## HALF STEPS, WHOLE STEPS and SCALE FORMULAS

If you arrived here because some of this "music theory" interests you, you might want to explore some of the music theory software that we have at school. Let me know and I will get you started. --MsM

According to the Harvard Dictionary of Music*, a half step (or semitone) is "one-half of a whole tone, the smallest interval in traditional Western music. ... The octave consists of twelve semitones and the diatonic scale includes two semitones." (The major scale and the natural minor are diatonic scales.) Click here for some scientific information about half-steps.

• Look at the diagram above and notice how the sharps and flats are laid out. A sharp is defined as the note that is one half step higher then the note you are starting on. A half step on the piano is the very next key. So, C# is the very next key to the right after the C and it happens to be a black key.
• Logically enough, flats are defined as the note that is one half step lower than the note you are starting on. So, Db is the next key to the left of D.
• Did you notice that C# and Db are the same key??? All notes have more than one name. The formal word for this concept is enharmonic tones. C# and Db are enharmonic tones (on the piano they sound the same, but have different names -- equal temperament...).
• Where is E#??? Where is Fb??? What is a double sharp??? What is a double flat??? If these questions interest you, click here.

## What is an Interval Formula?

Scales are built on interval formulas. Diatonic scales use only half steps and whole steps. When you get into the harmonic minor, melodic minor, blues scales & some of the "ethnic scales" like the Ahava Raba mode used in Klezmer music, then you run into scales that have whole steps, half steps, a step and a half, and more.

The first note of the scale is called its root:

• the root of the C major scale is a C
• the root of an Eb major scale is an Eb
• the root of a G minor scale is a G

Using white keys only, a major scale starting on C has no sharps and no flats. Look at the keyboard and see that this scale follows the whole step / half step pattern of

root, whole step, whole step, half step, whole step, whole step, whole step, half step
C    D        E         F       G        A        B        C

Major scale formula: R, W, W, H, W, W, W, H

Now, if you will use the keyboard and make a major scale starting on D, you will find that in order to key the whole step, half step pattern, you need to add an F# and a C#. Look at a key signature chart and you will see that is correct! This is how key signatures are arrived at - it really isn't just random! Try figuring out a few major scales using the keyboard, then check the key signature chart to see if you were correct.

Try these:

• E major
• Bb major
• Cb major
• F# major
• Gb major (notice F# & Gb have the same notes, but very different key signatures!)

### Scale Formulas

• Major Scale: R, W, W, H, W, W, W, H
• Natural Minor Scale: R, W, H, W, W, H, W, W
• Harmonic Minor Scale: R, W, H, W, W, H, 1 1/2, H   (notice the step and a half)
• Melodic Minor Scale: going up is: R, W, H, W, W, W, W, H
going down is: R, W, W, H, W, W, H, W
• Dorian Mode is: R, W, H, W, W, W, H, W
• Mixolydian Mode is: R, W, W, H, W, W, H, W
• Ahava Raba Mode is: R, H, 1 1/2, H, W, H, W, W
• A minor pentatonic blues scale (no sharped 5) is: R, 1 1/2, W, W, 1 1/2, W

### Diatonic Scale

A diatonic scale is one that has no added sharps or flats other than what is in the key signature -- the scale's interval pattern could be played using only the white keys on the piano. Diatonic scales are the major scales, natural minor scales (not the harmonic or melodic, because they add sharps!), dorian, mixolydian and all of the other so-called "Church Modes".       return to interval formulas

## Science

Again, quoting from the Harvard Dictionary of Music, "The exact measurement of a semitone (halfstep) varies slightly according to the system of tuning. In equal temperament, each semitone equals exactly 100 cents." --- Eeeek!! What are some of these words???

• "Cents" is the measurement unit devised to measure musical intervals.
• "Equal temperament" is the way a piano is tuned. Every half step is 100 cents, no more, no less.
• "Mean temperament" and the "Pythagorean system of tuning" both have unequal intervals. Some semitones are as small as 90 cents and others as large as 114 cents.
• Our ears do not hear in Equal Temperament; our ears want to hear the relationships between the notes to be different depending on how the note fits into the scale. Example 1: when a wind player or string player plays a melody, the note fingered as a D# or and Eb will actually be slightly different depending on whether it is a D# in the scale, or an Eb. Example 2: an F will be tuned slightly differently depending on whether it is the 3rd of the scale or the 5th of the scale. In an advanced player, the ear takes care of this discrepancy and the player doesn't really have to think about it.

So why have equal temperament?? Early in the history of keyboard instruments the player had to retune the harpsichord for each key -- in the key of C, an E was the third, but if we switched to the key of D, the E is now the second and needed to be tuned differently. Equal temperament was invented as a means of averaging out the difference and making everything just a little out of tune and a little in tune. Our ear gets used to this tuning and it works fine.

* Apel, Willi. Harvard Dictionary of Music, second edition. Cambridge, MA: Belknap Press of Harvard University Press, 1972. page 762.

## Key Signature Chart

Click here to go to a page with a key signature chart that will print out quickly.

Visit the Wayland Public Schools site http://www.wayland.k12.ma.us

Go to WMS Band introduction page

##### High School pages Local Concerts Grading Policies Private teachers
Wayland Middle School Bands, Wayland, MA muffitt@bandnotes.info