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Old Beginning Music Theory Workbook

David Raleigh Arnold

♭=flat. ♮=natural. ♯=sharp. ➀=first string

Introduction: The History of Western Music

In the West, ancient Christian monks devised a modal system in a completely new way. They did it without using a stringed instrument for reference. Because of that modal system, which cannot be demonstrated to have been derived from any other, notation became almost inevitable. For the first time in all of mankind’s experience, there arose a way of writing music as sound, not merely as instructions in how to play some stringed instrument.

Notation is the unique contribution of Western civilization to the cultural history of man. In that one way, Western culture was superior to all others. Western music has to some varying extent adopted all other music, and the same certainly cannot be said of any other musical culture. Ten thousand years from now, if we are remembered at all, it will be for having developed musical notation.

With notation, music with many different voices at once, (), became a practical possibility. A way of measuring the lengths of notes was strongly desirable for writing music in parts, and notation therefore became . Because written music is in time, and not merely sequential, it would be best if children were taught to read music as early as possible, before they were taught to read anything else.

At first, harmony was considered to be the interaction of voices and nothing more, and theories and practice of (writing voices which interact with each other) were in large measure the curriculum of Bach and his predecessors. During Bach’s lifetime Rameau published a Treatise on Harmony which propounded the notion of the fundamental bass, which we usually call the root today, and thus invented the idea of the chord as we understand it now. Soon, music began to be written accordingly, but for the Classical composer serious composition remained a matter of using the theories of the past as well as the new theories to write music as individual moving voices. In a very few years, some Romantic composers developed chord sense, which is the idea, implied by Rameau, that a chord can be a building block in writing music regardless of what happens to all the possible independent voices. From 1722 to before 1836 it came about that there were people hard at work with all three views, to varying degrees, of music composition and song writing.

If music theory is considered in this way, nothing much has happened since. Carcassi’s use of 9th and 11th chords shows that his writing of harmony has more in common with that of Duke Ellington or George Gershwin than that of his contemporary, Sor.

Progress in art is individual accomplishment, not style.

What is the purpose of music theory?

Musical theories are very important in the history of music, but this is not the point of studying, or rather practicing, music theory. Traditionally, that term refers to a series of writing exercises. Reading most web pages devoted to music theory is like reading an arithmetic textbook without doing any problems. It really doesn’t lead to understanding. Music theory can sometimes supply practical working guidelines, but one does not fit all. It is not about absolute truth, or what a given fragment of music really means. It is about point of view. It has been very wisely said that understanding is nothing more than seeing something from more than one perspective.

The traditional content is related to pitch and hardly at all to rhythm, but fortunately that is changing.

A Few Conventions for HTML

Individual note names will be small letters, c d e f g a b. A note name may be in quotes occasionally, as in a. Chord names or names of scales, modes, or tonalities will start with capitals. Sharps, flats, naturals, double sharps and double flats are as b#, bb, bn, bx, and bbb. The natural notes starting from the open sixth string are: e, f, g, a, b, c d e f g a b c’ d’ e’ f’ g’ a’ b’ c” d” etc.. For guitar, written middle c is c. Actual middle c is c’. The punctuation will often be omitted.

If you wish to become more familiar with bass clef, the sixth string e, may be placed on the first leger line below the staff. Draw your own clef, or use different paper.

Section 1: Scales and Modes

The Distance between Two Notes

You need some music manuscript paper.

Here is some in pdf format with treble octave clefs.

An is the distance between two notes, or two notes described by the distance between them. The distance is inclusive. When two notes have the same name and they are on the same line or space of the staff, the interval is not a zero, it is a prime. If these two notes are one fret apart because of flats or sharps, the interval is an augmented (or diminished if you prefer) prime. If the two notes are identical, the interval is a unison or perfect prime.

The distance from one note to the adjacent note, like d to e or g to f, is a . Among the natural notes, there are major and minor seconds. Notes which are a major second apart have one empty fret between them if they are both on the same string. Notes which are a minor second apart are on adjacent frets.

Writing whole notes at prime intervals using sharps

Now, leaving about one eighth of a line length between notes horizontally, write whole notes on each line and space starting with the c on the first leger line below the staff, which is written middle c, and ending with b, the middle line. Make them plain ovals, which are whole notes or semibreves. Put a bar immediately before each note, except the first.

Immediately after each note, using a sharp, write the note which makes an interval of an augmented prime. The sharp has to be between the notes, on the same line or space. Place a bracket over each pair of notes. Play it all on your instrument. Usually, playing intervals would mean playing the two notes at the same time. That is not necessary now.

Writing whole notes at prime intervals using flats

Start all over again, only this time use a flat to make each augmented prime. Again, the flat is between the notes, otherwise both notes would be flatted. (Shouldn’t the lower note be first? No, not necessarily.) Play the intervals. The augmented prime is not a possible interval in a chord name, but it is not very hard to find many in polyphonic music. One place to find one would be a piece that was , having simultaneous parts with different tonalities.

Writing whole notes at major second intervals

Start all over again, only this time you are to make major seconds above each note. When writing notes at the second to be played at the same time, the upper note is on the right unless the two notes are on different stems up and down. Consequently, the sharps of the second note appear before the first note, and only the fact that the sharp is on the same line or space as the second note indicates which note is to be sharped. You always have to look at seconds carefully when there are chromatic signs before them. Write finger numbers for each interval and play both notes of each interval together.

Writing whole notes at minor second intervals

Again, only this time you are making minor seconds above. Finger them and play them too, together if you are able.

Writing a C major scale and naming its tones

When referring to scales, a major second is called a whole step (w) and a minor second is called a half step (h). Which natural notes are a half step apart? You have to know that, forever. Remember it.

Write the note heads from c to c’ on the staff. The interval from c to c’ is an , eight notes. Make half notes about one tenth of the line length apart, so you leave some room at the end of the line. Draw stems. Stems originate counterclockwise from the note head. The stems are one octave long. They point straight up for notes below the middle line, straight down for the notes above. Usage varies for the middle line, but make them down for now.

Put the Roman numbers I, II, III, IV, V, VI, VII, and I above each note. I is called the . Label those tonic. V, the note five notes up, is called the . Bracket the tonic and dominant with a bracket below the staff and label the dominant below that. The dominant is the most important of these names to know.

The subdominant is five notes below the tonic. Bracket the tonic and subdominant with a bracket above the staff. Halfway up to the dominant from the tonic is the mediant. Halfway down to the subdominant from the tonic is the submediant. The note above the tonic is the supertonic. The note below is the subtonic. If VII is a half step below I, VII is a leading tone. Done. Now draw it pretty, with the notes, names, brackets, and numbers.

Writing the Major Scales with Sharps

The step scheme of a major scale in whole and half steps is wwhwwwh. That is the definition of a major scale. Some theory books make that two tetrachords wwh, joined by a whole step. Any connection of the modern scale with the ancient Greek tetrachord is dubious, to say the least.

Write the notes from c to c’ again. Put w between the notes c and d. Is c to d a whole step? Yes. Write a w between the notes d and e. Is d to e a whole step? Yes. Write an h between the notes e and f. Is e to f a half step? Yes. Write a w between the notes f and g. Is f to g a whole step? Yes. Write a w between the notes g and a. Is g to a a whole step? Yes. Write an h between the notes b and c’. Is b to c’ a half step? Yes. Done! Of course, this was our model, so there aren’t any sharps or flats. Remember, a major scale is wwhwwwh. If one of them doesn’t come out right, figure out why, always. That’s where you learn something.

Now find the dominant, V. Write a V below it. Start the next scale with that note. Draw it as before, but this time it’s from g to g’. Write a w between g and a. Is g to a a whole step? Yes. Write in the next w. Is a to b a whole step? Yes. Write in the h. Is b to c’ a half step? Yes. Keep going. Is e’ to f’ a whole step? No! It has to be a whole step, to be wwhwwwh, a major scale. How to you make it a whole step? You have to sharp the f’. Now, e’ to f#’ is a whole step and therefore f#’ to g’ is a half step. Put in the h. Done!

Now you start with the dominant of the scale you just finished, and make it wwhwwwh just like the others. Take any starting note above the middle line down an octave, or the notes will become very high.

Continue until you have all the major scales with sharps. When you end with all seven notes sharp what scale will that be? C# of course. How many will there be, not counting the C you started with? One for each note, seven. Many things in music theory are self checking like this. Again, if you mess up, find the error and fix it. Play it all. If you find mistakes, fix them.

Writing the Major Scales with Flats

Get another sheet. Write out the C scale as before, and locate the subdominant. You are going to go in the other direction from before, descending in five note intervals, or fifths. Draw the bracket. Put a IV under the subdominant, f in this case. Write the notes f to f’ and make the scale wwhwwwh. Keep going until all the notes are flat, the key of Cb. Look out! The one after F is not B, it is Bb. There will be seven scales with flats in them, so the last one will be...

If you feel that you didn’t get enough practice with this, do them all over with the scales descending instead of ascending. The step scheme is reversed: hwwwhww.

Writing the Key Signatures

The chromatic signs in key signatures are always placed on the staff in a standard manner. The sharps, with the octave, are f’ c’ g’ d’ a e’ b. The flats are in the opposite order: b e’ a d’ g c’ f. Write the key signature for each major scale at the end of each of the scales you have written before. Take a stab at drawing the clef. The quickest and easiest is to just make it a letter g. The ’g’ line goes through the center of the closed part of the letter. Then the appropriate sharps or flats go in order.

Transposition: Changing the Key

It is very common to want to change the key of something, to make it easier to play or sing or for some other reason. A transposition chart makes short work of it. For example, to move somthing from A to Eb, write out the names of the seven notes of each key and substitute. Here is a chart to move from A to Eb. Obviously, you have to transpose either up or down. You can’t do both at once.

Old key a b c# d e f# g# New key eb f g ab bb c d

We are working with key signatures, not keys, so the mode doesn’t matter. The simplest thing is to start with the notes of a major scale, but any lists of the notes in the new key and the old would do provided they were in the equivalent order. Don’t mess with minor, do major only, even if the music is ’really’ minor.

Do to each note in the new key anything done to it in the old key. For example, if you have a g in the key of A, since it has been lowered from g# you want a db in the key of Eb. The Bm chord in the old key is an Fm in the new key. An eb in the old key is a bbb (b double flat) in the new key. Preserve the note name. Any kind of f in the old key must be some kind of c in the new key. Always use the simplest chord name when transposing chords, however. An ax (a double sharp) in the old key will be an e# in the new key but an Ax chord will be an F chord. There is no E# chord. Make it an F chord.

Nothing could be simpler than the reason for being so totally inconsistent. Chord names are for grabbing quick, not for explaining things. You won’t find an E# chord in your chord chart, either. This system of naming chords was not developed for the convenience of teachers at universities. It is for ordinary readers of sheet music and musicians in a hurry, who hopefully are not interested in the latest cryptic symbols issuing from educators.

From my Nine Very Easy Guitar Duets, transpose the melody part of GuitarGo from C to G and from C to D. Again, you are working with a list of notes for each key signature, so it doesn’t matter what the mode is, or even if it has one. From my Ten Lessons transpose the Blues in G to A and E, the Blues in D to Bb and Ab, and the Blues in E to G and Eb. Play them to make sure they’re right. Better yet, transpose some things of your own choosing that you want to play. Of course there are other ways to transpose, but this method is quickest, easiest, least prone to error, and requires the least musical knowledge or experience.

The Modal System

A is two things: tonality and mode. The tonality is the note around which the music is organized. The mode is the system of notes. Key of C is short for tonality is C, mode is major A scale is a list of notes starting from the tonic, the tonality. As a practical matter, considering modes to be scales and the tonic to be the first note of each, there are only six possible modes using the natural notes we have. They make scales from c to c’, d to d’, and so on, but b to b’ cannot be made to work. A late medieval theorist cracked up a bogus Greek name for it, but no one ever wrote anything with it because it is impossible to do so.

The first task is to find the first mode. How do you start teaching singing without any reference point? You start with a tonic, and sing up a whole step, tonic, down a whole step. Then you add a half step more up and a half step more down, and so on. Every note above this tonic is the mirror inversion (not the ) of a note below. There is only one note from which you can do that, and that determined which was mode I. Figure it out. What note does mode I start with? What note has equal intervals up and down? Easy to find, it’s at X in wwwhwXwhwww so the step scheme of the first mode is whwwwhw.

In the fourteenth century the modes were renumbered. Since most guitarists are more likely to play Dowland than Ockeghem, it is probably better to look at the newer system.

There were twelve modes, in theory, one starting on each note from d to c’, which was called , and the next starting at the fourth below, which was called a mode. Count down four notes, inclusively, to get a fourth. Instead of using the last six, bbs could be substituted for the bs in the first six. The flat sign did not exist, but the bb, as a variation of the b (or vice versa) was available. (Germans use h for b and b for bb. Historically, the h represents a b with corners. The flat sign is a b with fewer corners.) The plagal modes were not quite like different scales because they had the same , or tonalities, as their authentic modes. Of course this is confusing in the extreme, and there is nothing for it but to start writing this all out.

Writing the Modes

Write the first mode, then its plagal a fourth below, which is mode II, then the next authentic a fifth above that, which is mode III. Use seven lines, but leave a bit more than half of each line blank for later use. Finish with mode VII, which starts with g if you haven’t made a mistake. Replace the II with minor and the VI with major. Write in the step schemes of all seven, but just write the h’s, you don’t need the w’s this time. Put a big letter A at the end of the authentic modes and a P at the end of the plagals.

Forget the fourth mode for a while. Now there are six. Bracket the first note with the third note in each remaining mode. There are three types of intervals of a third among natural notes, consisting in wh, hw, and ww. Therefore, if there is a half step in it, a third is minor. If not, it is larger, therefore major. Label those bracketed thirds as major or minor.

Mode I, minor, and mode III were once called the minor modes, and modes V, major, and VII were called the major modes, but that usage ended with the 19th century.

Skip a line, write the minor mode, and use a chromatic sign to change it to mode I, or rather change it so that it has the step scheme of mode I, so that it is the equivalent mode but in a different key. In other words, write the mode transposed. Use the step schemes which you have already written to figure out how. I’ve cheated, because I have turned plagal modes into scales, but we’ll see how that used to be done soon. Write the minor mode again and similarly change it to mode III. Write the major mode and change it to mode V, in the same sense. Write it again and change it to mode VII. This demonstrates that with the use of one sharp or one flat, the other four usable modes can be constructed from the minor and major modes, and that no other combination of two modes will do. That is why minor and major are still around and the other modes are gone.

It would probably make the most sense to call the mode built on the dominant of the major, mode VII, the dominant major mode, the mode built on the subdominant of the major the subdominant major mode, the mode on the dominant of the minor the dominant minor mode, and the mode built on the subdominant of the minor the subdominant minor mode. It’ll probably never happen, because half of the modes were plagal. Label them anyway, somewhere. Mode VII, built on the dominant of the major scale, is one that matters, a lot.

Adding the B flat

Back at the top of the page, copy the first seven modes again, using the rest of each line, which you left blank before. Flat all the b’s in the new set of modes. Write the new step schemes which they now have, referring to the first version. You only need the h’s. Make the comparisons. Which authentic mode, with a Bb, is equivalent to a minor scale? Which authentic mode, with a Bb, is equivalent to a major scale? Because IV is b to b’, mode III with a Bb is the useless and nonexistent one in the second column. Mode IV with the Bb is now usable though. At this point, we have five modes in authentic versions, that is, final to final, tonic to tonic, real scales. They are I, I with Bb, which is minor, III, V, and V with the Bb which is major. Where is the sixth? It’s the seventh mode, which is authentic. With that you have all of the usable modes as scales, from final to final, tonic to tonic. VII with Bb is equivalent to which mode? Mode I! If you started at VII with the Bb, you would repeat the first six if you went on, in theory. That’s a good reason to quit at VII.

With the Bb available there was no reason to go past mode VII, and it took out the problematic b to b’ mode. In fact, the (range) of the the b to b’ authentic mode was arbitrarily changed to c to c’, in theory that is. That’s it for the early system. It was good enough for Bach.

Now you can see, I hope, that once the sharp and flat were invented, it was inevitable that all modes other than the major and the minor would go to the scrap heap. They just aren’t needed.

Writing the Minor Scales in Sharp and Flat Keys

Over centuries, the meaning of the term minor modes has changed. Now it refers to arbitrary variations of the minor scale. The step scheme of a minor scale is whwwhww. Just to make it interesting, write harmonic minors instead. You raise the seventh degree a half step, so you have whwwh(wh)h. The purpose of that is to get a leading tone in the key to make available certain chords. Because you do it for the harmony you get, it’s harmonic minor. You start with a, not c. Again, the letter note names must always be in sequence. Write the key signatures after. They don’t include that raised seventh degree.

Now write the melodic minor scales. The idea was to close up that (wh) big step but keep the leading tone when ascending, so on the way up it’s whwwwwh. The leading tone is not also a following tone though, so on the way down you revert to the minor mode with natural notes, called the natural minor, wwhwwhw downward. Use key signatures writing them, but remember that these scales go both up and down and that accidentals are persistent.

The minor modes today really don’t represent a system. It makes more sense to regard these minor mode variations as temporary changes of mode to some other mode with the same tonality. There are no mode police around to make sure you stay in some mode or other, so even borrowing from another mode or key as a description seems to me to be a stretch. All kinds of (many modes at once) writing are common, and in early music each voice had its own mode and therefore to some extent its own tendencies toward tonality, so you would have to say that in early music polymodality was the rule, not the exception. From that perspective, speculation about what mode a modern harmonized melody is ’really’ in seems sort of inane, doesn’t it?

Play, review and check everything so far to make sure all is correct. If you skipped anything, especially the modal stuff, do it now. The practice will prove very helpful in the next section.

Section 2: Intervals and Common Chords


© 2005 David Raleigh Arnold. All rights reserved. 20050315

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