Ledzo notation

Ledzo notation is a JI and regular temperament notation proposed as an alternative to color notation by Vector Graphics. It aims to solve these observed problems with color notation:

• The association of intervals to "colors" in color notation is extraneous, unnecessary, and possibly counterintuitive for people with intuitive associations between colors and numbers/ratios/intervals/
• Spoken color notation sounds out of place in most contexts
• Color notation is 3-limit-centric
• Color notation attempts to be systematic but ends up being really patched together

Ledzo notation also retains these properties of color notation:

• No new symbols: all new accidentals are familiar characters; hence they are immediately speed-readable.
• Furthermore, they are all on the QWERTY keyboard, making the notation easily typeable.
• Every new accidental has a spoken name
• Most importantly, one can name not only notes but also intervals.

As opposed to creating new syllables and words to allow the notation to be spoken, the cross-linguistic nature is achieved in the written representation, by using symbols (the Latin alphabet and Arabic numerals) that are globally recognizable, and thus would have names in as many languages as possible, and that are already used in established xen notation systems for similar purposes.

The word "ledzo" is a portmanteau of "letter" and "monzo".

Mechanics

The monzo is a way of notating the prime factors of an interval. The exponent to which each prime is raised is listed in order of increasing magnitude of the primes. 17/16, for example, is 17 in the numerator and 2⁴ in the denominator, so can be written as 2^-4 * 17, or more fully 2^-4 * 3⁰ * 5⁰ * 7⁰ * 11⁰ * 13⁰ * 17¹, so the monzo is written [-4 0 0 0 0 0 1⟩.

(All ratios in simplest form do not include the same prime in both the numerator and denominator. For example, 3*5*2/3³ is the same as 5*2/3². This is because a number divided by itself equals 1.)

A simple approach would be just to read out the monzo itself, i.e. "minus four four minus one" for [-4 4 -1⟩. However, this has 2 problems,

1. it can be ambiguous whether, say, "negative one eleven" is [-1 11⟩ or [-111⟩
2. monzos that use very high primes can get repetitive and confusing: [-4 0 0 0 0 0 1⟩ would be said "negative four zero zero zero zero zero one".

However, there is a solution: warts!

Wart notation itself is not used here, as vals are outside the scope of this notation system (which deals in exclusively JI and abstract temperaments). However, wart notation does set a convention for referring to primes without just specifying their value (which would solve problem 2 but make problem 1 worse by introducing numbers that mean different things): use the other set of typable symbols we have an order and names for, letters! Each prime is assigned a letter in increasing order, so the prime 2 is called a, 3 is called b, 5 is called c, 7 is called d, etc. Each monzo entry is postfixed with its respective prime's letter, primes which aren't in the ratio are skipped, and primes with an exponent of 1 in the ratio are not given a number (primes with an exponent of -1 are just given a minus sign). So, the monzo [-4 0 0 0 0 0 1⟩ becomes the ledzo -4ag, and the monzo [-4 4 -1⟩ becomes the ledzo -4a4b-c.

(And if you are using a prime higher than 101, either something's seriously gone wrong or this notation probably won't be useful for your use case anyway.)

LEGACY: Intervals with no a specified are assumed to be octave-reduced, you can specify 0a if the distinction is significant. Due to this, octave complements can be specified by flipping the signs of the entries for each prime. For example, the major third is c, while the minor sixth is -c.

Reduction of intervals is no longer assumed.

Octave complements can be specified by flipping the signs of the entries for each prime, and adding 1 to the a entry. For example, the major third is -2ac, while the minor sixth is 3a-c.

To provide a sense of the interval's size, ledzo notation can be paired with WBN or another system for specifying interval regions in order to give a sense of the size of the interval. In the case of WBN, only the first step is taken, so intervals are only labeled "third", "sixth", etc. Additionally, WBN does not have a built-in system of interval arithmetic, so using interval arithmetic with ledzo+WBN intervals becomes harder.

In order to disambiguate from other notation systems, letters are always written lowercase (so that, say, a, when written, isn't confused for the note A).

Additionally, if the ledzo for an interval is a single letter with no number or minus sign, it takes a 1 before it, so that when spoken, ledzos aren't confused for note names.

Examples

Ratio	Monzo	Ledzo and WBN
2/1	[1⟩	1a octave
3/2	[-1 1⟩	-ab fifth
4/3	[2 -1⟩	2a-b fourth
5/4	[-2 0 1⟩	-2ac third
8/5	[3 0 -1⟩	3a-c sixth
6/5	[1 1 -1⟩	ab-c third
5/3	[0 -1 1⟩	-bc sixth
7/6	[-1 -1 0 1⟩	-a-bd third
9/7	[0 2 0 -1⟩	2b-d third
14/9	[1 -2 0 1⟩	a-2bd sixth
7/5	[0 0 -1 1⟩	-cd tritone
10/7	[1 0 1 -1⟩	ac-d tritone
81/80	[-4 4 -1⟩	-4a4b-c comma
64/63	[6 -2 0 -1⟩	6a-2b-d comma
531441/524288	[-19 12⟩	-19a12b comma
3/1	[0 1⟩	1b tritave

Interval arithmetic

Ledzo notation supports a form of interval arithmetic natively, since intervals are stacked just by adding their monzos together. For example, a major third plus a minor third is a perfect fifth, and that's represented as follows: [-2 0 1⟩ * [1 1 -1⟩ = [-1 1 0⟩. This translates intuitively into the ledzo: -2ac + ab-c = -ab (note that concatenation here does not mean multiplication in the logarithmic scale this is using, positive entries have a hidden plus sign that is not written!)

Regular temperaments

Regular temperaments can be notated by taking a, b, c, etc to stand for the tempered primes used in the temperament.