The word "cackle" can refer to a shrill, high-pitched laugh or the sound made by a hen. Here are some synonyms and an analysis:
giggle: This is a lighter form of laughter. It's often more subdued and can be associated with amusement or nervousness. For example, "The girls giggled at the joke." While not as shrill as "cackle", it shares the element of being a spontaneous expression of amusement.
chuckle: A soft, low laugh. It implies a sense of amusement or enjoyment that is more restrained than a full-on laugh. For instance, "He chuckled at the memory." Similar to "cackle" in that it's a form of expressing mirth, but without the high pitch or shrillness.
laugh: A more general term for expressing amusement through sound. It can range from a quiet chuckle to a loud guffaw. While not as specific as "cackle", it can be used in many contexts where there is mirth. For example, "They laughed at the comedian's jokes."
squeal: This can be a high-pitched sound, similar to "cackle" in pitch. However, it often conveys excitement or surprise rather than amusement. For example, "The children squealed with delight." But in some cases, it can be used in a way that is similar to "cackle" when there is a sense of wild or unrestrained expression.
cackle-like laughter: This is a descriptive phrase that directly references "cackle" while still providing some variation. It can be used when you want to emphasize the similarity to a cackle without using the exact word. For example, "Her laughter was cackle-like, filling the room."