How to make a sonority curve for a word where a 'syllabic consonant' is followed by a vowel or vice versa

Question

I've learnt that the number of peaks of sonority in a sonority curve determines the number of syllables in a word.
The number of syllables depend on the pronunciation. It may also be true for other languages, but consider English.
For example, if the word 'metal' is pronounced [metl̩], it's sonority curve would be:

In the second syllable, the peak is formed by a syllabic L, because syllabic consonants can form a syllable on their own.

But what if we have a vowel right after a syllabic consonant?

For example, if the word 'battling' is pronounced with three syllables: [bæ.tl̩.ɪŋ], how do I make a sonority curve for it?

I've made a curve for this but there are only two peaks in that curve and I don't know how else to make three peaks.

The red ones are the regular peaks but I need another peak at the blue dot.

Another example would be 'parent'. If I pronounce it [pɛːɹn̩t], there are two syllables in it. How do I make a sonority curve for it? I've made a curve, but it only has one peak:

Should it be like this:

We have a syllabic consonant right after a vowel.
How do I make sonority curves for these words?

Answer 1

Unfortunately, you're dealing with somebody's special theory of sonority, so you have to ask the instructor what's acceptable. This approach attempts to predict "syllabicity", or the fact of being a syllable peak, purely from the segmental classification as you've given it. This fails spectacularly when consonants in fact are the peak of a syllable, as happens in a number of languages. There is a solution in English, which is to re-transcribe syllabic sonorants so that they are written as schwa plus sonorant, thus [bætʌlɪŋ]. Syllable peak-hood simply cannot be predicted from segmental content.

In many languages, [i u] and [j w] are exactly the same segment, differing only in syllable function. By adding "vowel" versus "glide" into your segment clasification, you can get "i" as syllable peak and "j" as "not the peak but right next to it in sonority" (or, to use the shorter term, "glide"). Then by adding "vocalic liquid" and "vocalic nasal", up there with regular vowel, you can separate all of the common syllabic consonants, and not include schwa in your transcription of English. You'd also have to add "vocalic obstruent" because of languages like Berber where [x, z] can be syllable peaks, but I assume you don't have to deal with Berber.