Mike Battaglia's KISS notation
"KISS Notation" is a notation system for any rank-2 tuning system, originally proposed by Mike Battaglia on the Yahoo! Tuning list: https://groups.yahoo.com/neo/groups/TUNING/conversations/topics/105842
It could be easily extended to any epimorphic scale, rather than used only for MOS, but below is the original specification.
Numbers for note nominals were originally suggested in the "initial post" below, but in the follow-ups, I suggested going with letters instead to avoid confusion between absolute pitches and relative ones.
TO DO LATER: condense this, add pictures!
(Initially from https://groups.yahoo.com/neo/groups/TUNING/conversations/messages/105842, with some editing for formatting)
I've been working on a very simple, standardized notation scheme for any tuning system. It has standard nominals, a standard way to assign accidentals, generalizes the two-staff bass/treble clef notation, has a standard way to standardize the range of each staff, and has a standard way to set the absolute pitch of the staves (and hence the whole tuning system).
My intention is for it to be a very broad and simple blueprint, used for one to get set up quickly in any arbitrary temperament and start communicating with other musicians. I previously wrote about this and called it the KISS notation system; this is hence KISS 2.0. The whole thing is very simple and works as follows, simplified to just use MOS's for now:
- 1. Pick an MOS that you want to be "diatonic."
- 2. Ensure that every staff covers at least an octave by giving each staff round(N/2+1) lines for an N-note scale. So a 7 note scale would have 5 lines, an 8 note scale would have 5 lines, a 9 note scale would have 6 lines, a 10 note scale would have 6 lines. A 6 note scale would only have 4 lines, and a 5 note scale would also only have 4 lines.
- 3. Standardize one ledger line below the treble clef to represent the same pitch as one ledger line above the bass clef; I'll call this pitch the "Middle Note."
- 4. Pick a "privileged mode" of your scale.
- 5. Set the key signature up so that your privileged mode, when the Middle Note is used as the tonic, is represented by the lines and spaces of the staff with no accidentals.
ABSOLUTE PITCH RANGE
- 6. Standardize the tessitura of the entire notation system to be what it is now by setting the absolute pitch of the Middle Note to be 261.6 Hz.
- 7. Set the #/b accidentals to refer to the chroma L-s.
- 8. Set the nominals for your scale up to be ascending numerals so that the middle note is nominal "1."
And that's it. Some of it is so simple that it may not seem like it's worth writing about, but it's still probably good to write it explicitly somewhere.
This is not the end-all-be-all of notation. Depart from this and make your own tuning-specific tweaks to the notation if it serves you; use different accidentals for porcupine, or different nominals for meantone, or a different amount of lines if you think that 5 for meantone is too many, or a different absolute pitch if you want your playing to be in tune with 50 Hz electrical hum. I just hope that if you don't have strong feelings about one or more of those things, and want to just start talking about orwell MODMOS's and writing music, for instance, this can serve as a reference for some sensible "default" parameters to pick.
There's probably a nice way to generalize alto, tenor, baritone, etc clefs as well, but as a pianist I never come in contact with those, so I'm limiting my focus only to bass and treble clefs for now. I'd be interested if people have thoughts about that, though.
Follow-up: Using Letters instead of Numbers
(Originally from https://groups.yahoo.com/neo/groups/TUNING/conversations/messages/105844, reformatted for MediaWiki)
A few afterthoughts about my rationale behind some of this stuff...
On Tue, Jan 22, 2013 at 3:59 AM, Mike Battaglia <battaglia01@...> wrote:
- I've been working on a very simple, standardized notation scheme for any tuning system. It has standard nominals, a standard way to assign accidentals, generalizes the two-staff bass/treble clef notation, has a standard way to standardize the range of each staff, and has a standard way to set the absolute pitch of the staves (and hence the whole tuning system).
In general, I find that notation system design involves a lot of balancing between competing design constraints. For instance, as far as individual staves are concerned, we want to keep the whole thing as compact as possible, but contrarily, we also want to minimize the need to use ledger lines to avoid visual clutter. For the bass/treble staff split, our design goals are to cover a considerable amount of range with the two staves combined, but still also minimize the amount of ledger lines needed to play notes between the staves. Our existing notation system is one particular way to balance these competing constraints.
While there may be an argument to be made for overhauling our notation system in one way or another, for this particular notation, I sought when possible to satisfy these design constraints in a manner similar to how our existing notation system does it, since it's at least half-decent. The way I proposed setting the staves up thus generalizes what we have rather straightforwardly; you always end up with a ~3 octave range with the two staves overlapping at the same spot as before, and even a 10 note diatonic scale requires only 6 lines per staff. I approached designing the rest of the system in much the same way.
One thing which was a bit less clear to work out was nominals. Our existing notation system has a pretty strange brew of nominals, which don't generalize to other tuning systems in a way that easily obeys existing convention. For instance, we use nominals A-G, but the note upon which the entire notation is symmetrically built around is assigned the name "C". This note C isn't just the midpoint of the notation's range and the point about which the staves are symmetric, but it's also the tonic of C major, which for some reason has been standardized as having no sharps or flats.
If we were going to attempt to use letters as nominals for any arbitrary MOS, we might decide that the Middle Note is A, and things proceed from there. This is completely backwards-incompatible with what we have, though, as the old C becomes the new A, and the old C major is now the new A-C-E. We might decide to standardize the middle note as C instead, so that nominal name always start two before the Middle Note, but that's rather arbitrary, and any sort of cultural advantage of doing this goes straight out the window as soon as you start playing around in a decatonic scale or something like that.
I also found that using the letters that we already use in 12-EDO gets pretty damn confusing as soon as you use scales with more than 7 notes, and especially once you get up to 10-note scales and the like. I talked to a few of my musician friends about this, and found that all but one had the same opinion as me on that. Letters require you to first unlearn, and then memorize a bunch of mathematical relationships that aren't at all immediately obvious - for instance, how many people really know what the 14th letter of the alphabet is, modulo the letter "I", without internally translating into numbers first? Letters bring a lot of 12-EDO baggage that requires you to spend time unlearning things and adapting.
Numbers, on the other hand, are things we already know how to quickly reason with; they're fresh and intuitive. In blackwood's LsLsLsLsLs mode, 1-4-7 is a major chord, which makes it immediately obvious without any need to learn anything else that there are now two passing tones between the 1/1 and 5/4, and likewise between the 5/4 and 3/2. Likewise, 4/3 is now 1-5, which makes it obvious that there are now three passing notes between 1/1 and 4/3. It's very simple.
Another advantage to using numbers is that the resulting system has a bit of international usefulness to it. For instance, some cultures don't even use letters to begin with. If we want to use letters we can always extend past G, but someone who's familiar with fixed-do solfege has no obvious way of extending anything, so they'd have to switch to something else anyway. Numbers exist in every language, and are also mathematically useful.
There's only one real downside to using numbers for note names, and that's that we sometimes already use numbers to denote relative chords. So for instance, consider that we're in ordinary meantone, but using nominals 1-7 for the LLsLLLs mode, with the bottom note being the old C. So if we're in the key of "3 minor," aka E minor, I might tell you to go to the V chord. This doesn't mean to play a chord over the absolute pitch "5" in the system, but rather to go to the relative V chord of 3, which would have its tonic as 7. Then, over this V chord, I might tell you to play a fourth instead of the major third, making it a Vsus4 chord. However, when I say to play a fourth, I don't mean the absolute pitch "4", but a perfect fourth over the V chord; since the V chord has 7 as its tonic, the fourth would be 3.
When speaking, it's easy to avoid some of that confusion by using ordinal numbers to refer to chord extensions, e.g. "fourth" or "ninth", and being explicit when you use cardinal numbers as to what sense you mean. It shouldn't be too difficult to figure out what "the note 4" vs "the IV chord" refers to if you speak it. And when writing stuff out, it's even easier to avoid if you always write relative chords in Roman numerals, as we already do, and write absolute pitches in Arabic numerals. However, this is the tradeoff for using numbers; they now have to mean both relative and absolute pitches.
I still think that, all things considered, numbers are better than letters. For me, using letters A-G is a total dealbreaker. If the above downside to numbers is enough for you to not want to use them, another option would be to use nonstandard letters starting with I, or to use the Greek alphabet, etc. I think that nonstandard letters are less easily to reason with mathematically and require more learning, and that this is worse than having to be a bit more explicit when talking about numbers, and likewise with the unfamiliarity of the Greek alphabet to most musicians. Feel free to give your thoughts though.
Lastly, I want to note that while I tailored this post to MOS, this same notation system easily also applies to higher-rank Fokker blocks as well, or really any epimorphic scale if you want. The only thing that needs to change to use these is to define more accidentals than just #/b; for Fokker blocks, one should be defined for each chroma in the block. I'm very interested to see if this can be tied in with Sagittal notation by using as a base scale an 11-limit 7-note Fokker block which is a chain of seven 3/2's.
Follow-Up - Absolute Pitch Standardization
(Originally from https://groups.yahoo.com/neo/groups/TUNING/conversations/messages/105845, reformatted for MediaWiki)
This is all now so long that nobody will ever read it, but since I'm on a roll, I'll post it anyway, at least for future reference. There's one really longwinded thing that I want to address, and that's the concept of absolute pitch standardization.
On Tue, Jan 22, 2013 at 3:59 AM, Mike Battaglia <battaglia01@...> wrote:
- ABSOLUTE PITCH RANGE
- 6) Standardize the tessitura of the entire notation system to be what it is now by setting the absolute pitch of the Middle Note to be 261.6 Hz.
Absolute pitch standardization is deceptively relevant to the subject of notation.
All of the stuff with staves tells you the range of each staff, and the total combined range of the two staves, which is easily notated. I've standardized things so that each staff gives you about an octave, and both staves combined give you about three octave's worth of range. However, this doesn't say a thing about which actual register this "easy access" range covers. We ought to assume from the outset that register is important, and that composers pick the tessitura that they write things in for actual reasons, which could be compositional and/or psychoacoustic in nature.
Because of this, we want to ensure our 3-octave "easy access" range covers the portion of the frequency spectrum which is the most musically versatile and useful to write in, and which covers things like. For instance, we might assume that our notation wouldn't be as useful in capturing the registers that composers want to actually write in the most if the Middle Note were two octaves above middle C, or two octaves below. We can again assume that what we have now is near-optimal.
The easiest way to standardize the register is to pick a canonical note somewhere within the range of the notation, and assign it a standard reference pitch. While I tried to yield to existing convention as much as possible in this notation system, after some thought it's become clear that the the best option for arbitrary tunings is to standardize the Middle Note as 261.6 Hz, and to avoid having to pick a second note in the scale to take the role of "A" at 440 Hz.
The main reason is that if we want to have to pick this second note, there's no clear way to do it. In 12-EDO, we can easily standardize the range of the notation system by specifying the pitch of any note at all, because we know exactly how all of the notes relate to one another, so we know how to set it up a priori so that the registers and the whole thing will work out the way that we want. We just pick some other note, in this case A, and set that to be some pitch, in this case 440 Hz, so that all of the registers work out the way we want. However, for an arbitrary tuning, we don't know what other notes will be in the scale, so some sort of generalized approach is necessary.
Once we start trying to figure out how to standardize this procedure, we quickly realize that doing so requires us to implicitly standardize the Middle Note at 261.6 Hz anyway. If we accept that we want the Middle Note as the tonic for our privileged mode of choice, and we want the whole range of the notation system to be as close to to what it is now as possible, then the only possible way to meet these constraints while standardizing *some other* pitch to 440 Hz is to find the note in the scale such that tuning it to 440 Hz sets the Middle Note as close to 261.6 Hz as possible. This already requires us to specify the ideal tuning for the Middle Note is "as close to 261.6 Hz as possible," explicitly using that numeric value in the standard anyway, but then add in an additional extraneous step where we identify another note first in a tuning-specific way, and then tune that note to 440 Hz.
Another important reason is that many of the advantages of going with a standardized C261.6 for notation, but a standardized A440 for tuning, are specific to 12-EDO and meantone, and may not apply to arbitrary tunings at all. For instance, one of the reasons for tuning things to "A" is that "A "is an open string for the entire string section. However, for an arbitrary tuning, we have no idea how the strings will tune at all, and we certainly don't want to attempt to decide that this point. Thus, even if we attempt to standardize some other reference note in the tuning we have no guarantee that our other reference note will actually be any easier to tune to than the Middle Note itself.
This suggests that to finding the optimal "tuning note" for an orchestra is a task with quite different considerations from what we're discussing here, and that it will likely need to be done on a temperament-by-temperament basis. The criteria that go into picking a nice pitch to coordinate between tunings, for ease of notation and pitch standardization, are not the same criteria that go into determining the pitch out of that same tuning system that's optimal for the orchestra to tune to. Standardizing the Middle Note is useful for notational purposes and simultaneously provides a pitch standard for tunings in general. Assuming we do want to set the overall tuning up so that the Middle Note is 261.6 Hz, the question of which note in the scale to pick to best tune the orchestra will have to be worked out based on things like what the open strings of the violins are tuned to.
Finally, I note that having "middle C" be the same for all tunings is something which may have significance for those with AP. I certainly have a clear intuitive preference for this pitch being common to all tunings rather than A440, because our key signatures and my entire way of thinking about music builds out from C as the center. An informal survey of APers from Facebook's "Got Perfect Pitch" group yielded many similar preferences, though I note this is all a purely anecdotal account.