A View on Vowels in Khuzdul
To find the vowels of Khuzdul, we have to mostly work backwards, from the orthography towards the phones. That's because there is so little information about Khuzdul vowels (much like everything about the language). The orthographic vowels that show up in the primary sources are < i î e ê a â o u û >. This only lacks < ô >, but that shows up in a note Tolkien wrote about the etymology of the word "Lhûn", where he postulated a Khuzdul word "salôn". This shows that < ô > is probably included, or at the very least that he considered it.
There are many variations on vowel pronunciation in Hebrew's history, and most of those show vowel length as being phonemic. Adunaic had short, long, and "over-long" vowels of <i a u>, similar to Arabic's 3 vowel qualities. Nothing is said about the exact pronunciation of Adunaic's vowels, so I think length is probably contrastive for Adunaic as well. As a result, I'm guessing this is also probably true for Khuzdul.
In Appendix E, Tolkien writes that the vowels should be "approximately" pronounced as "i, e, a, o, u in English machine, were, father, for, brute". This was mostly as a guide for Sindarin, since that is most of the names in LotR. I have to admit that I'm exactly sure what sound the "e" in English "were" is supposed to be, but I would guess / e /. Quenya apparently has / e /, so that seems to be the most probable. That matches Hebrew as well, so Khuzdul probably has this sound. Putting these sounds in a chart, we get the following so far:
Vowels in English Orthography:
Vowels in IPA:
There were also two vowel symbols added by the Dwarves of Moria to the Angerthas, #55 and #56. At the end of appendix E, Tolkien says these were "used for vowels like those heard in English "butter", which were frequent in Dwarvish and in the Westron." There are runes on the title page of LotR which uses #55 in the English word "the", for which the "e" sound is most commonly the schwa, /ə/. Additionally, in the tomb inscription of Balin, #56 is used in the English word "son", such that #56 is /ʌ/. The Angerthas letters #55 and #56 "were in origin a halved form of #46". This sounds very much like "reduced" or "schwa" vowels that are commonly found in Hebrew, and the actual sounds are in line with Hebrew's reduced vowels as well.
In Hebrew, these sounds are represented by different letters than non-reduced vowels, which is apparently the case with the Angerthas runes. The big question is how these two sounds are represented in English orthography. Again, as above, Tolkien says that they are "frequent in Dwarvish and in the Westron." However, we don't see any English characters in Khuzdul samples other than the ones shown above. We also don't see Angerthas #55 and #56 anywhere other than in English words. So, if they are so frequent, why don't we see any evidence of them?
The most likely solution I can determine is that, in some situations, the existing vowels seen in the Khuzdul corpus represent the sounds of English "butter". In that light, the only systematic method I can find in languages that may have influenced Khuzdul is seen in Yiddish. There, we find the vowel phoneme /ɛ/, which is like "e" in "bet". However, in unstressed syllables it is realized as /ə/. Modern Hebrew also has /ə/ which tends to be realized as /ɛ/. In the Tiberian Hebrew vocalization of Biblical Hebrew (which eventually became something of a standard), there were three reduced vowels, once of which was /ɛ̆/. Khuzdul could conceivably follow the model of Yiddish here, such that the sound written as [e] be pronounced as /e/, like "e" in "mesa", when stressed, and then as either /ɛ/ or /ə/ when unstressed. Alternatively, it might be /ɛ/ when stressed and /ə/ when unstressed.
Angerthas #56 is represented by "u" in English "butter". However, the sound /ʌ/ is phonetically closest to /a/, not /u/. The other two reduced vowels in Tiberian vocalizations, /ă/ and /ɔ̆/, are right on either side of /ʌ/. I could see this sound being written as either [a] or [o]. Because /ʌ/ is unrounded, like /ă/ and unlike /ɔ̆/, I think the best choice is to write it as [a].
The information regarding vowel vocalization throughout Hebrew's history is extremely complex. From what I can gather, it seems that certain vowels, when reduced to an unstressed "schwa" sound, become a specific reduced vowel. If Khuzdul does follow the Yiddish model of certain short vowels having different pronunciations when stressed or unstressed, then the most likely pairings that I can see are that [i î e ê] become [e] when reduced and [u û o ô a â] become [a] when reduced. Generally speaking, if Khuzdul follows the Biblical Hebrew model, it would usually be the long vowels that are reduced to schwas in the course of inflection.
Based on this analysis, I would present Khuzdul's vowels as seen in the charts below.
To represent /ə ʌ/, I decided to use < ë ä >. Using <ĕ ă> might be better, but I hate to use non-ASCII characters. As it turns out, the IPA symbol for a centralized vowel is the dierisis. This is actually a good description or approximation of schwa sounds. Additionally, Tolkien was exacting on how words looked, and used minimal special symbols, but he did use ë in some places. So, it seems that using <ë ä> is a pretty good fit.
With that said, <ë ä> would only be used when one is absolutely sure that a given vowel is a schwa sound. Since we can't be sure, that results in "pretty much never". In any event, Tolkien didn't use these letters in any Khuzdul example. If one follows the analysis above, the sounds /ə ʌ/ can be written as <e a> in all circumstances. This ends up making the English transcription of Khuzdul words follow very close to the Yiddish model. To indicate this in the chart below, I have added (e) and (a) in blue to note that in general practice, these sounds can be written and (mostly) thought of as allophones of /e a/.
Vowels in English Orthography:
Vowels in IPA: