I created a heavily modulated and filtered bass sound on F-1 but cannot play anything other than the F-1 itself and its # as the rest have too many conflicting frequencies.
How can these two keys be used without boring the listener?
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A few ideas:
The most difficult but most flexible approach would be to continue playing with the synth programming until the synth sounds in tune on more notes, or program more synths to have similar sounds on different notes.
Use pedal point. A bassline using pedal point constantly plays the same note, regardless of the changes in harmony. Done well, which is easy to pull off in my experience, it adds a lot of color and tension to harmonies.
Use simple harmonies in your song. I've heard a lot of electronic music that has very simple chord changes, sometimes no chord changes for minutes at a time. These songs rely on strong synth programming and compelling rhythms to be interesting.
Use the synth sparingly and use a different bass for most of the harmony. I've heard songs where a synth is only used at the end of phases to play a riff that emphasizes the resolution at the end of a chord progression.
Kevin provided advice for making the most of the limitation. I'll offer a couple ideas related to escaping the limitation, in case that's also part of what you're after.
If the unwanted aspects occur only on two notes, they're not inherent to the waveform, so if you have the means to sample one note, you can then reproduce it with the sound you prefer on many notes via the sampler.
As to why only a couple notes work, there's not enough information for me to know but it can be something like a ring mod based on a fixed frequency being applied, a sound component which isn't changing pitch for different notes, and the patch incorporating a sampled waveform which changes depending on the note range (I have a Kawai K-4, for example, which uses some sampled wave sources and on some you can hear where they stitched different samples together). If you can identify the cause, you may be able to change it.
Limitations are always good for the creative process.
If you are limited to F and F# (from now on I'll call it Gb) in the bass you could create an interesting modal piece by alternating between the chords of Fm7 and GbMaj7 and play over them in the F phrygian mode (F Gb Ab Bb C Db Eb F - the same notes as Db major or Bb minor).
Alternatively (or additionally) you could use chords F (F7 might feel like it wants to resolve to Bb and break the 'modal mood') and GbMaj7 and use the F Phrygian #3 scale (F Gb A Bb C Db Eb F - the same notes as Bb harmonic Minor). This scale has quite an 'exotic' sound to western ears because of the augmented 2nd between Gb and A (3 semitones).
Last but not least, and even more exotic sounding, use Fmaj7 and Gbmaj7 in conjunction with the F Double Harmonic Scale (F Gb A Bb C Db E F).
You can drop the 7th from any of these chords or add the 9th, 11th or 13th just as it sounds good to you.
The Phrygian mode can sound quite Spanish / flamenco. The Phrygian #3 and Double Harmonic scales can be very reminiscent of East European gypsy scales, Yiddish, North African or even Indian music, so you could go for a World Beat sound.
As long as there is plenty of rhythmic interest then by mixing and matching the above scales and modes to create sections you should be able to create an interesting piece.