1. Five notes against two
Purpose: learn a reusable way to approach unfamiliar ratios. Preparation: understand a shared span and play 2:3 and 3:4 without losing the start of the next group. These studies are optional preparation for music that needs them, not a daily endurance requirement.
The RH quintuplet contains five equal eighth notes in the time of four ordinary eighths: one complete 2/4 bar. The LH plays two quarters over the same span.
Locate the second LH attack
Use ten equal small divisions. RH attacks occur at 1, 3, 5, 7, 9; LH attacks at 1 and 6. The second LH note therefore falls halfway between the third and fourth RH notes. It does not coincide with the middle note of the quintuplet.
Tap each stream separately, then combine one group on repeated pitches. Restore the five-note line only after the halfway point remains clear. The descending second bar has exactly the same timing as the ascending first bar.
Listening test: mute the LH mentally and hear five equal notes; then attend to the LH and hear an ordinary two-beat pulse. If the fourth RH note arrives early to join the LH, return to divisions 5, 6, and 7.
The suggested five-finger position is only a way to remove pitch difficulty. When a repertoire figure needs a crossing or a leap, practice that movement separately before asking it to carry a new rhythm.
2. Five notes against three
Both groups still fill one 2/4 bar. The LH quarter-note triplet now divides that span into three equal parts. Its notes are neither ordinary quarters nor an extra beat added to the bar.
Use fifteen divisions once, then hear the groups
With fifteen equal divisions, RH attacks fall at 1, 4, 7, 10, 13, and LH attacks at 1, 6, 11. Only the opening attacks coincide inside this bar; the next shared attack is the following downbeat.
The second LH attack is between RH notes 2 and 3, closer to note 3. The third LH attack is between RH notes 4 and 5, closer to note 4. These are useful orientation points, but neither gap should be approximated as an equal subdivision of a RH note.
- Speak or tap five equal notes across one span, then three across that same span.
- Combine repeated notes at a slow tempo, using the fifteen-part grid to place uncertain attacks.
- Play the printed pitches and connect both bars. Keep the quintuplet even on its return.
- Alternate attention between the two streams. Reduce the small count when each can be heard clearly.
Repair: if the quintuplet starts sounding like a triplet followed by a pair, stop and play the five notes alone. All five have equal duration. Any musical grouping must preserve that equality.
Variation: exchange the rhythmic roles on repeated notes. Do not transfer the descending line until the new hand assignment is secure.
3. A method for larger ratios
A common grid is a temporary measuring tool. Use the least common multiple of the two group sizes to find exact attack locations, then return to hearing the larger gestures. The table numbers divisions from 1; the next common arrival is one division beyond the table’s total.
| Ratio across one shared span | Equal divisions | First stream attacks | Second stream attacks |
|---|---|---|---|
| 4 against 5 | 20 | 1, 6, 11, 16 | 1, 5, 9, 13, 17 |
| 5 against 6 | 30 | 1, 7, 13, 19, 25 | 1, 6, 11, 16, 21, 26 |
For 4:5, neither middle pair quite coincides. For 5:6, the attacks repeatedly pass close to one another. Do not force these neighboring attacks together merely because that is easier to coordinate.
Choose the amount of precision the notation asks for
First identify whether the composer has specified equal tuplets, a cadenza-like flourish, or a freer ornamental group. Those are different musical situations. An exact ratio drill is appropriate for explicitly even simultaneous groups; it should not make a flexible ornamental passage mechanically rigid.
For a written 4:5 or 5:6 group, mark the shared endpoints, practice each line’s pitches separately, and try repeated-note tapping before restoring the texture. A slow sequencer playback can help you hear the alignment, provided its tuplets are entered correctly. Compare the individual streams as well as the combined sound.
Decide whether to practice it now
Work on a larger ratio when it solves a real passage or develops a clearly identified coordination skill. If you cannot yet maintain each line independently, return to 2:3 or 3:4. Difficulty alone does not make an exercise useful.
Completion check: explain the shared span, locate every true coincidence, and play several groups without lengthening the boundary. Fluent musical shaping and source-specific repertoire examples will need further work beyond these introductory studies.