[Note: This post is wrong. See this update.]
SHORT ANSWER
Movable Do with a Do-based minor provides an invariant mapping of solfa names to mode degrees.
Movable Do with a La-based minor, on the other hand, provides an invariant mapping of tonic solfa names to the positions of notes in a stack of tempered perfect fifths, which—combined with octaves—gives a unique name to each point in a two-dimensional tuning space [a space which includes standard 12-tone equal temperament tuning, and many other historically and culturally important tunings, too].
Using the "La-based minor" note-naming system allows any tonal structure (that is, any sequence or combination of tonal intervals) to retain its "shape" on each of JIMS geometric elements—that is, JIMS' staff, keyboard, and tonnetz—in every octave, key, and tuning . Using the "Do-based minor" naming system would change the shapes of these tonal structures with every mode. Hence, JIMS uses the "movable Do with a La-based minor" note-naming system, to keep its shapes invariant.
LONG ANSWER
All musical notational systems must provide naming systems for three separate concepts, one way or the other:
Absolute pitch
Mode degree
Locus: the unique position of a given note within the constellation of tonal intervals (called proprietas by Guido d'Arezzo, inventor of the musical staff, c. 1020).
Locus
I can't find an English word that means what Guido meant by proprietas—that is, "the unique position of a given note within the constellation of tonal intervals"—so I'm declaring locus to have that meaning. (Neither "function" nor "role" is quite right; worse, each of these alternatives already has far too many musical meanings.) In both Latin and English, locus means "place," which is evocative of "position in tonal space," which is ultimately what locus is all about. I'm going to italicize locus throughout this document, but only because this new musical meaning of locus is novel to this document. In subsequent postings & documents, I won’t italicize it, nor should you.
To hold one of these concepts to be invariant in a notational system , the other two must be allowed to vary. For example, the traditional staff holds the notation of pitch invariant. An A4 is notated in just one location on the treble clef, for example. One can determine the locus of that A4 from the key signature, but the tonic of that key, and hence A's mode degree, cannot be determined directly from the traditional staff or its key signature. That is, traditional notation barely notates locus at all.
Each of the three above-listed concepts is held invariant by one of the three versions of solfege:
Absolute pitch is held invariant by Fixed Do (Do == C)
Mode degree is held invariant by Movable Do with a Do-based minor (Do == tonic)
Locus is held invariant by Movable Do with a La-based minor (Do == the note that is a major second below Re)
"Re" is the only diatonic note around which the diatonic scale is symmetrical, so it is the reasonable point of reference for defining the locus of diatonic notes, and by chromatic extension, of all other notes.
The idea that "Re is the only note in the diatonic scale around which the diatonic scale is symmetrical" captures the essence of locus: that one can describe notes in terms of their relationships to the other notes of the diatonic scale, and by extension, to those notes' chromatic alterations, independently of their octave, key, mode, scale, or tuning.
WHAT IS A "NOTE"?
In JIMS, a note is a unique point in two-dimensional tuning space. Any given note N can be described by a two-dimensional point [j, k]. The origin note is the point [0, 0]. A reference frequency is associated with the origin note. The frequency of the note [j, k] is j*P8 + k*P5 cents higher than the pitch of the reference frequency, where P8 = the width of the tempered perfect octave, and P5 = the width of the tempered perfect fifth.
In the discussion below, I'll calculate intervals widths under the assumption that P8 = 1200 cents and P5 = 700 cents, which is equivalent to 12-tone equal temperament.
With these definitions,
- One executes a "key change" by changing the reference frequency.
- One executes a "tuning bend" by changing the width of P5 (and/or P8).
DO-BASED MINOR
Ask yourself this question: "What is the interval between Re and Mi in Do-based minor? Specifically, is it a minor second, or a major second? "
(I pause, while you actually ask & answer this question.....Are you done answering it? OK, moving on...)
The answer is, "It depends on the mode.
In major/Ionian, the Re-Mi interval is [-1, 2] (that is, (-1)*P8 + 2*P5 = (-1)*1200 + 2*700 = -1200 + 1400 = 200 = a major second), but
in minor/Aeolian, the Re-Mi interval is [3, -5] (that is, 3*P8 - 5*P5 = 3*1200 - 5*700 = 3600 – 3500 = 100 = a minor second)."
Now, look at Figure 8 in the Spectral Tools paper. It shows "the coordinates [j, k] of the Thummer's button lattice, when using its default Wicki note layout."
Looking at that figure, ask yourself, "which buttons should be labeled 'Mi'?" More specifically, using the numbering system shown in the figure,
should the buttons [j, 2] (for any j) be labeled "Mi", indicating that Mi is a major second higher than Re? Or
should the buttons [j, -5] be labeled "Mi," indicating that Mi is a minor second higher than Re?
The answer is, again, that "it depends on the mode." In Do-based minor, the naming of mode degrees is invariant, but the naming of [j, k] notes varies. In Do-based minor, Do means 1st degree, Re means 2nd degree, Mi means 3rd degree, and so on. The "Mi" label would have to move from one button to another every time the mode changed, in order to keep Mi's scale degree constant.
In short, for the Do-based minor note-naming system to hold mode degree invariant, it must allow the locus of a mode degree to vary.
LA-BASED MINOR
Looking again at Figure 8 from the Spectral Tools paper, and comparing it to Figure 3 in the Perception paper, we can see that
all of the notes labeled [j, 2] (for any j) in SpecTools' Figure 8 are labeled "Mi" in Perception's Figure 3, and
all of the notes labeled [j, -5] in SpecTools are labeled "Me" in Perception.
What, then, is La-based minor holding invariant?
Not pitch, which differs among keys.
Not degree, which differs among modes.
Not interval width, which differs among tunings.
Instead, the one and only thing that La-based minor holds invariant – despite changes to octave, key, mode, scale, and tuning – is the mapping between note-names and [j, k] points in tuning space.
Alternatively put, what "movable Do with a La-based minor" holds invariant is locus.
For example, in La-based minor, Mi is always a major second higher (that is, [-1, 2]) than the same octave's Re. Mi is never [3, -5] higher than Re. The note that's a minor second higher (that is, [3, -5]) than Re is Me (with an 'e'), not Mi (with an 'i'). The relationship between Re and Mi is invariant in La-based minor. The same is true for any other pair of note-names when using La-based minor: the relationship between them is invariant, no matter what changes might occur in key, mode, tuning, etc.
In short, for the La-based minor note-naming system to hold locus invariant, it must allow the mode degree of a given locus to vary.
JIMS
In designing JIMS, my guiding principle has been – as Guido d'Arezzo's was – that locus was the most important property of tonal music, so it should hold locus to be invariant.
If JIMS holds locus invariant, then everything else must be allowed to vary. Therefore,
When you change mode, the tonic "moves" to another keyboard button, staff location, and tonnetz location.
When you change key, the pitch of each note "moves" to another button, staff location, and tonnetz location.
When you change tuning (i.e., the width of the tempered P5),
the pitches of all notes (except the tonic) change, and
the widths of the intervals between all non-octave note-pairs changes.
To paraphrase the guiding principle of Western jurisprudence, "Let locus be invariant, though the heavens fall."
In JIMS' case, the "fall of the heavens" takes the form of incompatibility with traditional instruments. JIMS is great for isomorphic keyboards and the voice, but incompatible with essentially all other musical instruments. However, given the widespread availability of both vocal chords and the standard computer keyboard, both of which are entirely compatible with JIMS, there is considerable opportunity to benefit from JIMS' use.
Going back to the issue of notating the three separate concepts in music, JIMS uses three separate naming systems:
Absolute pitch: Traditional pitch names (A4 = 440Hz)
Mode degree: Arabic numerals (1 = 1st degree, 2 = 2nd degree, etc.)
Locus: Movable Do with a La-based minor (Re == the diatonic scale's note of symmetry)
Whereas traditional staff notation notates pitches directly, JIMS notates locus directly. However, JIMS provides much more context, including a stack of scale dots indicating the current scale, and an indication of the tonic's position within that scale (i.e., the current mode).
LOCUS AND THE ORIGINAL MUSICAL STAFF
When Guido d'Arezzo invented the musical staff c. 1020 C.E., he did not use it to notate absolute pitches (A4=440Hz) as we do today. Rather, he used it to notate locus (which he called proprietas, which is Latin for "properties"). Guido intended his staff to be used only by singers; it was the use of his staff by instrumentalists, later, that forced its switch from notating locus to notating pitch.
JIMS can be seen as a return to Guido's locus-based notation, with the addition of isomorphic keyboards and the tonnetz, both of which were invented centuries after Guido's death.
