Spoken language has a complex multi-dimensional compositional structure that enables the rich productivity so characteristic of this most important human biosignal. Even the utterance of a single vowel sound, such as [o] or [i], requires the talker to coordinate gestures of the lip and tongue with gestures of the respiratory-laryngeal system, to combine the timbre properties of the vowel with a specific voice quality and a specific melodic pattern. The latter can be chosen from some set of morphemes that differentiate utterances, as in the English sentences Oh? versus Oh!, or from some set of tonemes that differentiate words, as in the Mandarin Chinese words yī `one' versus yǐ `ant'. In either case, the utterance will have a kind of “simultaneous compositionality” that can be found also in the vocal communication systems of many other primates. Its importance for the human biosignal is evident in the fact that human infants begin to develop the capacity for it as soon as the larynx disengages from the nasopharynx at about 2 months. Of course, most of the words of all spoken languages have an even richer internal compositional structure that depends on the coordination of gestural complexes for contrasting timbre properties, to create a series of alternating consonant and vowel sounds, as in the “canonical” CV syllable type that infants with normal hearing begin to produce between 6 and 8 months of age, or the even more complex rhythmic structures that are used to make words in many languages. The development of writing systems was the first technical innovation for modeling this property of “serial compositionality” that is shared by all human languages, as well as by the vocal communication systems of some other primates (such as those of many species of gibbon). The history of the Interspeech conference series is closely intertwined with many other important technical developments that have greatly increased our ability to model both types of compositionality. Now is a fruitful time to apply these more recent technical developments in a concerted way to achieve a better understanding of the emergence of compositionality in phylogeny and ontogeny.