Borrowed Divisions: Arithmetic
Goal: To prepare combinations of differing borrowed subdivisions of the beat (triplets in simple meter, duplets in compound meter).
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 Rhythm Patterns/Borrowed Divisions Smorgasbord for ideas.
Addition
Can be sung as solos or harmonized in two or three parts. Can also be overlapped where one student sings eighths then triplets at the same time another student sings triplets then eighths.
Beginning
- Teacher sets up as above and prompts, “In key x, using scale y, in two-four, add one beat of triplets to one beat of eighth notes.”
- Students do so.
Intermediate
- Teacher sets up as above and prompts, “In key x, using scale y, in three-four, add two beats of triplets to one beat of eighth notes.”
- Students do so.
Advanced
- Teacher sets up as above and prompts, “In key x, using scale y, in four-four, add two beats of triplets on beats two and three to one beat of eighth notes on beats one and four.”
- Students do so.
Subtraction
Can be sung as solos or harmonized in two or three parts. Can also be overlapped where one student sings eighths then triplets at the same time another student sings triplets then eighths and/or dotted quarter notes.
Beginning
- Teacher sets up as above and prompts, “In key x, using scale y, in six-eight, replace one beat of dotted quarter notes with one beat of duplets.”
- Students do so.
Intermediate
- Teacher sets up as above and prompts, “In key x, using scale y, in nine-eight, remove two beats of dotted quarter notes and replace with two beats of duplets.”
- Students do so.
Advanced
- Teacher sets up as above and prompts, “In key x, using scale y, in twelve-eight, replace three beats of dotted quarter notes three beats of duplets.”
- Students do so.
Multiplication
Can be sung as solos or in two or three parts.
Beginning
- Teacher sets up as above and prompts, “Using scale x and repeating solfege, sing four measures of triplets in two-four.”
- Students do so.
Intermediate
- Teacher sets up as above and prompts, “Using scale x and repeating solfege, sing four measures of triplets in three-four.”
- Students do so.
Advanced
- Teacher sets up as above and prompts, “Using scale x and repeating solfege, sing four measures of alternating eighth notes and triplets within the measure in four-four.”
- Students do so.
Division
Turning notes into rests.
Beginning
- Teacher sets up as above and prompts, “Using scale x, in key y and in two-four, sing four measures of triplets.”
- Students do so.
- Teacher prompts, “Sing the same thing and turn the third triplet per beat into a rest."
- Students do so.
Intermediate
- Teacher sets up as above and prompts, “Using scale x, in key y and in three-four, sing four measures of triplets.”
- Students do so.
- Teacher prompts, “Sing the same thing and turn the second triplet per beat into a rest."
- Students do so.
Advanced
- Teacher sets up as above and prompts, “Using scale x, in key y and in four-four, sing four eighth notes and six triplets.”
- Students do so.
- Teacher prompts, “Sing the same thing and turn the second and fourth eighth notes and the third and sixth triplets into rests."
- Students do so.
More Advanced
Continue the same process turning other eighths and triplets in to rests.