Topics Focused on music theory

Dec 282011

What should a music teacher do first? I make no claims to pedagogical expertise. I do have some ideas, though. And I would very much like to know what you think, and what your ideas are.

I only know something about voice, piano, and guitar. But there are some principles, concepts, and ideas that would apply to any instrument.

From the teacher’s point of view, just as a doctor has to work with a patient’s tolerability to a treatment, she must be aware of the student’s tolerability. Continue reading »

Feb 282011

Guitar Harmonics

This is just a brief into/example. To really learn this, look for alternative resources. (See the “Links to Other Music Sites” in the sidebar).

The basic pattern for making a guitar sound harp-like is to alternate notes “chimed” at (usually) 12 frets higher than fretted with notes played in the normal fashion. Some guitarists use pick and 4th finger; some use thumb pick and fingers; some use just fingers. Each harmonic, though, requires at least 2 fingers of your picking hand to execute it.

As a starting point, play at the 12th fret, and do not make chord with your left hand (we will get to that). The openly played strings will be D, G, B, and E, low to high; the harmonics will be played on E (6th string), A, D, G, B. Then there is a role-reversal of sorts (this is just one way of doing it). You can see this in the following sketch in the 2nd bar.


Some of the great jazz practitioners of this technique that I am familiar with are Ted Greene, Martin Taylor, Lenny Breau, Phillip DeGruy, and many others. Most classical guitarists are use this technique as well, when appropriate. For example, a perfect use of this technique is used by Lagoya & Presti in their recording of Debussy’s Claire de Lune.

Ted Greene
Martin Taylor
Lagoya & Ida Presti


Ted GreeneMartin TaylorLenny BreauAlexandre Lagoya and Ida Presti

Jan 162011

Ihe sheet music excerpt below is the first few bars of the tune “On Green Dolphin Street”. Take a look at the last two measures before the final C chord (measures 9 and 10). The chord symbols say Dm7 and G7, or ii7-V7. Usually, in fake or real books, these are the chords you play (not what I put in the bass clef). In an actual playing situation, you certainly wouldn’t want to play the bridge the way I’ve suggested each time–it would get old very fast. Like spoken language, the more ways you can say the same thing, the more fluent you are with your vocabulary. In many cases, the dm7/G7  is just a subdivision of on what could be G7 for 2 measures. Here, those 2 measures are functioning as Dominant harmony. For a moment, leave the bass line out of the analysis, as it is just walking up from D to G. The inner 3 voices in the bass clef, have a similar pattern as the whole-note walk down in the post Alternative Voicing for Piano (1). The difference is that here they are moving up in minor thirds every other beat. They are, again, simple major triads in 2nd inversion, falling on the downbeats. On the offbeats, the chords follow the same pattern again as in the post Alternative Voicing for Piano (1). If you care to analyze deeper, beat 3 of measure 9 is an implied G7#9. Beat 1 of measure 10 is a G7b9+11 (aka Db/G7). Beat 3, measure 10 is a G13b9 (easy to do on a guitar, as well)..

The song is usually played with latin beat through the A section, and swing for the bridge (at the Dm7). Also, the bridge continues with a parallel section in the key of Eb (not shown).


Jan 092011

Equal Temperament

Pure Pitch vs. Equal Temperament

Equal temperament (ET) means the same distance between each of the 12 notes of the chromatic scale, as a ratio. Pure pitch is tuning slightly altered from ET. The reason it sounds more “pure” is that there is no degrading of the harmonic overtone series, as there is in ET; Although ET’s 4th, 5th, intervals are nearly identical to pure pitch; these intervals in ET do not degrade the harmonic series.

A Do-It-Yourself-er Finds a Problem

When I was young and foolish (now I’m old and foolish) I once tried to tune my piano by ear. I could tune any chord, say, an F, A, C, F to perfection and it would sound great…until I tried to play something. Most other chords sounded horrible. That same F chord on a properly tuned piano doesn’t sound quite as pure, yet we don’t notice.

An a cappella choir, an orchestra, or any configuration of instruments can sound pure chords in a piece of music that goes through many chords, changes key, and so on, as long as there is not a piano or fretted instrument in the ensemble. What about piano concertos? And other piano/instrument configurations?

The differences between pure pitch and ET are subtle. The other instruments can adjust pitch, and the “off-pitchness” of the piano is so slight that it could be looked at as different keys having different “colors”. Most people can’t notice a difference–this includes many musicians. The fretless and variable pitched instrumentalists use their ears to make subtle shifts (from a piano-pitch). For example, a Cb is flat from a piano B, assuming we are not in the key of E, F#, or B (perfect 4th/5ths, although for those keys, unless modulating, the note would be written as “B”, not “Cb”). Similarly, an A# is sharper than Bb. Some types of wind instruments, and single-reeded instruments can only approximate the actual pitch; the same situation as a piano.

Enter J.S. Bach

.Until the 18th century, the problem for music and composition was that until ET was discovered, a piece of music had a very limited harmonic pallet. The research, discovery, and gradual implementation,  must have facinated J.S. Bach. One of his major groundbreaking works is the “24 Preludes and Fugues for the Well-Tempered Clavier“. A series of pieces first in the key of C major, then C#major, then their corresponding minor keys. Twelve Major-key preludes and fugues, and twelve minor key preludes and fugues.

A Brief Introduction to the Math Behind It

There has been a lot of research into the history of ET. The formula needed to derive it, most likely discovered by trial and error, or applying the theories of the day to practice, was a major mathematical insight. The formula involves the 12th root of 2. Twelve, because there are 12 tones that span an octave, and two, becauuse the highest note is double the frequency of the lowest note. Also, we are concerned about the pitch-difference (measured in cycles per second) between 2 adjacent notes. The result is a decimal number, which is a multiplier constant applied to cycles/second.

The 12th root of 2 is algebraically equivalent to its inverse: [12√2 ➾ 2^(1/12) = 1.0594630943593]. The decimal number result is the multiplier. It can be thought of as a ratio,. Logarithmically, the distances between pitches are equal. We multiply, say, 440 * 1.0594630943593 and the result is 466.16 (rounded to 2 decimal places). This is the pitch for the next chromatic note up, in this case, Bb, or A#.

The table below shows the octave from A440–just above middle C–to  one octave higher, “A” at 880 cps. Incidentally, by tradition, A440 is the pitch the concertmaster plays for the orchestra to tune up to before they begin to play. The numbers in the 2nd column are the result of the adjacent note’s pitch times the ET multiplier, 1.0594630943593. For purpose of illustration, the result is rounded to 2 decimal places.

The practical implication of equal temperament is that each note is “compromised”, but just barely, for the best fit–and the fit works out for any piece of music no matter what key it may modulate to.¹

Equal Temperament

Note NameCycles per Second (Hz)

The Big Bangs of Music

Howard Goodall, a musicologist and composer, has an excellent DVD collection called “Howard Goodall’s Big Bangs”. His idea is that there were a number of historical events without which we wouldn’t have music as we know it today. These “Big Bangs” were originally aired on BBC TV.

The five “Big Bangs” were

  1. Notation and Sheet Music, so music could be distributed and assimilated across generations and geography;
  2. Equal Temperament tuning, so compositions of music could venture into many keys;
  3. Opera, which started to spread music to the masses, and large music halls were built;
  4. The invention of the Piano, which could handle all the elements of music, i.e., melody, harmony, and rhythm, and also began to bring music into the living rooms of the (initially, “upper”) middle class;
  5. The invention of recording–first wax, then tapes which could be played on the phonograph or tape deck. Recording technology made it possible to make records that could be mass produced from a master recording. About the same time, the microphone came to be in use, and that allowed for more control over balance to produce better ensemble recordings. An up-close-and-personal style of singing began to be developed, as opposed to the operatic type of singing, which necessarily requires volume and projection from the singer in order to be heard. Of course, before recordings, you could only hear music performed by live musicians.
You can watch a brief intro of Howard Goodall’s Big Bangs here:

Howard Goodall’s Big Bangs

Equal Temperament explained on WikiPedia

¹. In practice, when tuning a piano, the theoretical ET is not followed precisely. Called “scaling”, the notes are gradually stretched (the intervals widened) across the keyboard. A concert grand, because of it’s longer strings can be tuned in ET so that for example, a 3-octave unison pitch does not “beat”; whereas in smaller pianos, as the tuner moves higher up the keyboard the stretching increases. This is partly to compensate for the perception that higher notes sound flat. In the keyboard’s upper reaches, the ear can tolerate the “beating”, i.e., the off-pitchness found in the higher tones. The overtone series of the highest C, for example, soon get out of range for human hearing.

It’s just my opinion, but I believe an experienced tuner who tunes by ear will give the best results. This is due to the previous explanation, as well as the fact that every piano is different;  A machine can only measure empirical data, whereas the tuner can apply human perception.

Jan 032011

The Whole Tone Scale

The whole tone scale is a symmetrical scale. It’s called “whole tone” because it contains no 1/2 steps. You can start on any random note and play a scale up or down, always making the subsequent note two 1/2 steps (a whole step) away. The characteristic sound is hard to describe; however, you don’t get a sense of tension and resolution. (This is true of any symmetrical scale). You will hear it in movies and programmatic music. Thelonious Monk was one of the first major jazz pianists that seemed to be fascinated by it. Continue reading »

Dec 092010

Diminished Scales

Diminished scales are 8-note scales and they are symmetrical. There are only 2 (enharmonically) diminished scales: One  is based on a hale/whole step formula; the other based on a whole/half formula.

Eight note scales are easy to practice with a metronome. I have found that the symmetrical nature of chords or scales makes fingering easier on guitar or piano, possibly other instruments.

The half/whole type of diminished scale goes well over a dominant 7th chord; the whole/half type fits well when played over a diminished or half-diminished chord, e.g., a  2-5-1 in a minor key.

In the chart below, the 1st 4 bars are simply the C diminished scale (half/whole) ascending and descending. Continue reading »

Dec 092010

We derive our scales from looking at the 12 chromatic tones, picking a note to start, or name the scale, and follow the pattern of whole/half step intervals for the subsequent notes as described above. For clarity, we switch to the numbering system (and we assume more than one octave to play with), to derive these simple permutations, by picking the starting note as the “next” note.

1 – 8 / 2-9 / 3-10 / 4-11 / 5-12 / 6-13 / 7-14 / and (8-15), or

C to C’ / d to d’ / e to e’ / F – F’ / G – G’ / a – a’ / b – b’

In the sheet music below you’ll see this illustrated.

Continue reading »

Dec 082010

Major Scale

Although just about everyone knows or can recognize the major scale, I include it here to illustrate the whole-step/half-step sequence as a way of thinking about scales. Knowing the interval sequence of a scale will help you to understand any mode of the scale. Here is the whole/half step map (formula) to create the major scale: The first interval is the interval from whatever starting note you pick to the next note in the scale.

  • Whole / Whole / Half / Whole / Whole / Whole / Half.



One way to look at it is that the scale is derived simply by starting at the 6th degree of its relative major scale. Starting on any note, however, the Whole/Half step map below will produce this scale. The first interval is the interval from the starting note to the 2nd note:

  • Whole / Half / Whole / Whole / Half / Whole / Whole.

Continue reading »

Nov 282010

For rock and all of its offshoots you need not worry that much about voicing. Most of the time you are looking for a powerful sound, or at least to sound in character for the genre.

Classical guitar music is composed and written out. To play that correctly, the starting point is to first play the notes written correctly and in their prescribed position.

For jazz, however, The first thing to avoid, as a rule, are using the open chord forms that all beginner guitarists learn, such as E, A, D, G, and C, formed within the first three frets. Continue reading »

Nov 152010

One thing I discovered on my memorize-away-from-the-keyboard experiment is that I can look at a page of sheet music and hear it a lot better than I used to be able to. I am not to the level of looking at a score, atonal music, or music with a lot of accidentals and getting very far. Mozart and Beethoven (carefully chosen) are not too hard.

I’m starting with Beethoven’s variations in A on the song “Quant e piu bello l’amor contadino”. This is a fairly simple theme and the variations don’t look too formidable. I think I have the theme and 1st variation nailed. Next step is to try it blindfolded.