Lady Teleri the Well-Prepared
http://www.geocities.com/sca_bard/earlymusictheory.html
Motivation: Why should I care about music theory?
Studying the theory of music
in use at the time gives us some idea of how at least some people thought about music and composition. It can help us make educated decisions about
our own performances.
Scope: When and what are we covering?
The ideas presented today
were known to medieval
We will cover the physical
and mathematical foundations from which music and its theory naturally
springs. From there, we will cover the
modes and some of the rules governing monophonic music. At the end, we will touch on polyphony, since
it is very useful for arranging group singing or instrumental accompaniment.
Pythagoras’s hammers: The story goes that the great Greek philosopher, Pythagoras (yes, the
a2 + b2 = c2 one) was walking through a
marketplace when he heard blacksmiths at work.
Their hammers made different sounds when they struck the anvils; some
combinations of hammers sounded concordant, but others were discordant. Pythagoras investigated.
The 12-pound hammer and the
6-pound hammer sounded almost like the same note! Almost, but not really,
since the smaller one made a higher sound. The 8-pound hammer sounded particularly nice
together with the 12-pound hammer; the 9-pounder sounded pleasing with it as
well. But the 8 and 9-pound hammers
together sounded awful!
Pythagoras looked at the
ratios of the weights of the hammers.
The two that sounded best together were 6 and 12; that’s a
The most beautiful sounds
were related to each other in ratios 1:2:3:4!
This seemed very significant to Pythagoras!
And that
dissonant sound? It came from a ratio of 8:9, which does not
reduce at all. That ratio became the
“whole step” in the musical scale. From
these four “hammers,” the entire musical scale can be derived.
We’re not going to do that
math! But here are the results. These ratios, arranged as follows, form what
we call the diatonic scale. (All right,
some of the math is in the parenthesis.
But don’t worry about it too much.)
|
Ratio |
“Hammer
weight” |
Note name |
|
1:1 |
12 lbs |
Ut (modern
Do) |
|
8:9 |
10 2/3 lbs |
Re |
|
64:81 (82
: 92) |
9.481 lbs |
Mi |
|
3:4 |
9 lbs |
Fa |
|
2:3 |
8 lbs |
So |
|
|
7 1/9 lbs |
La |
|
128:243 (82·2
: 92·3) |
6.321lbs |
- (modern Ti) |
|
1:2 |
6 lbs |
ut |
The Monochord: The World’s Most Boring Instrument
Just as the ratios, above, can
be transformed into weights of hammers, they can also be applied to a length of
string or wire. Instead of 12 lbs, you
could have a string 12 inches long. If
you had a string only 6 inches long (and the same thickness and tension as the
first string!) it and the 12 inch string together would sound as harmonious
with the 6 lb hammer with the 12 lb hammer.
This super-harmonious interval is called the octave. Since there are five
notes between the 1:1 note (Ut) and its
next-most-harmonious friend, the 2:3 note (So), that’s
called a fifth. No surprise that the interval between Ut and Fa is a fourth.
(The Greek terms are diapason,
diapente,
and diatesseron. Impress your friends!)
The monochord was a soundbox with a single
string stretched over a moveable bridge.
An instructor would mark on the box the locations where the bridge
should be put to get a note. The way
they did this was interesting, but beyond the scope of this current class.
Strings make it easier to
talk about standing waves, which may make a little clearer why these numeric
ratios should affect the sounds we hear.
Demo Time!: If the demonstration goes well, you will have
seen standing waves in a string with one, two, three and maybe even four
“humps.” You will also have heard how, if
we shorten a string to half its length, it sounds the same as a string an
octave up. Similarly, that shortening a string to 2/3 of its length makes it
sound the same as a string a fifth up.
So there is no magic to
Pythagoras’s basic four “hammer ratios” – just some physics. These are natural harmonics in the
strings. The other ratios… seem to be
artificially contrived. Some of them are
rather close to some of the other harmonics, which is probably why a sixth,
say, doesn’t sound bad at all. But they don’t
have the pure sound of the fourth, the fifth, and the octave. The fourth, the fifth, the octave, and the octave+fourth, the octave+fifth,
and the double octave were the only intervals considered truly harmonious in
early period theory.
It’s a Small World: According to Guido d’Arezzo’s instructions
on constructing a monochord, there are just 21 notes. In “olden times,” they named them starting
with the letter A, going up to G, and then using small
letters, and then doubled smaller letters from aa to
dd. Later, in Guido’s time, they added a
new, lower note named Γ, gamma, as well as the note we would today call
b-flat (but only in the upper two octaves).
(We will discuss why under Modes, below.)
So, how do these letters
correspond to the “Ut Re Mi” above? Those names come from Guido as well; he had a
hymn, each line of which started with the next note up the scale. It’s a teaching mnemonic, exactly like “Doe,
a deer, a female deer.” His hymn started
on the monochord note C. So Ut = C, Re = D, and so on.
What about the notes that aren’t
there?
You could move the moveable
bridge on a monochord to anywhere, not just to the pre-marked positions. Why don’t we?
Why don’t we have all the notes on a standard keyboard? Or more? Other traditions, like Indian music, have
notes that Western music does not.
Blame Pythagoras and his
math. Insidious numerology!
The 8 and 9 pound hammers, So and Fa, defined an interval
that is called a whole step or tone.
They just do, okay? That is the basic physical definition of a
tone. If you look up at the table, you
will see that there is also a tone between Ut and Re. There is another
hidden between So and La, too! (If you don’t believe me, give them both a
common denominator of 9 and see what happens!)
And between Re and Mi, and between La and the Note Currently Known as
Ti.
But between Mi and Fa, and between “Ti” and ut,
there is not a tone. There is a smaller
step called a half step or a semitone.
This gives us a pattern of
tones and semitones, starting from Ut and going up to
the high ut:
Ut – Re Tone
Re – Mi Tone
Mi – Fa Semitone
Fa – So Tone
So – La Tone
La – Ti Tone
Ti – ut Semitone
That pattern, that
T-T-S-T-T-T-S pattern, is what we today call the major scale.
And medieval musical theory
couldn’t care less about it!
Hexachords: Six Notes All in a Row
Today, we think of notes as
being arranged in octaves. It makes
sense, doesn’t it? If there is a pattern
that repeats every seven notes, why not consider notes in groups of seven?
But it was not always
so. The ancient Greeks thought of notes
in “octaves” of four notes, called tetrachords. And the medieval people thought of them in
groups of six, called “hexachords.” The
hexachord had a pattern of T-T-S-T-T.
So when Guido describes the
four modes, he describes four different ways to span six notes in that
pattern. Consider the following notes
(with the distance between them, tone (T), or semitone (S), noted below):
Γ A B C D E F G a b c d e
T T
S T
T S T T
T
S T T
One hexachord is
highlighted. Consider the “mode” of
A. From A, we cover the hexachord by
going down a tone from A on the left, and up a tone, a semitone, and two tones
from A on the right. That same pattern
can be done at D and also covers a hexachord:
Γ A B C D E F G a b c d e
T
T
S T T S T T T S T T
These two have an affinity,
Guido tells us. Not only because the
patterns are the same, but A is a fourth below D and ‘a’ is a fifth above it,
so the octave is spanned. It’s nearly
magical!
For whatever reasons (they
are right now unclear to me), Guido stated other “patterns” for covering the
hexachord starting from B (which has an affinity with E) and C (which has an
affinity with F). He also stated the
pattern that goes with starting on G (down: tone, semitone, tone, tone; up:
tone), but specifically didn’t say that D has an affinity with G, even though
the same pattern works. Maybe because D
was already partnered with A? I’m
unsure. Also, it wasn’t apparently
permissible to start on Γ and just go up, or start on E and just go
down. The patterns had to come from A,
B, C (remember that Γ was a recent addition) and… well, G. Although if you went A, B, C, D, you’d get the G pattern…
Guido’s unexplained choices
aside: there were four officially recognized ways of traversing a
hexachord. Each of these corresponded to
a mode.
According to Guido, since in
music it is nice to be able to go up and down, the “core notes,” the ones that
would begin most songs and would end almost all of them, were the slightly
higher notes with affinities for A, B, and C: D, E and F. G was also included in this list of “finals.” So, if in theory
the modes came from patterns surrounding A, B, C and G, in practice we consider them patterns starting on D, E, F and G. And in practice,
Guido tells us, hymns rarely went a tone below these final notes, and usually
went an octave up; maybe a tenth at most.
(Aside: Wait, an octave? What happened to the hexachords? Answer: You were allowed to switch between
hexachords to increase range, but you had to do it in a certain way. Everyone thought octaves were the perfect
interval and certainly weren’t going to ignore them.)
Switching for the moment to a
more modern, octave-based notation, we write the final note, the name, and the tonal
patterns for each of the modes:
D, Dorian mode: T-S-T-T-T-S-T
E, Phrygian mode:
S-T-T-T-S-T-T
F, Lydian mode: T-T-T-S-T-T-S
G, Mixolydian
mode: T-T-S-T-T-S-T
These are also called the authentic versions of these modes. They (usually) start on the final note,
nearly always end on it, and typically rise an octave
above it and drop no more than a tone below it.
(Note that the “leading tone” in Lydian, the one that comes at the end
of the scale before a new F, is a semitone, so the Lydian was never supposed to drop below F.)
I said we’d get to b-flat,
and here we are. If you notice, the
first three intervals for Dorian, Phyrgian and Mixolydian modes all contain two tones and a semitone. Lydian does not! It contains three whole tones. This is the interval between F and b-natural
above it. It is not a perfect fourth –
it is the “tritone,” also called “the devil in
music.” It sounds very yucky and it was
advisable to avoid! By dropping the
b-natural to a b-flat, the perfect fourth could be obtained in Lydian
mode. This was know
at least as early as Guido d’Arezzo.
Plagal Modes
How many modes are
there? What about “Aeolian A”? You may have heard of Aeolian mode (it is the
same as our minor scale), Ionian mode (the major
scale) and Locrian mode (rarely found in nature;
start on B and go up). These were
defined sometime in the sixteenth century.
They were not recognized by theory in the era which this class covers. But
I said there were eight modes!
True. Different songs call for
different ranges; this is particularly true if you monastery is trying to set
music for boys’ and mens’ choirs! The plagal modes
were invented to cover this need.
We said the basic modes,
discussed above, are called authentic. Their final note is either
D, E, F, or G, and they can go an octave up and a tone below that note
in the course of the song. The plagal modes have the same finals: D, E, F,
and G. But their range is different. The
song may rise only a fifth above, but also a fifth
below the final. They can be referred to
as “plagal Dorian,” “plagal
Phrygian,” etc., or “hypodorian,” “hypophrygian,” etc.
Other than the range, there
is another difference between the authentic and plagal
modes. Each mode has a tenor or reciting tone. I quote Pat
Yarrow on this: “In authentic modes, the tenor is a fifth above the final (or
tonic). In plagal
modes the tenor is a third below the tenor of the corresponding authentic mode.
Whenever the tenor would fall on B, it moves to C.” A very typical chant structure in an
authentic mode is to begin at the final, rise smoothly up a fifth to the tenor,
and hover around that note, going up or down a bit, and then returning to the
final at the end – hence the term “reciting” tone.
|
Final |
Range |
Tenor |
Authentic or plagal |
Greek name |
|
d |
D-d |
a |
authentic |
Dorian |
|
A-a |
F |
plagal |
Hypodorian |
|
|
e |
E-e |
c |
authentic |
Phrygian |
|
B-b |
a |
plagal |
Hypophrygian |
|
|
f |
F-f |
c |
authentic |
Lydian |
|
C-c |
a |
plagal |
Hypolydian |
|
|
g |
G-g |
d |
authentic |
Mixolydian |
|
D-d |
c |
plagal |
Hypomixolydian |
Rules, Bah!
In most cases, Guido admits
that skilled composers can and have violated most of the rules he outlines:
there are hymns that go the interval of a tenth above the final in the
authentic mode, and those that go a sixth above in the plagal
mode. Authentic modes should be in the
higher registers and plagal in the lower, but those
rules get broken, too. One gets the
distinct impression, though, that Guido would advocate working within the
standard rules until one understood them, and then going outside of them when
one can a real reason to.
Well, yes.
Well, it will help you
understand medieval music, and possibly compose your own.
Note: There
is a whole world out there of “gapped scales” as they apply to modes which we
will not be covering. I have an
excellent handout by Adelaide de Beaumont (Lisa Theriot)
which goes into this a bit, and if I can get her permission, I will make it
available.
Early medieval music would
rarely have been written in a major or minor key. It would instead be written in one of these
modes. Accompaniment to it would not
have been modern I-IV-V chords; it would draw on the ideas of the “harmonious
intervals” of the octave, the fifth and the fourth. A simple drone an octave or two below the
melody’s final is a wonderfully period accompaniment.
What? Your recorder is in the key of D? No worries.
First, “A=440 Hz” is a 20th century convention. Second, it is pretty well-known that pitches
could vary more or less wildly between, say, pipe organs in different
churches. Pretend that your D is Ut, and your Dorian mode starts on E. Call it “Dorian E” to prevent confusion in
your ensemble; the harper in Dorian D may not
understand why the two of you are hopelessly out of tune, otherwise.
Some of Guido’s Other Rules
for Writing Music:
Guido’s Automatic Melodic Composition
His example is somewhat
confused in my translation; assuming a typo:
1.
Write down the gamut
from Γ (GG) to aa.
2.
Under the note
Γ, write the vowel “a.” Under the
note A, write the vowel “e.” Under B, “i,” and so on. (Example 1)
3.
Choose your mode
and starting note. Notice that this note
and the four above it contain all the vowel sounds.
4.
As you go through
your chant text, assign each syllable a note based on the vowel sound it
contains. (Example 2)
5.
Deviate from this
if you need to, say, bring the melody back to the final at the end.
6.
You never get
more than five notes in a song this way.
For more variety, write another line of vowels under the notes, this
time beginning with the vowel “o” under Γ.
(Example 3)
7.
Take an entire
octave thusly notated. You should get at
least three different possible notes for each vowel sound. Pick from these.
Example 1:
Γ A
B C D
E F G
a b c
d e f
g aa
a e i o
u a
e i o
u a e i o u a
Example 2
(after Guido; with “typo”):
G u Jo rum
F o to
E i ri
D e cte nes me
C a San han
Note: the vowels do not
correspond to those given in Example 1.
This is the “typo” for which I have no explanation.
Note: the lowest notes are
assigned to the more “closed” vowels, contrary to what one accustomed to modern
vocal production would expect.
Example 3:
Γ A
B C D
E F G
a b c
d e f
g aa
a e i o
u a e i o u
a e i o
u a
o u a
e i o
u a e i o u
a e i o
St. Ambrose’s basic principles of hymn-writing:
Divided into strophes
(stanzas) and each strophe has the same number of
lines (generally four) the same metrical pattern, and when present, the same
rhyme. Little is known about the music
of hymns until it was written down in 9th and 10th cen. The melody,
set syllabically, is repeated for each strophe and makes use of all church
modes.
Various repeat forms are
used, e.g., ABCA, ABAB, AABC
The interchange of text and music was fairly common in the hymns.
The earliest Western
polyphony was called organum.
It consisted of a primary voice, singing the melody, and an accompanying
or organal voice.
It seems mostly to be for two parts, although methods for writing up to
four parts are given. The first actual
example of three-voice music (besides the instructive examples in the
treatises) is found in Codex Calixtinus, c. 1140, Santiago de Compostella.
With two parts, the
accompanying (or organal)
voice can stand in one of four relationships with the primary voice as it moves
along. In unison, they sing the same notes. (If they keep this up for a
while, they’re not really parts at all!)
In parallel, they are
separated by some interval (usually a fourth or a fifth), and when the primary
goes up a tone, the organal goes up a tone. In oblique
motion, one voice is holding a note steady while the other is moving either up
or down. In contrary motion, when the primary voice goes up a tone, the organal voice goes down a tone, and visa versa.
Musica Enchiriadis c. 850 contains earliest known examples of Western
polyphony and has rules for writing organum. It puts the organal
voice in note-against-note counterpoint against the primary voice. In the middle of phrases, it tends to “strict
parallel organum,” where the interval between the
voices is either a fourth or a fifth (with the organal
voice below the primary). At this point,
fifths were favored. The phrases were to
end on unisons, so at the beginnings and endings of phrases there was oblique
and contrary motion as well. The fifth,
unison and octave were the most common intervals.
Guido d’Arezzo restated this, recommending first that one
place the organal voice a fourth below the primary,
then double that an octave up (a fifth above the primary) if desired. As soon as he writes this, however, he
dismisses it, saying that the fifths sound to “hard” and suggesting another
“softer” method instead. He permits the
intervals of the whole tone, the major and minor third, and the fourth, but
excludes the semitone and the fifth. He
says that the organal voice should stay below the
primary voice, and that they should never be separated by more than a
fourth. Convergence on the final tone (occursus) is
“preferably by a tone, less so by a ditone [major
third] and never by a semitone.” It is
“scarcely” made from a fourth. Guido’s
form of organum is called “free organum.”
b-natural does not have a perfect fourth below it. G is used instead. For chants that descend to F and/or end on G,
the organal will sound F to accompany G and ‘a’ at
“suitable places;” otherwise, “F in the chant is not accompanied by F in the organal voice.” (But
Guido does not say what should accompany it!)
But when b-flat is used in the chant, F is in the organal
voice (since it is a fourth below b-flat).
John Cotton mostly restates
Guido in his De Musica,
c. 1100. However, he has a marked
preference for contrary rather than parallel motion. He also suggests that octave finales are as good or better than unisons.
In addition to
note-against-note, there are examples of one voice holding long notes with the
other voice singing many.
Here are some early period
poems. Use Guido’s method of creating
melody as a guide to set them to music, then create an
organal accompaniment.
from “Dies irae,” St. Columba, c. 550
Day of the king most
righteous
The day is nigh at hand
The day of wrath and
vengeance
And darkness on the land
from “Written by Colman the Irishman to Colman returning
to his own land,” Colman, 9th cen. (Write music for one verse and use it for both.)
Vanquished art thou by love
of thine own land
And who shall hinder love?
Why should I blame thee for
thy weariness
And try thy heart to move?
Since, if but Christ would
give me back the past
And that first strength of
days
And this white head of mine
were dark again
I too might go your ways.
from “He complains to Bishop Hartgar
of thirst,” Sedulius Scottus,
c. 850
But with it all, there’s
never a drink for me
No wine, nor mead, nor even a
drop of beer
Ah, how hath failed that
substance manifold
Born of kind earth and the
dewy air!
from a 10th century love poem
Come, sweetheart, come
Dear as my heart to me
Come to the room
I have made ready for thee
Here there be
couches spread
Tapestry tented
Flowers for thee to tread
Green herbs sweet scented.
Warren Babb, trans. Hucbald, Guido and John on Music: Th.ree Medieval Treatises. trans. Warren Babb.
Harold
Gleason and Warren Becker. Music in the Middle
Ages and Renaissance. Music
Literature Outlines – Series I, 3rd ed. Frangipani Press:
Dom Anselm Hughes, ed. Early Medieval Music Up to 1300. New
Hendrik van der Werf. The Oldest Extant Part Music
and the Origin of Western Polyphony, vol. 1. Published by the author,
Helen Waddell, trans. Medieval Latin Lyrics. Penguin Classics:
Patricia Vivien Yarrow. “A Brief Introduction to
Modes in Early and Traditional European Music.” http://clem.mscd.edu/~yarrowp/MODEXh.html