Diamond-mos notation

Diamond-mos notation is a microtonal notation system designed for writing non-diatonic mosses.

Credits

User:SupahstarSaga (HEHEHE I AM A SUPAHSTAR SAGA#1163) contributed most of the key ideas. User:Godtone (osmium#0444) and User:Inthar (Inthar#5959) helped with designing accidentals for engraving software.

Motivation

So far all common microtonal notation schemes have been diatonic-based (e.g. ups and downs for edos, Sagittal, FJS). While these existing systems are useful in many contexts, they don't accommodate the logic of non-diatonic mos scales easily, requiring multiple accidentals to write these scales.

Diamond-mos notation is designed specifically to visually represent mos logic for compositions using mosses. It does not attempt to be a universal system that represents every microtonal theoretical school. It attempts to be back-compatible with standard notation and keep new symbols and memorization required to a minimum. In addition, diamond-mos includes visual aids and redundancies to make reading music in unfamiliar systems easier.

Concept

Diamond-mos notation is (usually) based in a specific edo at a given time. Within that edo, the naturals on the staff represent a specific mos. The mos pattern is written in edosteps at the top of the score, e.g. “33133131”. The natural staff notes are the specific mode written, starting from middle C.

Diamond-mos notation treats the diatonic mos (5L 2s) as special. If you’re using 5L 2s, diamond-mos looks like standard music notation. It uses standard clefs, #/b accidentals, and ABCDEFG note names. If the edo you’re using has 5L 2s, its notes can be used as an unchanging reference other mosses can be compared against. For example, you might say “P@ = Bb” to provide a reference for which note P@ is.

As a soft rule, the mode you write for notation should be the mode you’re writing in. This rule allows you to figure out the tonic just by looking at the key signature, as explained later. We call this a soft rule because there are situations where it makes less sense, such as if the tonic is ambiguous or shifting. If you break this rule, we would suggest making a note on the score when you do.

Staff

When using a mos different from 5L 2s, diamond-mos notation does not have traditional clefs. Instead, octaves on the staff are marked with diamonds on the left edge of the staff. The notes marked with diamonds are named J. The large diamond marks “middle J”, which is enharmonic to middle C. The point of the diamonds is to provide a visual reference for where the octaves are. It would be difficult to sightread otherwise, especially when using non-heptatonic mosses.

The mos pattern is marked on the score too. Just to the right of the diamonds, little boxes mark the steps of the mos. Hollow boxes represent small mossteps (s) and filled boxes represent large mossteps (L). To avoid clutter, only the small mossteps or the large mossteps are marked. The ones which get marked are the ones the mos has fewer of. (In the case of nL ns mosses, either is fine.) The boxes are another tool to make reading the score easier.

The staff can be extended up and down to have more than 5 lines when using non-heptatonic mos notation. When extra staff lines are used, at least one of the staff lines should be bolded. This doesn't add information but helps with readability.

Accidentals

There are three types of accidentals in diamond-mos. There are diatonic accidentals (#, b, …), mos accidentals (&, @, …), and ups and downs (^, v, …).

Diatonic accidentals are what you’d expect. # raises a note by a diatonic chroma, the difference between the diatonic whole tone and semitone. b lowers a note by the same amount. x and bb are the doublings of those. Half-sharp (t) and half-flat (d) represent changes of half a diatonic chroma, and are only allowed if #/b represent an even number of edosteps.

Mos accidentals are similar to diatonic accidentals, but for non-diatonic mosses. Mos accidentals are based on the specific mos used. & raises a note by a moschroma, the difference between the large mosstep (L) and the small mosstep (s). @ lowers a note by the same amount. e and a represent half of & and @ respectively, and are only allowed if &/@ represent an even number of edosteps.

The third type of accidentals are ups and downs. ^ raises by one edostep and v lowers by one edostep. There are double and triple versions, ^^/vv and ^^^/vvv, which represent movements of two and three edosteps respectively. Ups and downs can be combined with the other two types to create compound accidentals, such as ^# and vv@. 

The meaning of &/@ accidentals is dependent on the specific mos being used. &/@ have no meaning on the diatonic staff, so are not allowed in that case. Ups and downs can always be used. If the edo has 5L 2s, the meaning of #/b accidentals is fixed, not dependent on any context. However, in a non-diatonic mos context, #/b have no theoretical significance and could potentially be confusing. We heavily discourage #/b in a non-diatonic mos context for that reason. However, using them could be useful in very large edos where you need all the accidental options you can get.

When using 5L 2s in an edo where t/d are one edostep, we encourage using t/d instead of ^/v. This is to increase consistency with existing microtonal notation, such as notation for 17edo, 24edo, and 31edo. The same rule does not apply to e/a semi-moschroma accidentals.

A subtle but important feature of the standard is that accidentals are separate from each other and do not stack. If a note has an accidental in front, that accidental overrides any preceding accidentals. For example, if a note with a # is followed by a note with an ^, the sharp on the previous note does not apply to the new note.

Note names

When using a non-diatonic mos, the JKL… note names represent notes of the current mos. Only J has a fixed meaning (middle J = middle C); the remaining note names depend on both the mos pattern and the current mode of the mos (under our soft rule about what note names mean). For example, when you are writing in the LSLLSLSL mode of 5L3s, the “natural” note names JKLMNOPQ represent the scale LSLLSLSL on J.

Key signatures

To change the tonic, we use the key signature of mos accidentals to indicate how many chroma-positive mos generators the tonic is above middle J = middle C. (A generator of a mos is called chroma-positive if the larger intervals in each generic interval class of the mos result from stacking the generator upwards. For example, the chroma-positive generator of 5L 2s is the fifth, because the major third is 4 fifths up and the minor third is 3 fifths down.)

For example, if you want to notate the key of D in the 42442424 mode of 5L 3s in 26edo:

The chroma-positive generator of 5L 3s is the subfourth, 466.7¢.

D is reached by going 3 subfourth generators up from C.

This means that the key signature should have three &’s, notated on the appropriate notes to represent the 42442424 scale on D. 

The new tonic is K (which determined by the choice of mode), and the notes will be K L& M N O& P Q& J K. So the key signature will have an & on L, O and Q.

Similarly if your key is reached by stacking chroma-positive generators down, the number of generators taken to do so is the number of @’s in the key signature.

What notes you put the accidentals on in the key signature will depend on the mode, but the number of accidentals only depends on the number of generators up from J = C. This rule allows you to figure out the tonic just by counting the accidentals in the key signature.

If your key is not reachable from C by stacking a given generator in the edo (i.e. if the mos generator is contained in an edo subset), the key signature will indicate the number of generators (indicated by &’s or @’s) + the number of edo steps (indicated by shifting every accidental on the key signature by that fixed number of ^/v) the tonic is above C. For example, writing the key of F in 42442424 in 26edo will require one & on Q (to change the tonic from C to Fb), plus an ^ on every note in J...Q (to shift the tonic from Fb to F). Thus the key signature will be: ^& on Q, ^ on every other note.

The order of the accidentals of the key signature from left to right should follow the generator chain, just like how 12edo key signatures follow the circle of fifths.

= Modmosses

The mos mode written on the score, and represented by the boxes, cannot be a modmos. However, there is a way to make modmos key signatures to avoid cluttering the score with accidentals. First, we write the key signature corresponding to the tonic as if the mos is not modified. This is so that you can still find the tonic just by counting the accidentals. Next, we add the modmos alterations as extra accidentals to the right of the existing ones. These modmos accidentals have an exclamation point after them to indicate that they’re modifying the mos. As discussed before, all accidentals in diamond-mos completely override any preceding accidentals. This includes modmos accidentals in the key signature. [Show a picture of 34edo D Ionian v3 v6 v7 to illustrate this]

Edo superset notation

It’s often a good idea to notate small edos as subsets of larger edos. For example, notating 13edo as a subset of 26edo allows you to use ABC notation. To do this for a diamond-mos score, write “13edo, superset 26edo” at the top of the score. For the purposes of notation, the superset edo is the edo the score is in. However, you can’t stray outside the subset edo, because that wouldn’t make sense.

Using edo notation to represent non-edo tunings

Diamond-mos notation is based on an edo, but that doesn’t mean it can only represent music in edos. The tones of an edo in diamond-mos can be retuned to any scale of the same size. After the edo, write “Retune [scale]” at the top of the score, with the scale’s tones written out explicitly.

If you don’t use a tone of the edo at all in the score, you can write “X” in your scale, e.g. “7edo, retune 24:26:X:32:36:39:43:48”.

Sightreading

By looking at the boxes on the staff, you can quickly figure out what the interval between two notes is. What interval class (2-mosstep, 3-mosstep, etc) the interval is is easy to see; just count the distance between the notes. From there, look at the boxes on the left of the score contained by the interval. This will tell you what the interval would be if the notes were natural. If the interval contains one more filled box than it could in another position, or one fewer hollow box, it’s major. If the interval contains one more hollow box than it could in another position, or one fewer filled box, it’s minor. (Generators are called perfect/augmented/diminished, not major and minor, but it’s the same idea. Every interval class has at most two sizes.) Finally, look at the accidentals on the notes, including ones implied by the key signature. If the bottom note is natural and the top note has &, for example, the interval is one moschroma wider than the boxes suggest. Remember that a moschroma is the difference between any major interval and its corresponding minor interval.

This method allows you to figure out the size of the interval just from the visual aids on the score. You still have to remember how many edosteps each mos interval is, but that’s much less work than memorizing all the interval sizes on every scale degree.

How to choose between enharmonics

Especially when working in small edos, there will be cases where there is more than one way to write a note. For example, J& and K@ may be enharmonic to each other. The way to choose which to write is similar to how you would make the choice in a 12edo score. The spelling you choose should reflect the mos structure of the piece. More practically speaking, you should choose spellings which lead to arrangements of notes which don’t span too many generators. For example, in 12edo diatonic, “Bb Db F” spans four fifths (Db Ab Eb Bb F), while “Bb C# F” spans nine fifths.

Exceptions

There’s an exception to the rule that diamond-mos notation is always based on an edo. There is also the option of specifying a mos tuning which does not come from an edo. At the top of the score, write “Non-edo mos tuning” and specify the exact size of the generator. Instead of writing the mos mode in edosteps, use generic mos pattern notation, e.g. “LLsLsLs”. When using non-edo notation, ^/v have no meaning and are disallowed. The only available accidentals are the mos-based &/@/e/a accidentals (or #/b/t/d if using 5L 2s). Since non-edo notation doesn’t use a fixed, finite set of pitches, retuning to an arbitrary scale is disallowed.

It is acceptable to treat equal scales (sub-edos of the edo used for notation) as mosses for notation in diamond-mos, for example 9edo in 18edo. In that case, &/@ have no meaning and are disallowed. There are also no boxes on the staff.

It is acceptable to use 5L 2s-based notation even for edos where the best fifth is 4\7 or 3\5. This is for consistency with existing microtonal notation that allows this. In the case of 4\7, #/b technically alter notes by zero edosteps, so they’re disallowed. Equalized and paucitonic tunings are only allowed for 5L 2s, not other mosses.

It is possible to use non-octave equal tunings and non-octave mosses with diamond-mos. In this case, the diamonds mark the structural equave (the period or occasionally a multiple of the period). JKL note names repeat at this same equave. Explicitly write out what this equave is along with the tuning and mos pattern.