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Intervals: Arithmetic

Goal: To use arithmetic in order to focus on (but not limited to) larger intervals of sixths, sevenths, and octaves using major, minor, whole tone, and pentatonic scales.

Out of Tempo

All Levels

“Out of Tempo” exercises are not presented separately in Ear Training III and Ear Training IV improvisations. Should any exercise prove difficult performed in tempo, out of tempo versions of the materials in Ear Training III and Ear Training IV can be modeled on Ear Training I and Ear Training II approaches to out of tempo improvisations.

In Tempo

Teacher sets up a tempo, time signature, tonality, and rhythms.

Phrases can be four to thirty-two measures in length depending on the student’s ability.

Start with one note per measure and add different rhythms as students become more comfortable with the exercise. For more confident students, use non-repeating notes and rhythm patterns.

Students sing one, two or three notes per measure.

Speaking, singing, playing.

Refer to the Intervals Smorgasbord for ideas.

Addition

Beginning

  1. Teacher sets up as above and prompts, “Using scale x, sing the available sixths within the octave upward and downward.”
  2. Students do so.

Intermediate

  1. Teacher sets up as above and prompts, “Using scale x, sing the available sevenths within the octave upward and downward.”
  2. Students do so.

Advanced

  1. Teacher sets up as above and prompts, “Using scale x, sing octaves upward and downward.”
  2. Students do so.

Subtraction

Beginning

  1. Teacher sets up as above and prompts, “Using scale x, sing an octave from bottom up and then sing the next largest interval from bottom up.” For example, using the pentatonic scale, sing "Doe" to "Doe," then "Doe" to "La" because there is no "Ti."
  2. Students do so.

Intermediate

  1. Teacher sets up as above and prompts, “Using scale x, sing an octave from top down and then sing the next largest interval from top down.” For example, using the pentatonic scale, sing "Doe" to "Doe" down, then "Doe" to "Rey."
  2. Students do so.

Advanced

  1. Teacher sets up as above and prompts, “Using scale x, sing an octave from bottom up or top down, starting on a note other than 'Doe,' or the first scale degree. Then sing the next largest interval from bottom up or top down.” For example, using the pentatonic scale, sing "Rey" to "Rey", then "Rey" up to "Doe" or "Rey" down to "Fa."
  2. Students do so.

Multiplication

Repetition.

Beginning

  1. Teacher sets up as above and prompts, “Using scale x, locate sixths within the octave and sing each one four times up or down – one at a time.”
  2. Students do so.

Intermediate

  1. Teacher sets up as above and prompts, “Using scale x, locate sevenths within the octave and sing each one four times up or down - one at a time.”
  2. Students do so.

Advanced

  1. Teacher sets up as above and prompts, “Using scale x, locate 6ths, 7ths, and octaves, and sing up or down each one four times before moving on to another sixth, seventh, or octave.”
  2. Students do so.

Division

Beginning

  1. Teacher sets up as above and prompts, “Using the whole tone or pentatonic scale, sing an octave up from the first scale degree.”
  2. Students do so.
  3. Continue the process until all octaves are sung, vocal range permitting.

Intermediate

  1. Teacher sets up as above and prompts, “Using the whole tone or pentatonic scale, and divide the octave into two parts. Choose the interval that is close to the middle of the octave.” For example, using the whole tone scale, sing C up to F#, then up to C.
  2. Students do so.

Advanced

  1. Teacher sets up as above and prompts, “Using the whole tone or pentatonic scale, and divide the octave into three parts. Choose the intervals that are close dividing the octave into three equal parts.” For example, using the whole tone scale, sing up from C to E to G# to C or down from C to G# to E to C.
  2. Students do so.
  3. The process continues starting on a scale degree other than scale degree number 1. In this way, students experience dividing the octave in different intervals.