Meaning of "if not" in a sentence

Question

I don't understand the meaning of "if not" and grammar of the sentence in bold:

As early as the 14th century, the organ supplied polyphony, in which
case the odd-numbered verses of text were omitted altogether. This
role for the organ grew steadily in importance in the late
Renaissance.
Until the middle of the 17th century, organ versets
were based on the chants they replaced, thus invoking if not
presenting the appropriate words. During the next hundred years this
large organ literature made little reference to a cantus firmus,
reaching its artistic zenith and liturgical nadir in France.

[Source: The Harvard Dictionary of Music' By Don Michael Randel]

Love · Answer

If not, in gerneral terms represents a more extreme term than the former (in this case 'invoking'). Organ chants were based on the chants they replaced, thus invoking perhaps even presenting the appropriate words.

In this case, 'presenting' is more extreme than 'invoking' the appropriate words.

For example: Let's meet up tonight if not sooner (link).

Chappo Hasn't Forgotten Monica · Answer

This turned out to be a bit more complicated to explain than I had anticipated, so I can understand the reason to post your question on ELU! The combination of "if not", "invoking" (which lends a sense of something not present) and "presenting" (i.e. being present) takes some unravelling…

if not - ODO:

Perhaps even (used to introduce a more extreme term than one first
  mentioned):
  'hundreds if not thousands of germs'

invoke - ODO:

2.2 Give rise to; evoke:
  'how could she explain how the accident happened without invoking his wrath?'

So, your bolded portion could be reworded as:

organ versets replaced the chants, but as they were based on the words
  of the chants, the versets evoked [gave a sense of] the words without
  actually presenting them.

Meaning of "if not" in a sentence

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