User:Eufalesio/Proposal to the Functional Just System

Preamble

The Functional Just system is one of a kind. On the three systems of strict JI notation that I know: It, Johnson's system, and Helmholtz-Ellis; FJS is in my opinion, the most rigorous. It has been generalized to such an extent that it can provide a unique nomenclature to every single rational interval. Besides of course 0 and infinity. Its rigor, comes however with some unintuitive consequences, specially regarding the unintended consequences of a completely rigorous system.

The price of rigor

Whenever FJS produces a name for an interval, it is unquestionable. Exact. For example, 5/4 is M3⁵. 7/6 is m3⁷. 7/5 is d5⁷₅. Changing radius of tolerance (RT henceforth) may produce different names for certain intervals, for example; From the standard RT = 65/63, the formal comma for 31 is 248/243, but, from Flora Canou's proposal of RT = sqrt(2187/2048), the comma for 31 is 31/32. With the official RT, 31/16 is M7^³¹. With Flora Canou's RT, it is P8^31. In my biased opinion I think that Flora Canou's system, albeit possibly more unintuitive for some, minmaxxes simplicity and intuitiveness. It impedes primes from being assigned a formal comma in augmented/diminished intervals (aside from A4/d5).

Nontheless, the price of this simplicity, of this rigor, comes with lack of flexibility.

Particularly in the case that one composer might want to write using FJS, and is accustomed to writing the augmented sixth from Ab as F#, and coming from experience in meantone temperaments (including the well-established 12EDO), feels that the interval 7/4 from Ab should be spelt Ab-F# because it will resolve to G, not Ab-Gb as the correct FJS might proscribe, leading to the possibly good sounding, but unintuitively written movement Ab4₅,C5,Eb5₅,Gb5⁷₅ - G4,C5,E5⁵,G5. It is not a matter of correctness, but rather a matter of understanding and intuiting, both what the composer wants to write, and what the reader understands.

In my own experience, I feel that the interval 7/4 is ambivalent enough to serve as both a consonance like the major third, a type of minor seventh that might resolve down following a V⁷ - I perfect cadence, or a type of augmented sixth that resolves up. Being able to both describe 7/4 as a minor seventh and an augmented sixth is a very useful ability for a JI notation system, even if it is not entirely rigorous.

Introduction of the pythagorean comma as an optional formal comma

My proposal here is twofold: The solution to the problem I've stated above is introducing the pythagorean comma as a fixed formal comma for 3. This comma has the special property that it is redundant, but it can be quite helpful both to conceptualize convoluted intervals, and aid in writing and theory to reduce the amount of sharps/flats, and augmented/diminished intervals as much as possible.

Ideally, an implementation of this proposal would be a toggle. For example, you could notate the interval |27 -17⟩ as dd3, thus going up the scale like A4 Cbb5 D5 Cbb5 A4, if you wanted to compose in the 3-limit exclusively, but wanted an interval smaller than a limma. The alternative, would be A4 Bb4₃ D5 Bb4₃ A4, which might be an useful alternative

Implementation of interval regions for convoluted intervals

For lack of a better term, a convoluted interval is an interval whose notation does not match the expected pitch region. A great example of this is the schisma, which is a negative five-over diminished second, d-2⁵. Teeny alterations by this interval make intervals like Kirnberger's fifth, which sounds like a fifth, to be a diminished sixth. (d6⁵).

Another convoluted interval is 13/10, because people can't make up their mind on whether it is a kind of "third" or a kind of "fourth". It falls almost right in the middle of the extremes of these regions. The solution? Acknowledging that it could be both! d4¹³₅ , M3¹³_3,5.

For really convoluted commas, the solution is not that easy. The vavoom comma is still as unintuitive, officially 12d-7_5^17, as an alternative M2_3^24,5^17. My proposal to these convoluted intervals is to add before hand the interval region that it actually belongs to sonically + oid.

Using the completely arbitrary set of names for the regions according to the RT regions that each base Pythagorean nominal maps to: diesoid, limmoid, tonoid, melloid, tertioid, tessoid, diaboloid, pentoid, epogdoonoid, sixtoid, heptoid, septoid; plus the use of "compound" to add or subtract octaves, the vavoom comma is a type of diesoid major second, 13/10 is a type of tessoid major third. 88/63 is a type of diaboloid fourth. The Alpharabian minor ninth is a type compound diesoid limma.

As with the first proposal, it is completely redundant and optional. The names I chose for the interval regions are scale-agnostic enough to my ears to qualify.