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
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
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
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=pao major (\M), or "pao" for short, as long as we're clear we're talking chords. [4:5:6]
M3=pai minor (/m), or "pai" for short [10:12:15]
m3=tai major (fM), or "tai" [14:18:21]
M3=vai minor (^m), or "vai" [18:22:27]
And some seventh chords:
m3+M3+m3=minor seven (m7) [54:64:81:96]
M3=pao major seven (\M7), or "pao 7" [8:10:12:15]
m3=tao minor seven (tm7), or "tao 7" [12:14:18:21]
m3=nao minor seven (nm7), or "nao 7" [22:26:33:39] **here "n" is my shorthand for
and "u" for its inverse
M3=phai major seven (//M7), or "phai 7" [20:26:30:39] **here // represents
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]
But what about seventh chords with either one or no perfect fifths? My method is "name the third then the seventh"
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="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
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!
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