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TL;DR - I want my angelic race to have the ability to speak in a way in which up to three sentences are communicated at the same time. (Assume English for the sake of simplicity. I want physiology answers, not conlang answers, as evidenced by the question on whole and my tags.)

In my story, angels have the ability to express 3 sentences at the same time. If I were to write this out, it'd be a matter of [Main sentence], [Supertext], and [Subtext] all being in line with one another (like furigana in Japanese in terms of writing, but I know that's not how it works when spoken). These three sentences would be restricted to the same topic, but might be along different expressed ideas with the Main Sentence being the critical information and the Supertext/Subtext adding details to the sentence. For example:

The dog has brown fur.

The dog ran away.

The dog was barking.

In English, we'd simply say "The brown dog ran away while barking."

(And because I know somebody would ask "Why not just use English then?") Alternatively, using a more "high school" example:

Did you see what she was wearing?!

Jenny is such a b-.

She's totally sleeping with Mr. Caulton.

In this example, the person would be expressing three sentences all about how much they dislike "Jenny", but the three thoughts aren't all describing Jenny directly. Aside from making a paragraph saying these things (or a LONG compound sentence), English doesn't really have a way of properly expressing this. In the angelic tongue, this would be nicely condensed, though.

Now, I know asking for details on writing the language would more be a matter for the Conlang SE (maybe), but what I am curious about is this: What would be necessary for not only an angel to say these three sentences at once, but for somebody to be capable of understanding them clearly?

MAKING IT A SINGLE QUESTION: What would be the necessary requirements for this kind of speech to be communicated effectively?

I'll accept whichever answer is the simplest, but most effective. My current process involves manipulation of polyphonic vocalization, but I feel like there has to be a better way. I want my angels to still pass for being aesthetically human. They can't be distinctly alien in appearance to make this work. It needs to be up to 3 separate sentences spoken at the same time, but needs to also allow for only one or only two sentences as well. (Meaning, just because they can speak three sentences at once, that doesn't mean they are physically required to. It should be something they can control.)

To answer questions:
- I need it to be a single mouth if at all possible. If there HAVE to be multiple mouths, I could always give the angels an inherent glamour effect, but would prefer to minimize the amount of magic used to keep them looking human.
- The voices can be somewhat differently pitched, but preferably not in a way where one sounds like Crocodile Dundy and the other like Mickey Mouse (and the third like it's coming straight from the depths of hell). I'd prefer if they were all "normal" voices, even if one is higher pitched and another lower.

Angels are clearly part bird. More specifically, they have a syrinx instead of a set of mammalian vocal cords. As a result, they can effectively produce two different sets of tones at the same time by controlling the air flow in and out of each lung.

More exotically, you could bring their trachea further up into their chest and give them a "fake mouth" for each tube - that way, they could produce multiple labials and fricatives in parallel, coming from deep inside their (broader than a human's) chests. You should be able to get three vocalizations in parallel with this arrangement.

As for understanding them... it'd probably take specialized training to parse out individual sentences - ever try to listen to pay attention to multiple people speaking at once? -
but it should be just dandy for making them sound all echo-y and choral.

This answer was to an earlier version of the question before I understood what Sora was looking for. Please do not vote for it. It is retained only as a reference. Thanks.

Recognize that you'll have limits. You won't be able to arbitrarily express any three ideas, even if related

What you have to work with falls into several categories.

Spoken Inflection How you express a single word can express additional context. We actually do this already. Even with a simple word like "no," the vocal intonation can express humor, incredulity, definitiveness, alarm, etc. Curiously, Engligh-speaking humans have never codified this behavior to produce more efficient language — but that might be because expressing it in written form (through the use of diacritics) makes the written language much more complex.

Expanded Conjugation and Declination Other languages (e.g., Finnish, which I speak (well, not as well as I did 30 years ago)) use complex conjugation and declination combined with suffixes to substantially modify the base meanings of words. This process would, for example, convert the verb "to jump" to the noun "one who jumps" to the verb "to be one who jumps" to the adjective "of the nature of being one who jumps," etc. Friends and I once tried to calculate the number of legitimate words with unique meanings based on the ways words could be conjugated, declined, and modified with suffixes. We stopped counting at 250,000 and moved on to something else.

Secondary Vocalization Have you ever heard someone speak or sing in two tones at once? I have. It's called polyphonic and could be used to literally carry two levels of meaning simultaneously. For example, if we use only the word "no" as an example and the handful of intonations mentioned above, you could express 4*3=12 possible unique meaning combinations at one time. You could really run with this one, angels being what they are, they could have as many vocalizations as you want. Give them four and you have (per my example) 4*3*2*1 = 24 unique meanings at one time. (That mathmatical progression is called a factorial or n! meanings for every "n" vocalizations.)

Those three examples are purely vocal. If you include non-vocal possibilities such as body language or telepathy, your abilities improve considerably.

Where you have limitations is that, theoretically, none of the examples above can give you infinite expression. In other words, unless divine ability exists, the angles can do something unbelievably amazing, but not miraculously godlike. For most stories, such limitations are usually good things as they make the "superpower" more relatable to the reader.

You want music.

I can sing you a song. I am conveying two messages when I do that - the melody of the song, which carries information in the form of pitch, key, transition and tempo. And you hear my words, the meaning of which changes because of the music that is with them. I can sing to you and play my guitar at the same time. I am conveying three messages when I do that: the song as above and the harmony of the guitar, which frames the message in the song and might itself become the message. I can tap my foot for #4. #5: I can move my body, sway and close my eyes, or glare and furrow my brow as I sing - all messages, all conveying information.

That is 5 messages and I am just a guitar player hoping for a coin in my case. Your angels are musicians par excellence. 5 messages would be beginners stuff to them.

These angels have much more control over their vocal chords than we do. They are able to create arbitrary sound waves using their voice boxes, much like a speaker, by adding together sound waves. Their auditory processing centers are also specialized to be able to separate out distinct sentences from complicated waves. Essentially, they are able to perform complex Fourier Analyses. This would result in some rather interesting abilities, from a human perspective. Like the ability to simultaneously enjoy three separate podcasts at once. They would likely be extraordinarily talented musicians and vocalists, because of the increased vocal and sound processing abilities. They can literally speak three sentences at once, because they can produce the necessary wave forms, just as any old speaker can reproduce three sentences simultaneously, and their highly adept acoustic processing brains can separate and interpret those multiple sentences at once.