# Notation

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```					Tonality
tutorial
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Macromedia Breeze

   The Macromedia Breeze Presentation will advance automatically, but you
can stop and start the presentation at any time by touching the space bar on
and PageDown keys. The index to the right is clickable and the whole
presentation is searchable by keyword, for instance, ―interval.‖
Instructions

   Now practice stopping the presentation by tapping on your spacebar. Start
and stop it again (space – space). Now try PageDown and PageUp.

   In the Tests and Quizzes folder you will find Exercises 1-7, which apply to
these tutorials. You’ll also be asked about the content of these tutorials in
the self-paced Chapter quizzes, as well as on the midterm and final.

   Now resume the presentation by tapping on your spacebar.
Tonality Tutorial

   This chapter introduces you to a certain amount of the grammar of music:
scales, keys, intervals, triads--some of the things that go to make up
called ―harmony and theory‖), it should take you 30 minutes or so to look
over and refresh your memory. If it’s all new to you, it will probably take
something over an hour, and you’ll want to come back and review it from
time to time. At a couple of junctures, you’re asked if you want to pause and
do a few minutes’ worth of exercises. If so, click the button. You’ll also be
Note from the Professor

   Note: In an introductory course in musical repertoire or music appreciation,
you don’t usually have to get much beyond knowing that there are scales,
the difference between major and minor, and what cadences and
modulations sound like. Go as far as you need to in this unit and then feel
free to abandon ship--at least for now. Once their curiosity is aroused,
that structural attribute of music. If that’s the case with you, finish the unit. If
you know your intervals and triads, and can build (and play) all the major
and minor scales, you’ll be way ahead of the competition.
Tonality

   The harmonic system in predominant use from, say, 1600 to the present is
called tonality. In it the octave is divided into twelve half-steps: major and
minor scales are built from standard patterns of half- steps and whole steps.
The first pitch of the scale, and whether its mode (i. e., its pattern of half-
steps and whole steps) is major or minor identify its key. There are twelve
major keys and twelve minor keys--one each for each half-step.
Major scale

   The MAJOR SCALE divides the octave into a pattern with half-steps
between scale degrees 3-4 and 7-8. The only place on the keyboard where
this occurs without the use of black keys is from C to C: this is the C-major
scale.
Half steps, 3-4 and 7-8

   In order to form major scales starting on pitches other than C, notes must
be raised (sharped), or lowered (flatted) to create the pattern that puts half
steps between 3-4 and 7-8. This requires use of the black keys. For
example, to build a major scale on the pitch D, it is necessary to raise
(sharp) the notes F and C to position the half-steps between scale degrees
3-4 and 7-8.
E-flat major scale

   To build a major scale on the pitch E-flat, it is necessary to lower (flat) the
notes A and B (in addition to the original E-flat, of course) to position the
half-steps between scale degrees 3-4 and 7-8.
   NOTE: You must have one of each letter name. Thus the half-step between
3 and 4 is G – A-flat , not G – G-sharp.
Major scale

   And so on. You can now construct a major scale on any of the twelve
pitches. The technique is the same for every scale: always put half steps
between scale degrees 3-4 and 7-8.

   EXERCISE 4 in “Tests and Quizzes” concerns major scales.
Enharmonic equivalence

   By the way: throughout this discussion we are assuming the equivalence of,
say, C-sharp and D-flat. That is, for all intents and purposes the scales of C-
sharp major and D-flat major sound the same--as do the pitches, say, B-
sharp and C natural. The technical term for this is enharmonic equivalence.

   Enharmonic equivalence is undeniably true of a keyboard, where a B-sharp
and a C-natural are always the same, because you can only strike the one
key. On orchestral instruments and with the voice, however, you’re always
adjusting pitches according to the surrounding context, such that B-sharp in
C-sharp major is in fact a rather different pitch than C-natural in C major.
Minor scale

   The MINOR SCALE divides the octave into a pattern with half-steps
between scale degrees 2-3 and 5-6. The only place on the keyboard where
this occurs without the use of black keys is from A to A: this is the A-minor
scale.
Half steps, 2-3 and 5-6 … but

   For the so-called natural minor scale, that’s all you have to do: put half
steps between scale degrees 2-3 and 5-6.

   The strongest directional pull of tonality, however, is the half step from scale
degree 7 to 8. (We’ll come to call this ―leading tone‖ to ―tonic.‖) So usually
you throw in the half step between 7-8 for the minor scale as well. This is
achieved by sharping the 7th scale degree. We call this variety of minor
scale the harmonic minor scale.
Major vs. minor

   What really makes the difference in flavor between the major and minor
modes is the third scale degree: E-natural in C major, E-flat in C minor.

   Three Blind Mice in major

   Three Blind Mice in minor

   Minor is darker, deeper in color, and often used for sinister or melancholy
effect. Kings are crowned and God is praised in major. Death scenes are
usually in minor.
Melodic minor scale

   One more complication: good melodies don’t usually skip 1 1/2 steps, as
you would have to in A harmonic minor from F-natural to G-sharp.

   So you often correct a minor tune by throwing in sharp-6 as well as sharp-7,
and reverse the process going down (because you don’t need the pull of
sharp-7 to 8 going down from 8). The result is a melodic minor scale,
ascending and descending.
Natural minor, harmonic minor, melodic minor

   Summary: All minor scales, in theory, have a half-step between scale
degrees 2-3. Natural minor has half steps between 2-3 and 5-6.

   Harmonic minor has half steps between 2-3, 5-6, and 7-8 (sharp the 7th).

   Melodic minor sharps 6 and 7 ascending and returns them to their natural
state descending.
Raised seventh degree

   I wouldn’t worry too much about these various forms of minor just now. The
important thing is to sense the difference in major and minor. Just don’t
freak out when you see, say, G-sharps in A minor: it’s just a matter of
raising the 7th scale degree to make it a leading tone.
Minor scales

   And so on. You can now construct a minor scale on any of the twelve
pitches. The technique is the same for every scale: always put half steps
between scale degrees 2-3 and 5-6. For the harmonic minor, sharp the 7th
degree to make a leading tone. (This requires, for the flat keys, raising a flat
to a natural, as in the cases of C and F minor, below.)

   EXERCISE 5 in “Tests and Quizzes” concerns minor scales.
Key signature

   In order to avoid the necessity of writing sharps before all the Fs and Cs in a
passage in D major (for instance), the composer writes the regularly
recurring sharps at the beginning of each staff (to the immediate right of the
clef sign) in a key signature. This key signature means that all encounters
with the specified pitch (in any octave) are to be so altered, unless marked
by an accidental sharp, flat, or natural, or until the key signature is changed.
Order of sharps and flats

   To figure out a key’s signature, you just count the number of sharps or flats
in the scale and place them into the signature--in the order shown.

   (Use the natural minor--half-steps between 2-3 and 5-6--when counting
sharps and flats in minor scales.)

   Most people just memorize the scheme, which adds sharps and flats in a
predictable order.
Relative minor, relative major

   As a matter of fact, if you work out the key signature for a D major scale
(half-steps 3-4 and 7-8) and that for a B minor scale (half-steps 2-3 and 5-
6), you’ll find that they both have two sharps. Each key signature, in short,
serves for both a major and a minor key.

   You call B minor the relative minor of D major and, likewise, D major the
relative major of B minor.
Major, minor

   The sharps and flats of the key signatures occur, as you just learned, in a
predetermined order. Each of the twelve key signatures serves for two keys:
a major key and its relative minor. Thus C major and A minor (the relative
minor of C major) share the same key signature, i.e., no sharps or flats. To
determine whether the key signature indicates its major or its relative minor
key, listen, first of all for the characteristically bright sound of major or dark
sound of minor. Then look at the final bass note in the composition or
movement: it will almost invariably be the tonic (key-note) of the
composition. For example, a piece with no sharps or flats in the key
signature ending with an A in the bass is probably in A minor; if it ends with
a C in the bass, it is probably in C major.
Key signatures: all

   Here are the twelve possible key signatures. Note that the flat keys and the
sharp keys intersect at 6 flats = 6 sharps.

   These are quite easy to memorize if you stop at first to analyze the system.

   There are some nifty properties to key signatures that can save you a lot of
time:
  To get the major key from a sharp key signature, go up a half-step from
the last sharp.

   To get the major key from a flat key signature (from two flats on), go
back one flat.

   To get the minor key name, ascertain the major key name and go down
three pitches (counting both first and last, and including the sharps or
flats in the key signature).
Circle of Fifths

   A useful tool in understanding the major-minor tonal system is a diagram
known as the circle of fifths (next screen). Each segment of the circle is a
5th higher than the pitch to its immediate left: you start at the top with no
sharps or flats and proceed clockwise by adding sharps and
counterclockwise by adding flats. The flat keys and sharp keys intersect at
the bottom (G-flat major = F-sharp major). You’ll also notice that:
  the major keys are on the outside, the relative minor keys on the inside;
  each key’s dominant key is that of the next wedge over, clockwise: G is
the dominant of C is the dominant of F, and so forth;
  keys which are adjacent, e.g., C major and G major, or keys which share
the same key signature, e.g., G major and E minor (its relative minor)
are said to be ―closely related keys.‖ Keys which lie opposite in the
circle, e.g., A Major and E-flat Major, are distantly related.
   EXERCISE 6 in “Tests and Quizzes” concerns key signatures.
Circle of Fifths
Parallel minor

   You’re probably wondering what to call the relationship between D major
and D minor. It’s called parallel (as opposed to ―relative‖): C minor is the
parallel minor of C major.

   Another way of expressing this relationship is to say simply that there’s
been a shift of mode from the major to the minor of C. You talk, generally,
about ―the major mode‖ and ―the minor mode‖--not to be confused with the
Renaissance church modes (Dorian, Plagal, etc.), about which you may be
hearing before we’re all done.
Intervals

   An interval is the distance between two pitches. It is calculated by counting
the number of tones of the diatonic scale it includes. (Count on your fingers,
if you like, and include the first and last pitch in your tally.) The interval C to
D is a second; C to E, a third, and so on. The names of the intervals are:

unison, second, third, fourth, fifth, sixth, seventh, octave
Perfect, major, minor intervals

   The quality of seconds, thirds, sixths, and sevenths is described as either
major (M) or minor (m) according to the scale that contains them. Fourths
and fifths are the same in both major and minor scales, so they are called
perfect (P). (There are some other intervals, but they need not concern
students at this level.)

   Every note of the major or minor scale may be imagined to have a triad (a
3rd and a 5th) built upon it. The most important triads of every major and
minor scale are those built upon the first degree (I=tonic triad) and the fifth
degree (V=dominant triad). The harmonic progression V to I is known as a
full cadence and is important in establishing the key-area (or tonality) of a
composition. More on this in a minute.

   Triads are described according to the intervals they contain, again based on
whether the pitches are contained in the major or minor scale constructed
from the lowest member of the triad: its root. Thus a triad on C, built from
members of the C major scale, will be C-E-G: the C-major triad.

   Built from members of the C-minor scale, you would get C-E -G: the C-

   The pattern of triads, whether major or minor, is the same for every major
scale.

   The quality of the triad built on VII, consisting of two minor thirds (B-D and
D-F) is called diminished.

   A different pattern obtains for the minor scales, but (likewise) is the same
from scale to scale.
Root position, inversions

   So far all the triads you’ve seen have been root position, that is, in 1-3-5
order with the 1 on bottom. Obviously, in real music the triads can be
distributed in any inversion. The trick in identifying them is to reduce them to
root position.

   All the triads to the left above are C-major triads. The example to the right
Functional Harmony

   The triads have a functional, or hierarchical relationship, culminating in the
strong pull of dominant to tonic. The conventional order is something along
the following lines:

   I = tonic
   VI = submediant
   IV = subdominant
   V = dominant
   I = tonic

   At the right is an elementary chord progression. The functions associated
with the chords appear beneath the staff.
Roman numerals for chord description

   In case you hadn’t yet noticed, we typically call chords by Roman numeral
based on the scale degree of their roots and individual pitches by Arabic
numeral, again based on scale degree.

   The second triad (A-C-E) in the left-hand example is called vi because its
root, A, is the sixth scale degree of a C-major scale. (It’s a lower case vi
because its quality is minor.) The third triad (F-A-C) is built on F and is
major, thus IV.
Seventh shord

   You may also have noticed in our chord progression the curious dominant
chord G-B-D-F and its Roman numeral with superscript 7.

   This is a seventh chord.
Seventh chords

   A common enhancement to triadic harmony is to add a fourth pitch to the
triad, up another third, thus: root + 3rd + 5th + 7th. For obvious reasons you
call these kinds of four-note chords seventh chords.

   The most common of these chords is the one where you add a 7 to the
dominant chord (V). This becomes a so-called dominant seventh chord, and
makes the pull to tonic even stronger.
Tritone

   The reason the pull is so strong has to do with an interval called the tritone
(same as a diminished fifth, scale degrees 4 and 7), where the strong
dissonance urges itself to resolve outward: the 7th upward to 8, the 4
downward to 3. When combined with the bass motion V - I, you can’t get
any more conclusive.

   The ancients called this interval diabolus in musica: ―the devil in music.‖

   For your present purposes, the most important aspect of functional harmony
to grasp is the concept of cadence. A cadence is the process of coming to
harmonic rest on (usually) the tonic triad. The most common sort of
cadence is called authentic, and involves going from the dominant triad to
tonic. The strongest kind of authentic cadence (called a ―perfect‖ authentic
cadence) has leading tone to tonic in the uppermost voice and dominant to
tonic in the lowermost, that is: 7-8 in the soprano and V-I in the bass:

   A perfect authentic cadence is a powerful event because it harnesses two of
the strongest pulls in music, leading tone to tonic (7-8) and dominant to
tonic (5 down to 1) in the two outer voices; if you add the dominant 7th
(here, F on a G-major V triad), the effect is stronger still. You sense a
feeling of arrival, of coming home, as it were.
Modulation

   Once you’ve established the tonic (usually through the kind of chord
progressions we just saw), you can modulate to a new key. Very often your
first modulation is to the dominant key (i.e., to the key of the triad based on
the dominant pitch of the original scale: G major from C major). The easiest
way to do this is by pivoting to the new key on a triad common to both keys.
The triad of C major, for example, is tonic (I) in C and also sub-dominant
(IV) in G.
Pivot points

   Cadences also confirm that a modulation to a new key has occurred. A
composer who has modulated from C major to G major over the course of a
musical passage will go to great lengths to prove all this with cadential
material. You’d spot the process of moving toward G major when the F-
sharps begin to occur—F-sharp being the difference between C major and
G major--and know for certain that’s what happened when you see or hear
Enough said

   I think that’s enough harmony for right now. You now know pretty well how it
all fits together. The rest is a matter of building on these principles, and of
course of learning to hear what these things sound like. Your Music
Department doubtless offers classes in harmony which will give you enough
vocabulary to compose serious songs for your rock band, church choir, or
whatever. I commend such classes to your attention. Well done. Now do the
last exercises for this chapter and go back go to the home page.
   EXERCISE 7 in “Tests and Quizzes” concerns keys and intervals.