User:Eufalesio/JI functions

Preamble

Functional harmony is great. It makes pitches go places predictably and we can know more or less what they are. Functional harmony had only been working within 12edo or meantone however, and as the scope of music gets exponentially more complex when dealing with close approximations of JI approached from too many different points of view, the tower of Babel is destroyed.

There had been many proposals to add more functions. However, all of them are idiosyncratic. Most important problem is, unless you're somebody famous: no one will learn your functions. I stopped myself from making a set of functions based on Ultimate because at the end of the day no one would understand me, and no one will give a shit about what I think to be right.

However, there is one language everyone in the Xen spaces speaks: JI. Be it ratios, EFR, applied to RTT, fudged, tempered, tackled directly, NEJI'd we all refer to this absolute reference when dealing with most frequency ratio-based tunings. I say then that to produce an understandable set of functional harmony... you need JI, thus infinite functions! WHat the FUCK?

JI functions

Purpose

Communication. Speaking to musicians who know what JI is, and making everyone understand themselves again. Belief is poison. If people start to say things for how they want them to be or for how they feel they are, misunderstandings and disputes will ensue. Thus, intuitiveness is NOT the aim, but consistency with current practices, and most importantly: falsifiable. (see addendum)

What they are

It seems utterly idiotic to have to know infinite functions. It seems that I've become insane, but bear with me for just a moment. You don't need to know all functions. You pick and choose what you need. But what exactly are you picking...?

What you pick is a JI ratio (octave-reduced most often but needn't be). Any. Whichever you need. Every single JI ratio has a unique function. Function being, a role. Music bound by functional JI harmony makes each JI ratio go places predictably, be it arriving or parting, just as functional harmony in 12edo has its pitches go places.

Disclaimer: What those roles are is NOT on the scope of this article. That is composition. How you handle each JI ratio, that's for you to figure out, either rigurously described, or intuitively applied. No, what I am showing you here, is how to adress these functions and communicate what you need.

How to write them

This is the gist of it:

• a/b is a JI ratio. Nothing else.
• ~a/b is a function. It is the unique role of the ratio a/b, and it can be assigned to any pitch reasonably close to it. We say pitch X works as ~a/b, or that pitch X ~a/b's (conjugate a/b as if it was a verb; ~a/b's, ~a/b'd, ~a/b'ing...).
• ~a/b~c/d~e/f... is many functions. A pitch with with these targets can take any of function of the JI ratios a/b, c/d, e/f... etc.

Any pitch, JI or not, can take many functions.

• 7/4 is ~7/4 trivially, but you can make it ~7/4~16/9.
• 3\12 is ~6/5~32/27~7/6 always due to 7-limit patent val temperament.
• 14\53 is ~6/5~65/54 always due to 2.3.5.13 patent val temperament, but you can make it ~32/27.

Remember that I've said that any function can be assigned to any pitch reasonably close to it. This is important, as you can distinguish many ways to make that function in your music.

• True function: ~a/b is always a true function of a/b. A pitch that is close enough to be deemed the main realization of a/b, be it through a temperament or through a fudger, also has ~a/b as its true function. You can expect this pitch, even if it is not exactly a/b, to ~a/b.
• Tempered function: ~a/b is a tempered function of a pitch if it is not the main realization, but one that comes at the expense of being part of a regular temperament. For instance, in meantone, the major third (see addendum) ~5/4's, but that is not its true function. The major third (81/64)'s true function is ~81/64, but in meantone, it has the tempered function ~5/4 by definition.
• False function: ~a/b is a false function of a pitch if its not the main realization, and it is not even one that may come from a regular temperament. For instance, using the minor seventh of 31edo, which does not temper out 64/63, to ~7/4. This can be used even in JI to do fudging, comma drift, pivots, pedals, modulations... whatever you desire!

Addendum

Falsifiability

We're artists, and intuition is for the art. For explaining these technical things, falsifiable language is necessary, independent of belief. Forgetting about bespoke categories or arbitrary conceptualizations, and sticking to the real math. After we're done explaining, we can continue to talk about intuition, how these things feel, and how we give functions to pitches to compose delectable music.

Regarding vocabulary

There is confusion on whether or not things like "minor second", "major third" or "fifth" are taken to be JI ratios, or mappings of a temperament or an edo. Context can easily clear the ambiguity, but, since context may not always be available, to disambiguate, I propose the use of the written JI ratios 256/243, 81/64, 3/2 to refer exclusively to the JI ratios, and the names for the maps.

Regarding mappings, in the context of a temperament of course a/b won't be the exact JI ratio, but its monzo, analyzed through the mapping matrix of the temperament. You can always say "our a/b", or "a/b map" to be more precise. Layered precision, that's what's is best for everyone.

This will be helpful in the following examples:

• "I need a flatter meantone in this movement so that the ~10/9 really comes out of that major second."
• "We need our 25/24 to ~256/243 here to compensate for the comma drift of the last passage."
• "Yeah, this high D is actually ~147/128 too. This is because 159edo tempers out the gamelisma." (slightly modified, quoting Aura in one past call I had with him long ago)
• "Hold that 11/8 A4, we'll make it ~7/4 later on to make a spicy pedal!"