Is mapping sound frequencies to the vertical axis universal? Shrill notes are said to be "high", and rumbles are said to be "low".  Humans seem to metaphorically map frequency to the vertical axis, and in the cultures that I know of, high frequency is considered "up".


Is this a universal phenomenon? Are there any cultures that map it upside-down, or in a completely different way? (For example, smaller objects tend to produce higher pitches. Are there any cultures that say that a pitch is "small" instead of "high"?)
Does this convention pre-date the scientific understanding of sound waves?
If it is universal (or nearly so), is there any neurological hard-wiring that would cause that mapping?
Humans technically don't perceive frequencies, they perceive pitch. According to Wikipedia:


  the idiom relating vertical height to sound pitch is shared by most languages.


citing a 1930 article by Pratt, which in turn says that:


  Stumpf has found that adjectives meaning high and low (or words closely related in meaning) have been applied to tones in almost every known language.[2]  But why should tones be characterized as high or low? Do these characteristics refer to differences in spatial height and depth? The answer to this second question has been almost without exception in the negative. A high tone does not mean a tone which is high in space. The phrase is merely figurative, and must be accounted for in terms of secondary criteria such as, e.g., the apparent localization of high vocal tones in the head and low ones in the chest.  The composer Berlioz makes sport of such explanations and reminds his readers that high and low tones for the pianist lie in the horizontal directions of right and left and that the violoncellist must reach downward to produce high tones, and suggests that those composers of opera who use descending passages for a person falling downstairs have stupidly transferred to the tones the arbitrary downward character of the printed notes on the staff. And yet Stumpf, convinced that there can be no intrinsic height and depth in tones, has felt obliged to argue that here again some associative mechanism, strangely obscure and elusive, has been at work. Even Wundt was forced to agree with Stumpf in calling these terms metaphorical when applied to tones [3], and most psychologists who have given the matter any thought have expressed similar views [4].
  
  [2]:   C. Stumpf, Tonpsychologie, 1883, I, 192 ff.
  
  [3]:   W. Wundt, Grundzüge der physiologischen Psychologie, 1910, 11, 78.
  
  [4]:   J. Redfield, Music: a Science and an Art, 1928, 42 ff.


All these sources are pretty dated alas. 

There are however some newer papers on this; e.g. Fernandez-Prieto et al. (2017) which do qualify such statements, e.g.:


  Higher frequency and louder sounds are associated with higher positions whereas lower frequency
  and quieter sounds are associated with lower locations. In English, ‘‘high’’ and ‘‘low’’ are used to
  label pitch, loudness, and spatial verticality. By contrast, different words are preferentially used, in
  Catalan and Spanish, for pitch (high: ‘‘agut/agudo’’; low: ‘‘greu/grave’’) and for loudness/verticality
  (high: ‘‘alt/alto’’; low: ‘‘baix/bajo’’). Thus, English and Catalan/Spanish differ in the spatial
  connotations for pitch. To analyze the influence of language on these crossmodal associations, a
  task was conducted in which English and Spanish/Catalan speakers had to judge whether a tone
  was higher or lower (in pitch or loudness) than a reference tone. The response buttons were
  located at crossmodally congruent or incongruent positions with respect to the probe tone.
  Crossmodal correspondences were evidenced in both language groups. However, English
  speakers showed greater effects for pitch, suggesting an influence of linguistic background.


They also cite an older review paper (Cassanto, 2013) which highlights a similar studies (co-authored by Cassanto) on Dutch vs Farsi:


  Like English, Dutch describes pitches as ‘high’ (hoog) or ‘low’ (laag),
  but this is not the only possible spatial metaphor for pitch. In Farsi, high
  pitches are ‘thin’ (nāzok) and low pitches are ‘thick’ (koloft). Dutch and
  Farsi speakers’ performance on non-linguistic pitch reproduction tasks reflects
  these linguistic differences (Dolscheid, Shayan, Majid, and Casasanto
  2013). Participants were asked to reproduce the pitch of tones that they
  heard in the presence of irrelevant spatial information: lines that varied in
  their height (height interference task) or their thickness (thickness interference task). Dutch speakers’ pitch estimates showed stronger crossdimensional
  interference from spatial height, and Farsi speakers’ from the
  thickness of visually presented stimuli. This effect was not explained by
  differences in accuracy, or in musical training. When trained to talk about
  pitches using Farsi-like metaphors (e.g. a tuba sounds thicker than a flute)
  for 20-30 minutes, Dutch speakers’ performance on the non-linguistic
  thickness interference task became indistinguishable from native Farsi
  speakers’. Experience using one kind of spatial metaphor or another in language
  can have a causal influence on non-linguistic pitch representations. [...]
  
  infants as young as four months old are sensitive to the height-pitch mapping
  found in Dutch-speaking adults (but not in Farsi-speaking adults), and
  also to the thickness-pitch mapping found in Farsi-speaking adults (but not
  in Dutch-speaking adults; Dolscheid, Hunnius, Casasanto, and Majid 2012).
  There is no need, therefore, to posit that using linguistic metaphors causes
  people to construct these mappings de novo.


A somewhat similar paper on infants, perhaps with stronger claims is from Walker et al., 2010:


  Stimulation of one sensory modality can induce perceptual experiences in another modality that reflect synaesthetic
  correspondences among different dimensions of sensory experience. In visual-hearing synaesthesia, for example, higher pitched
  sounds induce visual images that are brighter, smaller, higher in space, and sharper than those induced by lower pitched sounds.
  Claims that neonatal perception is synaesthetic imply that such correspondences are an unlearned aspect of perception. To
  date, the youngest children in whom such correspondences have been confirmed with any certainty were 2- to 3-year-olds.
  We examined preferential looking to assess 3- to 4-month-old preverbal infants’ sensitivity to the correspondences linking
  auditory pitch to visuospatial height and visual sharpness. The infants looked longer at a changing visual display when this was
  accompanied by a sound whose changing pitch was congruent, rather than incongruent, with these correspondences. This is
  the strongest indication to date that synaesthetic cross-modality correspondences are an unlearned aspect of perception.


Cassanto and colleagues also develop a Hierarchical Mental Metaphors Theory (HMMT) according to which 


  correlational mental metaphors develop in two stages, the second of which may extend throughout the lifetime.
  The mental metaphors that adults typically use are specific instances
  of more general families of mappings. These families may be evident in
  behavior from infancy, and reflect regularities in humans’ experiences of
  the physical and social world, many of which may be universal. As children
  are exposed to peculiar aspects of their languages, cultures, or even their
  own bodies, certain mappings from a given source-target family become
  strengthened through repeated use, which weakens ‘sibling’ mappings as a
  consequence. The result is that people tend to think in language-specific,
  culture-specific, or body-specific mental metaphors: Relativity emerges
  from universals. [...]
  
  One prediction of HMMT is that specific source-target mappings
  should be easy to activate via linguistic training so long as they are members
  of one of the families of source-target mappings encoded in our minds
  (over phylogenetic or ontogenetic time) on the basis of observable sourcetarget
  correspondences in the world. Mappings that are not members of a
  pre-linguistically established family (and that do not reflect correlations in
  our experience) should be relatively hard to activate via training, because
  these mappings would need to be created, not just strengthened.
  
  In a test of this prediction, Dutch speakers were trained to used a thickness-
  pitch mapping that is the reverse of the mapping found in Farsi, and in
  the natural world: thin=low and thick=high. These ‘reverse-Farsi’ trained
  participants received the same amount of training as the participants trained
  to use the Farsi-like mapping. Whereas Farsi-like training had a significant
  effect on participants’ nonlinguistic pitch representations, reverse-Farsi
  training had no effect (Dolscheid et al. 2013). Thus, brief linguistic experience
  caused participants to use the thickness-pitch mapping that reflects
  correlations between thickness and pitch in the world (and is evident in prelinguistic
  infants). Yet, the same amount of linguistic experience was not
  effective at instilling the opposite thickness-pitch mapping, which has no
  obvious experiential correlates, and is therefore not predicted to be among
  the pre-linguistically established space-pitch mappings.


That's referring to the same paper as the first citation in the previous quote. Oddly this experiment isn't summarized in the paper's abstract, but does appear in the paper (as Experiment 4). Cassanto has a free pdf of that paper as well. 

A 2014 PNAS paper by Parise et al.
proposes that the low-high pitch metaphor has a natural basis in the elevation of the sources of such sounds in nature.