I'm curious about the comparative reliance on tonality in Asian languages. By this I mean not the number of tones, but the frequency of tonal versus non-tonal words in communication. (When I say non-tonal here, I mean words that are neutral tone and have no other tonal homophones) Among large tonal languages, Vietnamese sounds like the most extremely reliant on tonality, and Thai sounds the least. I imagine Vietnamese has much more words that would be indistinguishable if it wasn't for tonality than Thai. Based on this, as a speaker of only non-tonal languages, I believe Vietnamese would be by far the most difficult for me to learn. Roughly speaking, how often do tonal words appear in Thai? Would you say every 5th word is tonal, or every 20th word, or every other word? Thanks
This may not be the type of answer you were hoping for.
You are correct in asserting that most tone languages can be analyzed from a phonological perspective as including words or syllables that lack a specification for tone. Some examples:
Mandarin can be analyzed as having "neutral" tone specified on things like the sentence-final particle -le.
Many African languages can be analyzed as having single-tone systems, with select syllables being marked for high tone and the rest being unmarked.
"Standard" Japanese, which is usually analyzed as having a lexical accent system (a subclass of tone systems), includes words in which one of the syllables is marked for a lexical pitch accent and the rest are not. Some words are analyzed as completely "unaccented" (i.e., none of the syllables in them are marked for accent).
However, in all such cases, there are "rules" that govern what pitch contours are given to the words and syllables that are analyzed as tonally "unmarked". They are analyzed by phonologists as "unmarked" only because their pitch contours are predictable. In other words, a language learner need not memorize any inherent tone shape associated with them, but the learner does need to be aware of the "rules" that govern their pronunciation, which is invariably dependent on the inherent tonal features of neighboring syllables that are marked for tone.
For example, in Mandarin, the fact that the particle -le is unmarked for tone doesn't mean that you can get away with pronouncing it with whatever pitch contour you feel like. What pitch contour is appropriate for the particle depends entirely on the tone of the syllable before it in the sentence. If that penultimate syllable is high or falling, the particle must be lower relative to the penultimate syllable. If the penultimate syllable is low or rising, the particle must be higher.
In African languages that are analyzed as marking select syllables as "high", usually all the rest of the syllables are pronounced with pitches that are lower compared to those that are tonally marked. Those other syllables could often just as easily be analyzed as being marked as "low", but to do so would be redundant.
In Japanese, the traditional analysis of its tonal system assumes that accented words include a syllable that is marked for a locus of a tonal shift from high to low. Syllables leading up to the marked syllable are given a relatively high pitch, and syllables following the marked syllable are given a relatively low pitch (it's a bit more complicated than that, but that's the basic system). In words that lack an accent-marked syllable, the pitch gradually rises through the end of the word.
I raise all these examples to illustrate the fact that getting the pitch of "unmarked" elements right is just as important as getting the pitch of marked elements right in tone languages. If anything, it can be even trickier for a non-native speaker to learn how to pronounce such unmarked elements. I've witnessed this firsthand watching American college students attempting to learn how to pronounce sentence-final particles in Mandarin--even after getting the basic four tones down, they often struggled with that elusive "neutral" tone.
So I know I didn't directly answer your question in terms of helping you find a tone language with a relatively large number of tonally "unmarked" words, but my point is that you may not want to use that criterion to predict which tone languages will be easier for you to learn.
EDIT in response to comments under the original question:
After reading some of the comments above, I think it's important to point out two things.
One--in the languages you mention, tone is contrastive at the level of the syllable, and while most "words" (in the morphological sense) are monosyllabic, many vocabulary items that you as an English speaker would consider to be "words" are compounds comprising multiple syllables. In these polysyllabic compounds, the tone of each syllable must be taken into consideration. I am not an expert in Thai or Vietnamese, and I know next to nothing about Burmese, but in Mandarin, there are almost no vocabulary items that lack tonal specifications. Either they are monosyllabic and the tone on that single syllable is specified, or they are compounds and just a subset (usually just one) of the syllables in the compound gets "stripped" of its underlying tone. The exception to this is monosyllabic particles, but they can't occur in isolation either. So the likelihood of encountering a standalone vocabulary item that is entirely unspecified for tone in Mandarin is pretty much nil. My limited understanding of Thai is that every syllable bears a tonal specification. In Thai orthography, which marks tones, some syllables don't bear a tonal marking, but (as in the examples above) there are rules that govern what tone such unmarked syllables bear. I found this Thai-language-learning website that lays out those rules.
Two--just because a word happens not to have any alternate-toned near-homophones (what linguists call minimal pairs), it does not mean that it's not important to be aware of its tonal specifications. There's not a perfect analogy in English, but think about a word like beige. I can't think of any other monosyllabic words in English that start with [b] and end with [ʒ]. But that doesn't mean that it doesn't matter how I pronounce the vowel in that word. A learner of English would have to learn how to pronounce the word with the correct vowel.