Re: Stevin and solmization (for Manuel Op de Coul)

Hello, there, and this will be a note on a topic peripheral to tuning
but possibly of interest to those taking an interest in the excellent
Huygens-Fokker archives available online, and specifically the
theories of Simon Stevin on singing and 12-note equal temperament
(12-tet) around 1600.

Please let me warmly thank Manuel Op de Coul and the other people who
have made this material available.

In Fokker's discussion of the musical writings of Stevin, I was
fascinated to see that Stevin was evidently using a solmization
system much like one that I am experimenting with now. Since Fokker
comments on a feature of this system which he seems to find unfamiliar
but I may be able to explain more clearly, I thought that I might
attempt such an explanation even though the topic is a bit tangential
to tuning proper.

Fokker raises a question about examples from Stevin of this kind:

G A B C D E F G
ut re mi fa sol la sa ut

Here Fokker remarks that if the solmization syllable ut were
associated with C, as is common in many more recent systems, then the
syllable sa (a minor seventh above ut, and a fourth above fa) would be
associated with Bb instead of F. Here I would like to explain how
Stevin's syllables seem quite natural to me as an extension of the
usual medieval and Renaissance hexachord system.

In the 16th century, the usual system of solmization was based on
hexachords, with the characteristic scheme described by Guido d'Arezzo
around 1025 or 1030. Here (T) shows a whole-tone, and (S) a semitone:

ut re mi fa sol la
T T S T T

In the classic hexachord system developed by Guido and his successors,
the standard system for Gothic and Renaissance musicians, the regular
gamut is made up of a series of three types of interlocking
hexachords: natural (C-A); hard, using B-natural (G-E); and soft,
using Bb (F-D). As one sings beyond the range of a single hexachord,
it is necessary to make a "mutation" from one hexachord to another:

G-E C-A F-D
hard natural soft note name

E5 E la -- -- Ela
D5 D sol -- la Dlasol
C5 C fa -- sol Csolfa
B4 B mi -- -- Bmi
Bb4 B -- -- fa Bfa
A4 A re la mi Alamire
G4 G ut sol re Gsolreut
F4 F -- fa ut Ffaut
E4 E la mi -- Elami
D4 D sol re la Dlasolre
C4 C fa ut sol Csolfaut
B3 B mi -- -- Bmi
Bb3 B -- -- fa Bfa
A3 A re la mi Alamire
G3 G ut sol re Gsolreut
F3 F -- fa ut Ffaut
E3 E la mi Elami
D3 D sol re Dsolre
C3 C fa ut Cfaut
B2 B mi Bmi
A2 A re Are
G2 Gammaut Gammaut

Certain theorists of the Renaissance such as Bartoleme Ramos (1482),
however, proposed that a solmization system based on octaves rather
than hexachords would make many mutations unnecessary, thus
simplifying the singer's task and also making the system more
user-friendly, as one might now say, for beginners.

Note that the regular hexachord gamut, as shown above, includes not
only the seven diatonic notes but also Bb, the step B/Bb (or in German
conventions H/B) being a fluid or flexible one. Thus one way to expand
the hexachord system to an octave-based system is to add a seventh
flexible degree to each of the three usual hexachords.

Let us follow Stevin's convention, also followed for example by the
Italian composer and theorist Adriano Banchieri (1614), of naming this
fluid seventh degree si/sa, with si a semitone below ut, and sa a
semitone above la. We then derive a heptachord gamut -- or possibly an
"octachord" gamut -- of three heptachords or octachords, each
repeating itself through as many octaves as desired:

durum naturale molle note name
hard natural soft
G-E, C-A, F-D,
F/F# Bb/B Eb/E
..
E5 la mi si Esilami
Eb5 -- -- sa Esa
D5 sol re la Dlasolre
C5 fa ut sol Csolfaut
B4 mi si -- Bsimi
Bb4 -- sa fa Bsafa
A4 re la mi Alamire
G4 ut sol re Gsolreut
F#4 si -- -- Fsi
F4 sa fa ut Fsafaut
E4 la mi si Esilami
Eb4 -- -- sa Esa
D4 sol re la Dlasolre
C4 fa ut sol Csolfaut
B3 mi si -- Bsimi
Bb3 -- sa fa Bsafa
A3 re la mi Alamire
G3 ut sol re Gsolreut
F#3 si -- -- Fsi
F3 sa fa ut Fsafaut
E3 la mi Elami
D3 sol re Dsolre
C3 fa ut Cfaut
B2 mi Bmi
A2 re Are
G2 gammaut Gammaut

Here I have followed the usual convention by which the lowest note the
standard gamut is taken as G2, known as Gammaut (Gamma-ut), from which
the name "gamut" derives. However, any of the heptachords may be
continued into upper or lower octaves as desired.

In such a system, if we wish to sing an octave-species or mode of G-G,
the Seventh Mode of usual medieval and Renaissance numbering systems
or Mixolydian in the usual nomenclature, we may interestingly begin
either on ut, as in Stevin's example, or on sol. Here I show the
syllables and corresponding notes in each of the basic three
heptachords or octachords for each of these choices:

Mixolydian, ut-ut

T T S T T S T
ut re mi fa sol la sa ut
G-F/F#: G A B C D E F G
C-Bb/B: C D E F G A Bb C
F-Eb/E: F G A Bb C D Eb F

Mixolydian, sol-sol

T T S T T S T
sol la si fa re mi fa sol
G-F/F#: D E F# G A B C D
C-Bb/B: G A B C D E F G
F-Eb/E: C D E F G A Bb C

It will be seen that the ut-ut form of Mixolydian has for its seventh
degree the sa flavor of the fluid degree sa/si; while the sol-sol form
has for its third degree the si flavor of this fluid octachord degree.

If a given melody in this mode requires no inflections at all, then
either solmization should be equally satisfactory. However, one might
prefer ut-ut for a melodic line calling for a flexible degree F/F# (as
happens, for example, in some pieces centering on G around 1200); or
sol-sol for a melodic line calling for a flexible degree B/Bb, as
happens in some liturgical chants.

If such cases, if F# or Bb respectively is the only inflection
required, then a solmization in ut-ut or sol-sol will permit singing
the melody within a single octachord without a need for mutations.

(Although Stevin may not address this point, for other inflections one
can follow the usual hexachord convention that one sings an accidental
sharp as mi and a flat as fa, thus momentarily shifting to some other
octachord. Typically sharps apply only to the note immediately
inflected, or to repeated notes or notes in the same cadential
figure. Flats may sometimes imply a more prolonged mutation to a new
hexachord -- or here octachord. Thus an indicated Bb or B-fa may
effectively "naturalize" Eb-sa as a possible inflection, for example
to avoid a following direct tritone leap of Bb-E.)

To conclude, Stevin's G-G octave-species with an ut-ut solmization
results quite naturally when we derive an octave-based solmization
system from the conventional hexachord system, where the syllable ut
may regularly represent the note C (natural hexachord C-A), F (soft
hexachord F-D) or G (hard hexachord G-E).

Here it may be added that a hexachord or octachord system of this kind
shows only whole-tones and diatonic semitones, and thus may adopted to
Pythagorean tuning, as assumed by Guido (T=9:8, S=256:243); or to
Zarlino's syntonic diatonic (T=9:8 or 10:9, S=16:15), or to some form of
meantone or to Stevin's system of 12-tet.

Most respectfully,

Margo Schulter
mschulter@value.net