Size Ratio ===

	The largness or smallness of the steps is called the step size ratio. The quality of the step ratio, whether it’s hard or soft or equalized is categorized according to the Temperament Agonstic Mos NAMing System (TAMNAMS). The ways to modify mosses are documented on the page “Operations on MOSes”. Things like chromatic alterations, subdividing scales (muddling), or neutralizing the scale are all documented. There’s a lot you can do with MOS. Things like splitting the scale into a hexachord or a tetrachord allows you to transpose the same melodic segment. You can morph between scales, say between hard diatonic and soft diatonic. It’s worth looking at the MOS family tree as well for reference, illustrating MOS’s recursive structure. The Math behind MOS is actually quite simple, but that’s not what we’re hear to cover.		The largness or smallness of the steps is called the step size ratio. The quality of the step ratio, whether it’s hard or soft or equalized is categorized according to the Temperament Agonstic Mos NAMing System (TAMNAMS). The ways to modify mosses are documented on the page “Operations on MOSes”. Things like chromatic alterations, subdividing scales (muddling), or neutralizing the scale are all documented. There’s a lot you can do with MOS. Things like splitting the scale into a hexachord or a tetrachord allows you to transpose the same melodic segment. You can morph between scales, say between hard diatonic and soft diatonic. It’s worth looking at the MOS family tree as well for reference, illustrating MOS’s recursive structure. The Math behind MOS is actually quite simple, but that’s not what we’re hear to cover.

Fmlatghor at 21:16, 25 February 2026

2026-02-25T21:16:39Z

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	=== Subharmonic Series ===		=== Subharmonic Series ===
	Let’s take a step back. You know about the harmonic series, sometimes called the overtone series, since harmony results from the alignment of partials. Now we have to introduce the second-to-last fundamental idea of JI, and that is the subharmonic (or undertone) series. if we go to scale workshop and choose 1 as the lowest number and 8 as the highest, we get this scale: 2/1, 3/1, 4/1, 5/1, 6/1, 7/1, 8/1.		Let’s take a step back. You know about the harmonic series, sometimes called the overtone series, since harmony results from the alignment of partials. Now we have to introduce the second-to-last fundamental idea of JI, and that is the subharmonic (or undertone) series. if we go to scale workshop and choose 1 as the lowest number and 8 as the highest, we get this scale: 2/1, 3/1, 4/1, 5/1, 6/1, 7/1, 8/1. Note the scale shape:
	[[File:Harmonic series 2-8.png\|thumb\|Harmonic Series segment 2-8\|none]]		[[File:Harmonic series 2-8.png\|thumb\|Harmonic Series segment 2-8\|none]]
	~~Note the scale shape: [[harmonic_series_2-8 1.png]]~~ As we go up the scale, intervals get smaller and smaller. The subharmonic series is the Mathematical opposite of this, and the same 1/8 range gives us this undertone series segment: 8/7, 8/6, 8/5, 8/4, 8/3, 8/2, 8/1. It’s important to know that although we’re calling this the undertone series, the undertone series does not have a physical acoustic basis.		As we go up the scale, intervals get smaller and smaller. The subharmonic series is the Mathematical opposite of this, and the same 1/8 range gives us this undertone series segment: 8/7, 8/6, 8/5, 8/4, 8/3, 8/2, 8/1. It’s important to know that although we’re calling this the undertone series, the undertone series does not have a physical acoustic basis.
	[[File:Subharmonic series 7-1.png\|thumb\|Subharmonic Series segment 7-1\|none]]		[[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).		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).

Fmlatghor at 21:15, 25 February 2026

2026-02-25T21:15:58Z

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	=== Lattices ===		=== Lattices ===
	[[File:Wilsonic lattice.mp4\|thumb\|Wilsonic lattice\|none]]		Lattices are another way to visualize JI: [[File:Wilsonic lattice.mp4\|thumb\|Wilsonic lattice\|none]]
	~~Lattices are another way to visualize JI:~~

	=== Conclusion ===		=== Conclusion ===
	In essence, JI is when intervals can be defined as frequencies that align and form whole number ratios. The tonality diamond is the conceptual layout of an odd-limit.		In essence, JI is when intervals can be defined as frequencies that align and form whole number ratios. The tonality diamond is the conceptual layout of an odd-limit.
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	[[File:7-Limit Tonality Diamond.png\|none\|thumb\|7-Limit Tonality Diamond]]		[[File:7-Limit Tonality Diamond.png\|none\|thumb\|7-Limit Tonality Diamond]]
	We can remove the 6 (i.e. the 3) and we get this:		We can remove the 6 (i.e. the 3) and we get this:
			[[File:2.5.7 Subgroup.png\|none\|thumb\|2.5.7 Subgroup Tonality Diamond]]
	[[~~Pasted image 20260223170844~~.png]]

	And if we input A = 1, B = 1, C = 7, you get these intervals.		And if we input A = 1, B = 1, C = 7, you get these intervals.
			[[File:2.5.7 Subgroup Triadic Diamond.png\|none\|thumb\|2.5.7 Subgroup Triadic Diamond]]
	[[~~Pasted image 20260223170924~~.png]]

	Same shape, but different harmonic structure: 2.5.7 subgroup.		Same shape, but different harmonic structure: 2.5.7 subgroup.

	Now let’s go back to the infinite JI lattice and look at the small interval 50/49. I said any interval that goes thru that line before, but that was just to evoke mental imagery. Really, it’d go through every other interval on the line the same distance apart, e.g.:		Now let’s go back to the infinite JI lattice and look at the small interval 50/49. I said any interval that goes thru that line before, but that was just to evoke mental imagery. Really, it’d go through every other interval on the line the same distance apart, e.g.:
			[[File:Infinite JI Lattice 50-49 comma.png\|none\|thumb\|Infinite JI Lattice 50/49 comma]]
	[[~~Pasted image 20260223170932~~.png]]

	That’s essentially a subgroup temperament. Not sure the name, but the names aren’t important.		That’s essentially a subgroup temperament. Not sure the name, but the names aren’t important.
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	I took the 7-limit diamond and removed a prime to get the 2.5.7 subgroup, then generated a lattice using the intervals those primes generate and found a small interval I wanted to squash out. That interval is called a comma. Squashing it out is called “tempering it out,” and what results is the temperament.		I took the 7-limit diamond and removed a prime to get the 2.5.7 subgroup, then generated a lattice using the intervals those primes generate and found a small interval I wanted to squash out. That interval is called a comma. Squashing it out is called “tempering it out,” and what results is the temperament.

	Now let’s look at 27edo. Let’s take the 27edo tempered representation of the 2.5.7 subgroup intervals. Hmm, if we superimpose the JI lattice together, we see the spot where 50/49 would be 1\27 not 0\27. 27edo does not temper out 50/49, but what does it temper? Whatever JI intetrval the red dots of the tempered lattice overlay are related to the commas that 27edo tempers out. [[~~Pasted image 20260223171009~~.png]] Here we see that 27 tempers out 128/125 and you can make music in that temperament in 27edo by using 4:6:7 and 1/(7:6:4) chords.		Now let’s look at 27edo. Let’s take the 27edo tempered representation of the 2.5.7 subgroup intervals. Hmm, if we superimpose the JI lattice together, we see the spot where 50/49 would be 1\27 not 0\27. 27edo does not temper out 50/49, but what does it temper? Whatever JI intetrval the red dots of the tempered lattice overlay are related to the commas that 27edo tempers out.
			[[File:Infinite JI Lattice 128-125 Comma.png\|none\|thumb\|Infinite JI Lattice 128/125 Comma]]


			Here we see that 27 tempers out 128/125 and you can make music in that temperament in 27edo by using 4:6:7 and 1/(7:6:4) chords.

	=== Temperaments Ext. ===		=== Temperaments Ext. ===

Fmlatghor at 21:10, 25 February 2026

2026-02-25T21:10:44Z

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	=== Subharmonic Series ===		=== Subharmonic Series ===
	~~[[File:Harmonic series 2-8.png\|thumb\|Harmonic Series segment 2-8]]~~		Let’s take a step back. You know about the harmonic series, sometimes called the overtone series, since harmony results from the alignment of partials. Now we have to introduce the second-to-last fundamental idea of JI, and that is the subharmonic (or undertone) series. if we go to scale workshop and choose 1 as the lowest number and 8 as the highest, we get this scale: 2/1, 3/1, 4/1, 5/1, 6/1, 7/1, 8/1.
	Let’s take a step back. You know about the harmonic series, sometimes called the overtone series, since harmony results from the alignment of partials. Now we have to introduce the second-to-last fundamental idea of JI, and that is the subharmonic (or undertone) series. if we go to scale workshop and choose 1 as the lowest number and 8 as the highest, we get this scale: 2/1, 3/1, 4/1, 5/1, 6/1, 7/1, 8/1. Note the scale shape: [[harmonic_series_2-8 1.png]] As we go up the scale, intervals get smaller and smaller. The subharmonic series is the Mathematical opposite of this, and the same 1/8 range gives us this undertone series segment: 8/7, 8/6, 8/5, 8/4, 8/3, 8/2, 8/1. It’s important to know that although we’re calling this the undertone series, the undertone series does not have a physical acoustic basis.		[[File:Harmonic series 2-8.png\|thumb\|Harmonic Series segment 2-8\|none]]
	[[File:Subharmonic series 7-1.png\|thumb\|Subharmonic Series segment 7-1]]		Note the scale shape: [[harmonic_series_2-8 1.png]] As we go up the scale, intervals get smaller and smaller. The subharmonic series is the Mathematical opposite of this, and the same 1/8 range gives us this undertone series segment: 8/7, 8/6, 8/5, 8/4, 8/3, 8/2, 8/1. It’s important to know that although we’re calling this the undertone series, the undertone series does not have a physical acoustic basis.
			[[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).		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).

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	=== Lattices ===		=== Lattices ===
	[[File:Wilsonic lattice.mp4\|thumb\|Wilsonic lattice]]		[[File:Wilsonic lattice.mp4\|thumb\|Wilsonic lattice\|none]]
	Lattices are another way to visualize JI:		Lattices are another way to visualize JI:

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	== Equal Divisions of the Octave (EDOs) ==		== Equal Divisions of the Octave (EDOs) ==

	Last time we covered JI and mentioned the harmonic series. 2/1, 3/1, 4/1, 5/1, 6/1, 7/1, 8/1 is the segment we looked at last time. See how the steps get smaller, but in a regular way. ~~[[harmonic_series_2-8 1.png]]~~ Technically speaking this is a type of equal division, the Arithmitic Equal Division, where we instead multiply the base frequency by a set amount. That amount is (2/1)^(1/x), where x is our number of equal divisions. Henceforth, we will refer to logarithmic equally divided octaves as EDOs. Unlike JI intervals that can be described as x/y ratios, the intervals of EDOs will always be irrational numbers when you try to express them as ratios. Instead we decide to use a unit called the cent, which divides the octave into 1200 logarithmically equal parts. That way we can say that the cent values and it’s easier to describe them. With EDOs there are two main categories, prime and non-primes. Non-prime EDOs are composite (by definition), which means you can think of them as being made up of smaller EDOs interwoven within each other. If an EDO is some multiple of a prime EDO, it can be called a superset of that prime EDO (e.g. 22edo is a superset of 11edo, and 11edo is a subset of 22edo).		Last time we covered JI and mentioned the harmonic series. 2/1, 3/1, 4/1, 5/1, 6/1, 7/1, 8/1 is the segment we looked at last time.
			[[File:Harmonic series 2-8.png\|none\|thumb\|Harmonic Series segment 2-8]]
			See how the steps get smaller, but in a regular way. Technically speaking this is a type of equal division, the Arithmitic Equal Division, where we instead multiply the base frequency by a set amount. That amount is (2/1)^(1/x), where x is our number of equal divisions. Henceforth, we will refer to logarithmic equally divided octaves as EDOs. Unlike JI intervals that can be described as x/y ratios, the intervals of EDOs will always be irrational numbers when you try to express them as ratios. Instead we decide to use a unit called the cent, which divides the octave into 1200 logarithmically equal parts. That way we can say that the cent values and it’s easier to describe them. With EDOs there are two main categories, prime and non-primes. Non-prime EDOs are composite (by definition), which means you can think of them as being made up of smaller EDOs interwoven within each other. If an EDO is some multiple of a prime EDO, it can be called a superset of that prime EDO (e.g. 22edo is a superset of 11edo, and 11edo is a subset of 22edo).

	== Subgroups ==		== Subgroups ==
Line 67:		Line 70:

	Temperaments are based on approximating some JI thing, which can be a subgroup. Standard meantone is a 5-limit temperament, but orgone is a 2.7.11 temperament. A way you can think of it is that meantone uses 4:5:6 and 1/(6:5:4) based harmony and orgone uses 8:11:14 and 1/(14:11:8) based harmony. The limit or the subgroup it’s referring to is the JI structures and harmony the temperament is trying to represent. Most temperaments refer to the JI structure you are approximating and bending in an equal division.		Temperaments are based on approximating some JI thing, which can be a subgroup. Standard meantone is a 5-limit temperament, but orgone is a 2.7.11 temperament. A way you can think of it is that meantone uses 4:5:6 and 1/(6:5:4) based harmony and orgone uses 8:11:14 and 1/(14:11:8) based harmony. The limit or the subgroup it’s referring to is the JI structures and harmony the temperament is trying to represent. Most temperaments refer to the JI structure you are approximating and bending in an equal division.
			[[File:Triadic Diamond.png\|none\|thumb\|Triadic Diamond Wilsonic]]
	Take the triadic diamond:		Take the triadic diamond:
			[[File:Infinite JI Lattice.png\|none\|thumb\|Infinite JI Lattice]]
			Then extrapolate it infinitely.

	~~[[Pasted image 20260223170637.png]]~~		The further you go, the further you get from 1/1, and no matter how far you go, you’ll never get back to 1/1. But sometimes you’ll get an interval that’s very close to 1/1 in sound, but not in number. Take the meantone comma, 81/80, and draw a line through 81/80 and 1/1 infinitely. Now imagine we take 81/80 and we make it equal to 1/1 and we spread the 21 cents out throughout the whole lattice. The point is that every interval on the line, between 1/1 and hte small interval 81/80, becomes equal to 1/1 when 81/80 does. Now the lattice loops instead of going on forever. Now when you use major and minor chords to get to the spot where 81/80 would be, you instead land on 1/1, and all of the major and minor chords are slightly out of tune. Whenever an EDO approximates some JI intetrvals, you can make the same triangular grid with the EDO intervals instead of the JI ones and it’ll show you what commas it tempers out visually. Now what does all this mean? Let’s take the 7-limit diamond.
			[[File:7-Limit Tonality Diamond.png\|none\|thumb\|7-Limit Tonality Diamond]]
	~~Then extrapolate it infinitely. [[Pasted image 20260223170725.png]]~~ The further you go, the further you get from 1/1, and no matter how far you go, you’ll never get back to 1/1. But sometimes you’ll get an interval that’s very close to 1/1 in sound, but not in number. Take the meantone comma, 81/80, and draw a line through 81/80 and 1/1 infinitely. Now imagine we take 81/80 and we make it equal to 1/1 and we spread the 21 cents out throughout the whole lattice. The point is that every interval on the line, between 1/1 and hte small interval 81/80, becomes equal to 1/1 when 81/80 does. Now the lattice loops instead of going on forever. Now when you use major and minor chords to get to the spot where 81/80 would be, you instead land on 1/1, and all of the major and minor chords are slightly out of tune. Whenever an EDO approximates some JI intetrvals, you can make the same triangular grid with the EDO intervals instead of the JI ones and it’ll show you what commas it tempers out visually. Now what does all this mean? Let’s take the 7-limit diamond. [[~~Pasted image 20260223170748~~.png]]

	We can remove the 6 (i.e. the 3) and we get this:		We can remove the 6 (i.e. the 3) and we get this:

Line 113:		Line 116:

	== Tonnetz ==		== Tonnetz ==
			[[File:Tonnetz Lattice.png\|thumb\|Tonnetz Lattice]]
	A nice way to lay out chord progressions is through a tonnetz lattice. Tonnetz originates from Neo-Riemannian Music Theory and the Mathematical toys of figures like Euler. [https://generalizedtonnetz.nickvuci.com/ Play with Vuci’s Generalized Tonnetz here]. 1. Input the EDO, here we do 12edo 2. Input the intervals for a minor triad, X: 7, Y: 3 3. For the diatonic scale, input the relative scale degrees in the drop down as “2,2,2,1,2,2,1” (which is MOS 5L 2s 2:1) 4. Highlight “0” if you want to 5. Input the minor triad for the red chord overlay: 0,3,7 6. Input the major triad for the blue chord overlay: 0,4,7 7. Then click on triangles and try out that progression in your DAW.~~[[Pasted image 20260223171154.png]]~~		A nice way to lay out chord progressions is through a tonnetz lattice. Tonnetz originates from Neo-Riemannian Music Theory and the Mathematical toys of figures like Euler. [https://generalizedtonnetz.nickvuci.com/ Play with Vuci’s Generalized Tonnetz here]. 1. Input the EDO, here we do 12edo 2. Input the intervals for a minor triad, X: 7, Y: 3 3. For the diatonic scale, input the relative scale degrees in the drop down as “2,2,2,1,2,2,1” (which is MOS 5L 2s 2:1) 4. Highlight “0” if you want to 5. Input the minor triad for the red chord overlay: 0,3,7 6. Input the major triad for the blue chord overlay: 0,4,7 7. Then click on triangles and try out that progression in your DAW.