The Honkyoku pieces (solo traditional pieces written by monks) have no fixed rhythm. The rhythm is based on the length of the breath, the amount of notes you have to play in one breath and how you use your breath. The tradition transmitted from a master gives  indications how to place the tones within a breath. It can vary from a player to the other, but also from one day to the next.
How about silence?

In Western music, notes and rests are measured with beats and bars, and you learn to count silences. When you play in an orchestra, you often have to count a lot of bars of rests, especially if you are a brass, percussionist or harp player. Silence means being quiet. Stop playing. It gives shape to the music, but also is understood as the end of the music (when the audience starts to applaud). Breath shouldn’t be heard. For wind players, taking a breath should be as quick and discrete as possible, it’s not part of the music.

In a shakuhachi honkyoku, breath and silence are both part of the music. In the silence disappears the tone and the next one is being prepared. It isn’t measured either, but this doesn’t mean all silences should have the same length, or that it doesn’t count. Silence is definitively something to pay attention to. It contains a secret rhythm. Its quality is as important as the quality of the tones. Silence is the connection between two tones or two musical phrases. It is the “ma” (間) of the music. If the concentration drops during the silence, it creates a break in the piece. Silence is not empty, useless or just convenient to take a quick breath. Silence contains all sounds. The quality of your next phrase will depends largely on its preparation during the silence before, as the quality of your tone is influenced by the quality of your inhalation.
Listen to your silences: are they long, short, always the same? What do they contain? Are they part of your music?

” Silence is eternal. It exists before there is sound. It is just the pristine ground of sound. It is the empty base of existence. With the mastery of breath, each moment is spontaneously reacting to the conditions. With mind concentrating on breath, every thought is a breathing action. Based on non-duality of mind and breath, every thought will realize the sound of unity. When the breath is exhausted, all notion and conceptuality is cleared, one moves from sound to silence and the mind itself is experienced. ”

Dr. WONG Wah-Sang in “Blowing Zen on the Shakuhachi”

Giving full attention to the silence during your practice is a way to meditate with your shakuhachi.