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Intervals: Arithmetic

Goal: To use arithmetic concepts in order to hear, recognize, and perform intervals.


Out of Tempo

Addition

Interval number

Beginning (Up)

  1. Teacher counts speaking, “1 to 2 = a second.”
  2. Students repeat.
  3. Teacher counts speaking, “1 to 3 = a third.”
  4. Students repeat.
  5. This process continues until “1 to 8 = an octave.”

Intermediate (Down)

  1. Teacher counts speaking, “8 to 7 = a second.”
  2. Students repeat.
  3. Teacher counts speaking, “8 to 6 a third.”
  4. Students repeat.
  5. This process continues until “8 to 1 = an octave.”

Advanced (Up and down)

  1. Teacher counts speaking, “1 to 2, 8 to 7 = a second.”
  2. Students repeat.
  3. Teacher counts speaking, “1 to 3, 8 to 6 = a third.”
  4. Students repeat.
  5. Process repeats until “1 to 8, 8 to 1 = an octave.”

Subtraction

Interval type: Major, minor, perfect, the tri-tone using half steps

Beginning (Subtracting 1 half step from an interval)

  1. Teacher speaks, “2 half steps – 1 half step = 1 half step = a minor second.”
  2. Students repeat.
  3. Teacher speaks, "3 half steps – 1 half step = 2 half steps = a major second.”
  4. Students repeat.
  5. Process repeats using the following intervals: minor third, major third, perfect fourth, augmented fourth, perfect fifth, minor sixth, major sixth, minor seventh, major seventh, perfect octave.

Intermediate (Subtracting 2 half steps from an interval)

  1. Teacher speaks, “3 half steps – 2 half steps = 1 half step = a minor second."
  2. Students repeat.
  3. Teacher speaks, “4 half steps – 2 half steps = 2 half steps = a major second."
  4. Students repeat.
  5. Process repeats using the following intervals: minor third, major third, perfect fourth, augmented fourth, perfect fifth, minor sixth, major sixth, minor seventh, major seventh, perfect octave.

Advanced (Subtracting three half steps from an interval)

Process repeats as above.

Multiplication

Teacher demonstrates, prompts, students repeat.

Beginning

  1. 1 x 1 = 1 = minor second.
  2. 1 x 2 = 2 = major second.
  3. 1 x 3 = 3 = minor third.

Intermediate

  1. 2 x 1 = 2 = major second.
  2. 2 x 2 = 4 = major third.
  3. 2 x 3 = 6 = augmented fourth.

Advanced

  1. 3 x 1 = 3 = minor third.
  2. 3 x 2 = 6 = tritone.
  3. 3 x 3 = 9 = major sixth.

Process repeats covering all intervals within the octave.

Division

Beginning

  1. Divide the octave in 6 = 2 = major second.
  2. Divide the octave in 12 = 1 = minor second.

Intermediate

  1. Divide the octave in 3 = 4 = major third.
  2. Divide the octave in 4 = 3 = minor third.

Advanced

  1. Divide the octave in 2 = 6 = tritone.

In Tempo

Sung using two-four, three-four, four-four, six-eight, or cut time, one note per measure.

Addition

Beginning: Upward (Major and minor key)

  1. Teacher sings "Doe Rey," then speaks “major second” in tempo (adding a measure in time).
  2. Student repeats.
  3. Teacher sings "Doe Mi," then speaks “major third” in tempo (adding a measure in time).
  4. Student repeats.
  5. Process continues until "Doe Doe" (upward).

Intermediate: Downward

  1. Teacher sings "Doe Ti," then speaks “minor second” in tempo (adding a measure in time).
  2. Student repeats.
  3. Teacher sings "Doe La," then speaks “minor third” in tempo (adding a measure of time).
  4. Student repeats.
  5. Process continues until "Doe Doe" (downward).

Advanced: Singing up and down, in tempo, adding extra beats to stay in time.

  1. Teacher sings "Doe Rey, Doe Ti."
  2. Student repeats.
  3. Teacher sings "Doe Mi, Doe La."
  4. Student repeats.
  5. Process continues until "Doe to Doe" (up) and "Doe to Doe" (down).
    1. Some students may not have the range to do this therefore starting on another solfege (key) is possible.

Subtraction

Using half steps

Beginning Upward (Shrinking an interval by half step)

  1. Teacher sings "Doe Rey (natural)," then "Doe Rey (flat)."
  2. Students repeat.
  3. Teacher sings "Doe Rey Mi (natural)," then "Doe Rey Mi (flat)."
  4. Students repeat.
  5. Process repeats for all intervals upward.

Intermediate Downward

  1. Teacher sings "Doe Ti (flat)," then "Doe Ti (natural)."
  2. Students repeat.
  3. Teacher sings "Doe La (flat)," then "Doe La (natural)."
  4. Students repeat.
  5. Process repeats for all intervals downward.

Advanced (Subtracting different numbers of half steps)

  1. Teacher sings "Doe Sole," a perfect fifth.
  2. Teacher prompts, “Subtract 3 half steps from Sol."
  3. Student sings "Doe Sole," then sings down "Sole (flat), Fa, Mi."
  4. Student then sings "Doe Mi."

Multiplication

Repetition in pairs of notes

Beginning

  1. Teacher sings, "Doe Rey."
  2. Teacher prompts, “Repeat four times.”
  3. Student repeats, "Doe Rey" four times.
  4. Process continues with other single pair of notes.

Intermediate

  1. Teacher sings, "Doe Rey," then "Rey Doe."
  2. Teacher prompts, “Repeat four times.”
  3. Student repeats, "Doe Rey Rey Doe" four times.
  4. Process continues with other single pair of notes.

Advanced

  1. Teacher sings, "Rey Doe Doe Rey".
  2. Teacher prompts, “Repeat four times.”
  3. Student repeats, "Rey Doe Doe Rey" four times.
  4. Process continues with other single pair of notes.

Division

(Even or uneven divisions of intervallic space)

Beginning (Seconds divided by half step)

  1. Teacher sings, "Doe Rey."
  2. Teacher sings, "Doe Doe (sharp), Doe (sharp) Rey," dividing the original "Doe Rey" in two half steps.
  3. Students repeat.

Intermediate

  1. Process repeats dividing any interval in to combinations of half steps:
    1. Example: Teacher prompts, “Divide a perfect fifth into two intervals by singing by half steps.”

Advanced (Dividing the octave ‘diatonically’ introducing counting inclusively, repeating a ‘starting note’ resulting in the octave = to 9 notes)

  1. Teacher sings, "Doe Doe" (an octave).
  2. Teacher sings, "Doe Rey Mi Fa Sol," then "Sol La Ti Doe." Then speaks, “The octave divides into a perfect fifth and a perfect fourth.”
  3. Students repeat.
  4. This process continues for all intervals.