The language consists of 3 different tones: high, medium, and low. There are trills (~), chords(=), neutral tones (-), rising tones (<), falling tones (>), and rising/falling tones (^). Would the combinations of the whistles make words, or would the collection of tones limit it to a 6 letter alphabet?
Real-world "whistled languages" are generally adaptations of spoken languages, taking some aspect of the phonology (tone, prosody, sometimes formants) and conveying it through whistling. A lot of information is lost, but there's enough redundancy that native speakers can still understand—much like how English-speakers can understand whispering, which discards most information about voicing, or styles of singing that discard some formant information.
It sounds like that's not what you're looking for, though. The rest of the answer is built on the assumption that you're trying to make a language which is only whistled, rather than using whistling as a lossy way of conveying a spoken language.
Languages with very few phonemes exist in real life, so an inventory like that is plausible enough. If you want to figure out what the words would look like, I think this is a place where some basic information theory could be useful.
The core idea behind information theory is that you can consider a lot of different things (including a spoken language) as a series of discrete signals, transmitted one at a time from a source to a receiver, and then use some mathematical structures to analyze that. If you've got three tones and six contours, and each phoneme consists of exactly one tone and one contour, then you've got eighteen possible signals. Not knowing anything else about your language, this means you can transmit a bit over two bits of information per phoneme. Each English word generally conveys about eight bits of information (given context and such)—so completely ignoring phonotactics, syntax, and various other aspects of your language, you could theoretically encode the same amount of information as an English word with four phonemes.
Once you add those aspects in, the information load per phoneme will probably decrease somewhat (since all the signals won't be equally probable in all contexts), so I'd expect it to take about six of your tone-plus-contour phonemes to convey the equivalent of one English word. That sounds like a plausible word size to me.