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Scales: Arithmetic

Goal: To hear, sing, and play church modes, octatonic scales, and twelve-tone rows.

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, three, or four notes per measure.

Speaking, singing, playing.

Refer to the Scale Smorgasbord for ideas.

Addition

Beginning

  1. Teacher sets up as above and prompts, “I am singing or playing church mode x. Student A sings up the first four notes. Student B adds the second four notes. Then both sing down the scale together.”
  2. Both students do so.

Intermediate

  1. Teacher sets up as above and prompts, “I am singing or playing octatonic scale x. Student C sings up the first five notes. Student D adds the next group of four notes. Then both sing down the scale together.”
  2. Both students do so.

Advanced

  1. Teacher sets up as above and prompts, “I am singing or playing a twelve note row. Student E sings the first four notes, student F sings the next four notes, and student G sings the final four notes.”
  2. Both students do so.

Subtraction

Beginning

  1. Teacher sets up as above and prompts, “Here are all the notes of church mode x. Remove one from the scale and identify which scale degree it is that you are removing.”
  2. Students do so.

Intermediate

  1. Teacher sets up as above and prompts, “Here are all the notes of octatonic scale. Remove two non-adjacent notes from the scale and identify which scale degrees you are removing.”
  2. Students do so

Advanced

  1. Teacher sets up as above and prompts, “Here are all the notes of a twelve tone row. Remove three non-adjacent notes from the row and identify which numerical pitch you are removing.”
  2. Students do so.

Multiplication

Beginning

  1. Teacher sets up as above and prompts, “Using Church mode x, sing two notes at a time in non-adjacent leap pairs.”
  2. Students do so.

Intermediate

  1. Teacher sets up as above and prompts, “Using Octatonic Scale x, sing three note combinations in non-adjacent leap combinations.”
  2. Students do so.

Advanced

  1. Teacher sets up as above and prompts, “Using all twelve pitches, sing four note combinations in mostly non-adjacent leap combinations.”
  2. Students do so.

Division

Beginning

  1. Teacher sets up as above, pulsing an octave and prompts, “Here is octave x. Student A sings all the notes of church mode x in non-adjacent combinations.”
  2. Student A does so.

Intermediate

  1. Teacher sets up as above, pulsing an octave and prompts, “Here is octave x. Student B sings all the notes of octatonic scale x in non-adjacent combinations.”
  2. Student B does so

Advanced

  1. Teacher sets up as above, pulsing an octave and prompts, “Here is octave x. Student C sings all twelve notes of inside the octave in mostly non-adjacent combinations.”
  2. Student C does so.