(This is a detailed analysis of a discussion in the HEWM notation definition. Here I use the vector notation which is explained on that page.)
Where we use positive and negative integers (including zero) to represent the exponents of prime-factor 3 in a Pythagorean tuning system -- and assuming it to be octave-equivalent so that it describes pitch-classes rather than specific pitches, and we can ignore prime-factor 2 -- this is how the standard Pythagorean tuning is notated (as an incipient "cycle of 5ths"):
Eb Bb F C G D A E B F# etc. ... -6 -5 -4 -3 -2 -1 0 1 2 3 ... etc.
German musical nomenclature is quite unorthodox in including the 8th letter, H, for , so that B stands for [-5], and the b symbol is used with all 6 of the other letters but not with the letter B, as portrayed on this lattice diagram:
Eb B F C G D A E H F# etc. ... -6 -5 -4 -3 -2 -1 0 1 2 3 ... etc.
This has the effect of destroying the regularity of the application of letter-names to the diatonic scale steps as in the standard system in the first lattice.
The association of the L and s step-sizes with the progression of letters is as follows:
standard non-German nomenclature A B C D E F G (A) 2 -5 2 2 -5 2 2 L s L L s L L German nomenclature A B C D E F G H (A) -5 2 2 2 -5 2 4 -2 s L L L s L
The shifting of position of B from [+2] to [-5] merely alters the arrangement, without changing the basic patter of L and s.
But note that the progression from G to H is a pythagorean interval called the ditone, the equivalent of a major-3rd. Then the progression from H back to the first letter A, is -L, that is, a whole-tone downward.
Examination must also be made of the relationship of these letters to their neighbors in the scale, that is, in terms of pitch-height.
In fact, this nomenclature arose as a result of the standard medieval scale containing both a B and a Bb in the "8ve" below "middle-C", which in turn was a holdover from ancient Greek theory. The Greek "Perfect Immutable System" contained both diezeugmenon (disjunct) and synemmenon (conjunct) tetrachords, which, if we call the reference note mese by the letter A, contained respectively the notes B (paramese) and Bb (trite synemmenon) below "middle-C". (See Monzo, Tutorial on ancient Greek tetrachord-theory for a detailed explanation.)
The tetrachords were all assumed to have the same intervallic structure, ascending tone - tone - semitone. Here is the central "8ve" of the Greek system in both types of nomenclature, with proportional spacing to show tones and semitones:
standard non-German nomenclature meson synemmenon __________ ____________ | \/ | A Bb C D E F G A B C D E \/ |____________| tone diezeugmenon of disjunction German nomenclature meson synemmenon __________ ____________ | \/ | A B C D E F G A H C D E \/ |____________| tone diezeugmenon of disjunction
The German nomenclature merely sought to give each pitch-class that ocurred in the system a unique name. Later, when the letter b was employed to effect mutation into other, more distant tetrachords (or hexachords), the German nomenclature was never modified to accomodate it, and its use as a flat sign was simply extended to the other 6 letters while retaining the H/B distinction for what everyone else calls B/Bb.
standard nomenclature A Bb B C -5 7 -5 German nomenclature A B H C -5 7 -5
Much is often made of the fact the these four letters spell the last name of the great German composer Johann Sebastian Bach, and Bach himself created musical motifs from these four notes to spell his name, most notably in the motive which permeates his Die Kunst der Füge [The Art of Fugue]:
B A C H -m2 +m3 -m2
I used the generic interval symbols m2 for "minor 2nd" and m3 for "minor 3rd" here instead of the specific Pythagorean ones, because by Bach's time meantone -- which completely reverses the effect of the diatonic and chromatic semitones, making the flats higher than the sharps -- had already been firmly established as the standard tuning for over a century, and Bach himself was helping to establish the well-temperaments (such as Werkmeister's) which would dominate European tuning for about the next century, until the preeminence of 12-edo beginning around 1900. Examples of musical motifs spelling B-A-C-H in 12-edo tuning can be found in scores by Schoenberg, Berg, and Webern.
German writings about music still use this nomenclature today, so it is applicable to all of the tunings I discuss here.
This confusion of meaning for the letter B (for the note Bb in German but B in all other languages) is the cause for a serious typographical error in the English translation of Schoenberg's Harmonielehre (see the opening discussion in Monzo, Searching for Schönberg's Pantonality).
I had a glance at your pages this PM and -- amid many nice things -- wandered over to your page on German note names. The presentation doesn't quite work for me, for two reasons:
(1) the characterization of "German" notation as "unorthodox". This is far from the case, as both common note naming systems stem from a common tradition, but branch off into two significant streams. I find it hard to consider the Lowlands-German-Eastern Europe branch as any bit less "orthodox" than the Anglo-Latin branch; in fact, you could make the far better case for the former being the leading branch, based on the repertoire represented and on the fact that in Latin countries, Solfege syllables are more important than letter names.
(2) you don't bring across the essential notion of hard and soft "b's", one of which has survived as a lower-case h (because of the similar squared shape). This is critical in the history of pitch notation, especially in the 14th and 15th centuries, when (a) a given voice in a composition was notated in terms of either the collection with the hard b or the collection with the soft b, and (b) the choice of hexachords used in performing a given piece were determined by the hard/soft question. Because of the equal weight given to both collections (and leaving aside the fact that the notation using a letter "h" instead of the squared "b" would only be introduced later), it is impossible to say that the collection "without accidentals" was either the one with the soft b or the one with the hard b. And, if you think of it a bit, it has no relevancy to the relationship of the accidentals to the pythagorean sequence of fifths whether one starts the sequence on Bb/b or ends it on B/h.
I personally use English note names, but I can't see the path that leads to the b/h notation as anything other than an equally legitimate one to ours, even if one which reflects a slightly different view of a common tradition and seems, superficially, to lack our logic.
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