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HomeMusic Theory: Scales: Scale Formulas

"Scale Formulas"
Scale Math for to Build Your Knowledge

This page is a quick reference to build scales with the math formulas. 

Types of Scales

There are several types of music scales they are listed here with basic definitions. 
  • Chromatic Scale

  • Major Scale
  • Minor Scale (harmonic, natural or relative, and melodic)

  • Diminished and Polytonal Scale

  • Whole tone and Pentatonic
  • Blues and Gospel Scale

  • There are others but that's for advanced theory we will wait to explore those.

Music scales are the structures that make up which notes are used and how chords are constructed within all the key signatures. 

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Constructing Music Theory Scales with Scale Formulas

We use the 12 notes with their corresponding sharps and flats to build a music scale.

The music theory scales are developed mathematically using whole, half and minor third steps.

If we assign a numeric value to each note, this provides an interval to each note. with in the math pattern producing the scale formulas.

Notation for exploring scales.

W = whole step: such as C to D
H = Half Step: as in C to D#
m3 = minor 3rd which is a whole and a half step together: C to Eb

We will use the S to indicate the 'Start' position or tonic root, then add the steps to construct a scale.

Keyboard Notes

Chromatic Scale:

The Chromatic scale is produced using all half steps and uses all 12 notes. This is the most basic of all scales. Used to make a type of fill in special applications to move from one point to another.

S -  H - H -  H - H - H -  H - H -  H - H -  H - H - H
C - C# - D - D# - E - F - F# - G - G# - A - A# - B - C

Scale Formulas

Each is given in Interval steps and notes for the scale primarily starting on C.

Major Scale:

Mathematically the major scale is produced using the following formula.

1 - 2 - 3 - 4 - 5 - 6 - 7 - 8
S- W - W - H - W - W - W - H  

C - D - E - F - G - A - B - C

Harmonic Minor Scale:

Mathematically the Harmonic minor scale is produced using the following formula. (flat the 3rd and 6th) Notice the 3 half steps (m3) in the 6th to 7th position.

1 - 2 - m3 - 4 - 5 - m6 - 7 - 8
S- W -  H - W - W - H - m3 - H  
C - D - Eb - F - G - Ab - B - C

Natural or Relative Minor Scale:

Mathematically the Natural minor scale is produced using the following formula. (flat the 3rd, 6th, and 7th) the A natural minor is shown here also, it has no sharps or flats the same as the C major scale. 

1 - 2 - m3 - 4 - 5 - m6 - m7 - 8
S- W - H  - W - W - H - W  - W  
C - D - Eb - F - G - Ab - Bb - C
A natural minor
A - B - C - D - E - F - G - A

Melodic Minor Scale:

Mathematically the melodic minor scale is produced using the following formula. (flat only the 3rd )

1 - 2 - m3 - 4 - 5 - 6 - 7 - 8
S- W - H - W - W - W - W - H 
C - D - Eb - F - G - A - B - C

Diminished Scale:

Mathematically the diminished scale is produced using the following formula of  whole - half combinations. Notice we have 9 notes to complete the sequence instead of 8. This music scale is used in jazz a lot.

1 - 2 - m3 - 4 - d5 - +5 - 6 - 7 - 8
S- W - H  - W - H -  W  - H - W - H
C - D - Eb - F - Gb - G# - A - B - C

Polytonal Scale:

Mathematically the Polytonal scale is an inversion of the diminished scale.  Using the the half - whole repeating pattern. 1, b3, 4, b5, 5 b7

1 - m2 - m3 - b4 - #4 - 5 - 6 - m7 - 8
S- H - W - H  - W  -  H - W -  H -  W
C - Db - Eb - Fb - F# - G - A - Bb - C

Pentatonic Scale:

Mathematically the Pentatonic scale is produced with 5 notes using the following formula. Using the 1, 2, 3, 5, 6 

1 - 2 - 3 - 5 - 6 -  8
S- W - W - m3 - W - m3
C - D - E - G - A -  C

Blues Scale:

Mathematically the Blues scale is produced with 6 notes using the following formula. Using the 1, b3, 4, b5, 5 b7

1 - m3 - 4 - d5 - 5 - m7 -  8
S- m3 - W - H  - H - m3 - W
C - Eb - F - Gb - G - Bb - C

Gospel Scale:

Mathematically the Gospel scale is produced with 6 notes using the following scale formula. Using the 1, 2, b3, b4, 5, 6

1 - 2 - m3 - d4 - 5 - 6 -  8
S - W - H  - H - m3 - W - m3
C - D - Eb - E - G  - A - C 

Keyboard Notes

So there is just a quick review to math of scale formulas. Go to the lessons to explore further the scale development and use.

Learning Resources

Scales Workshop

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Learning scales is intimidating for the beginning student.

Don't let them get the best of you.

This Getting It Down Cold Key and Scale Workshop easily leads you through the process step by step.

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Lessons in Scales:

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Next: Chords



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