My general approach to notation: pitch, intervals, chords

[cam.taylor]

There has recently been a lot of debate over facebook on how pitches, intervals and chords can and should be notated generally, in a kind of metatuning so all this information can be carried over to another tuning, from 19-prime limit or 2.3.11 JI to 81EDO to George's 29HTT to 72EDO to 22EDO, etc. Debate will probably continue, but I would like to share my preference for a singe all-encompassing notation system that doesn't lose too much in translation.

Sagittal notation is a fantastic basis for pitch, and it is upon Sagittal that my extended notation is based, where intervals and chords all derive from how something looks on the page and its relationship with JI. What this means is that tunings that fail to closely approximate JI perhaps don't do so well here. Tunings where the fifth is far from just (outside the regular diatonic range from 7EDO to 5EDO) or without a real fifth have the same problems with intervals and chords they have in Sagittal pitch notation. Solutions to these problems includes treating them as subsets of more JI-friendly good-fifth systems, or using a different notation that reflects their unique JI subset nature (i.e. a non-3 group), as the Sagittal approach relies upon 3 to be reasonably approximated, and works best in the regular diatonic range.

**Note: If you're unfamiliar with Sagittal flags, their meanings or their ASCII shorthand forms, review them using the main website, and if you're unfamiliar with their names or ASCII longhand forms, scroll over the "Smilies" bar on the right. An explanation of some accidentals including name and symbol will be provided :D**


For pitch, I use Sagittal notation as accurately as necessary, usually up to the Athenian level or sometimes higher but without accent marks unless absolutely necessary. After having a wee while to mull over Sagispeak, I'm starting to speak of pitches using some of the Sagispeak names for accidentals. Everything is exactly as on-the-page, with all nominals representing a chain of best fifths, and the apotome representing 7 of these fifths upwards and downwards
An 81:80 comma below E will be notated "E " and spoken "E pao", a 64:63 comma above F# (F) is notated F and spoken "F tao-sharp", a 33:32 comma below A is notated A and spoken "A pakao" or "A kapao" (kapow) or "A vao" for short.

This generalises to intervals in the expected way. What may be unexpected is that mixed Sagittal actually breaks things up a wee bit simpler for interval names, as we will shortly see. All intervals only involving the apotome accidentals can be written and spoken in the usual way. D-F in any tuning is a "minor third", shorthand m3. G-F# is a "major seventh", or M7. An interval notated D-F is a "minor third" lessened the "tao" accidental, so it's a "tao minor third". One can write this interval shorthand by writing the Sagittal symbol for the alteration or its shorthand followed by the interval type, in this case, m3, or tm3. With the interval G-F, a Sagittal novice might think we're dealing with a 135:128-altered minor seventh, but of course in mixed Sagittal this is G-F, so we have the accidental " pao" lowering a major seventh, or, a "pao major seventh", shorthand M7 or \M7. This is of course the same as saying "15:8", as every Sagittal interval is also a just interval.
Every interval should be of the format [accidental][fifths-interval].

So how does this approach generalise to chords? Much the same way.
Any pythagorean chord (i.e. unaltered by any accidentals other than apotomes) is as in common practice. A major third followed by a minor third gives us our usual "major" triad, even though it has the fairly complex ratio 64:81:96 in JI. This means that 4:5:6, 14:18:21, 22:28:33 and 10:13:15 are all kinds of major chords (respectively pao major, tai major, ranai major and phai major) but none of them are "the" major chord unless those respective commas are tempered out, as for the pao major triad in meantones, tai major in archy, ranai major in gentle, etc.

I realise many people want the "major" triad to correspond to 4:5:6 with 5:4 major third and 6:5 minor third but for that we must have a disjunct between our on-the-page Sagittal notation and our pitch/interval/chord names. Using the fifths-names-plus-alterations approach seems to tie ratio, closest fifths-position, accidental, on-the-page notation and spoken names for pitches, intervals and chords all together into one system.

Here's how I've thought it best to name chords, though there are a lot of chords to get through:
Chords of 3 or more notes assume a perfect (unaltered) fifth unless otherwise stated, and take their name primarily from the first third.

**Note for short names: In general, we can even abbreviate "minor" or "major" and just use the alteration names as a short name (if we assume the commas are going one-way to make the simplest thirds, like /m3 [6:5] and \M3 [5:4] and not \m3 [2560:2187] and /M3 [6561:5120], ditto tm3 and fM3, etc.**

M3+m3=major ( ) [64:81:96]
m3+M3=minor (m) [54:64:81]
M3+ m3=pao major (\M), or "pao" for short, as long as we're clear we're talking chords. [4:5:6]
m3+ M3=pai minor (/m), or "pai" for short [10:12:15]
M3+ m3=tai major (fM), or "tai" [14:18:21]
m3+ M3=vai minor (^m), or "vai" [18:22:27]
etc.

And some seventh chords:

m3+M3+m3=minor seven (m7) [54:64:81:96]
M3+ m3+ M3=pao major seven (\M7), or "pao 7" [8:10:12:15]
m3+ M3+ m3=tao minor seven (tm7), or "tao 7" [12:14:18:21]
m3+ M3+ m3=nao minor seven (nm7), or "nao 7" [22:26:33:39] **here "n" is my shorthand for and "u" for its inverse **
M3+ m3+ M3=phai major seven (//M7), or "phai 7" [20:26:30:39] **here // represents **
M3+ m3+ M3=vao major seven (vM7), or "vao 7" [44:54:66:81]
M3-m3-M3, "patao (major) 7" [26:32:39:48],
M3-m3-M3, "ranai (major) 7" [22:28:33:42]
etc.

But what about seventh chords with either one or no perfect fifths? My method is "name the third then the seventh"

M3+ m3+ m3="pao tao 7" [4:5:6:7], a "pao (major)" chord with a "tao (minor) seventh", the short names really coming in handy.
m3+ M3+ m3="tao vai 7" [6:7:9:11], a "tao (minor)" chord with a "vai (minor) seventh", as above.

I have mapped out a few more chords but I found one problem was names becoming perhaps a little heavy for chords using two or more of the double-sided flags like , and

But what about [5:6:7:9] you say?
It's a "pai7 nao-flat 5" which may look ridiculous in words, but in symbols it's just (/7 b5), not much more of a mouthful than your average fake book chart!

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Hopefully from these basic guidelines people can name any pitch, interval or chord they like, as long as they have a ratio interpretation or a Sagittal notation to go off!

In my opinion, this system works pretty well. And though names and possibly even symbols may change, it represents a logical approach to an integrated microtonal notation.

Let me know what you think.

P.S. That got a lot longer than I'd anticipated... There may some typos

Thanks,
Cam

[cam.taylor]

Ah, as I'd feared one of the flags has become a smiley. ")|(" (with colons) is the "ranai" comma, 896:891, between 81/64 and 14/11 major thirds. This symbol is not yet enabled in the forums, but hopefully it can be understood.

The other one is "||\" (with colons), the "pao-sharp" [135:128], or the apotome 2187/2048 minus the "pao" comma 81:80. These were not meant to turn into smilies.

[cam.taylor]

I should perhaps add, this is hardly something I should be calling "my" approach. I didn't come up with any of the groundwork, which is all there in the current Sagittal framework. All I'm really doing is taking that framework to its logical endpoint, and for that I must praise the brilliant work that George and Dave have done on the system that is Sagittal. Thank you.

[Dave Keenan]

Hi Cam. Thanks for your post, and thanks for the kind words. I have added all the Sagittals required by your post now, and many more. And I have edited your post so it was re-rendered and now shows them.

All single shaft symbols are available now, and the diacritics (accents). But of the multi-shaft symbols needed for the pure-Sagittal, I have so far only added the 72edo set (which also notates many smaller EDOs and 11-limit JI). All can be seen via the "View more smilies" link on the posting page. I note that there are two pages of Sagittals within the "more smilies" window.

I note that unfortunately our readers can't see the "Smiley bar" with the Sagittals, unless they hit the "Reply" button. But it's well worth doing that, and mousing over them to see their names. You can back out without posting when you're done.

Some examples with schisma diacritics, just because I can. Enclosed in [ pre ] ... [ /pre ] tags to preserve multiple spaces so they can line up vertically.

   diaschisma,        code   ":.:" ":/|:"   without the quotes or the space
   syntonic comma,    code         ":/|:"   without the quotes
   Pythagorean comma, code   ":':" ":/|:"   without the quotes or the space

[cam.taylor]

Thanks Dave! Man is it great to have full access to Sagittal symbols in a text format!


If anyone has any doubts or questions about this system, I'd be more than happy to render any musical examples in words, symbols or on a stave, be they pitches, intervals, chords, melodies or sequences in whatever relevent tuning or left in a translatable "meta" JI notation.

[Dave Keenan]

The problem I have with it is that I am still going to want to say things like (for e.g. 22edo): That pao-major third sounds like an ordinary major third, and that major third sounds like a super-major third. To disambiguate, we're going to have to use terms like "notational major" versus "acoustic major".

[cam.taylor]

Or simply, "the pao major chord sounds like a pao major, but the major chord sounds like a tai major"...

[Dave Keenan]

I suspect I just proved my point. I don't think you understood what I meant when I wrote:

(for e.g. 22edo): That pao-major third sounds like an ordinary major third, and that major third sounds like a super-major third.

With your system, to make my meaning clear, I'm forced to use something longwinded like this:

"That interval that's notated as a pao-major third sounds like an ordinary major third, and that interval that's notated as a plain major third sounds like a super-major third."

As opposed to:

"That pao third sounds like a major third, and that natural third sounds like a super-major third".

[cam.taylor]

Perhaps I didn't fully get that comment. But what does "ordinary major third" mean here? Surely not 81:64, which it would usually in Sagittal...
If it means 5:4, then that's a "pao (major) third" by default (using Sagittal's JI logic). What does "supermajor third" mean here? Does it mean 9:7? Then that's a tai major third? So what is a "major third"? Simply the notation for 81:64, surely? Or am I missing something?

[cam.taylor]

So, back to your comment further above, "notational major" is ALWAYS simply "major" in my system. By acoustic major if you mean 64:81:96, then that's "major" again, but if you mean 4:5:6, then that's "pao (major)"...