I believe the title says it all. On Wikipedia it shows the five species but I have the idea that there is another type of counterpoint not based on those species. Is that vague idea of mine correct? I believe I got that idea reading this sentence:
Species counterpoint can at first seem very limiting, but through its practice, it can be used to create very exciting melodies and counter-melodies.
From this answer.
The linked answer almost answers this question as well, I think. "Species counterpoint" is the name given to counterpoint composed in the styles described by Fux in Gradus ad Parnassum (though the idea is apparently older). The species were intended to be a didactic tool to teach the student how to write counterpoint. The approach is sufficiently effective that modern university students of eighteenth-century counterpoint frequently still use the species to understand how to write and understand that style of counterpoint.
Real eighteenth-century composers may or may not have used the ideas of Fux when learning to write counterpoint. It is probably not accurate to say that an accomplished composer wrote a piece with a particular "species" in mind, though it is possible that a species could describe part or all of a contrapuntal work. For the fifth species ("free" counterpoint, following the stylistic guidelines of the time) to describe such a work is a tautology, of course, but the other species could be used to describe parts of certain works.
I like the other answers here, but I want to stress that species counterpoint developed out of analysis of existing music. Palestrina had never heard of species counterpoint. He just wrote music. Fux and others like him then looked at Palestrina's music and said, hmm, well, he does this and this in these circumstances, so we'll call that the first species. And he does this other thing and this other thing in these other circumstances, so we'll call that the second species. Then we'll use these species to teach counterpoint. Palestrina himself breaks the "rules" of species counterpoint all over the place.
To my mind species counterpoint gets (some of) the form of Renaissance music without the soul. Sort of like reading a basic book about massage and going into business as a masseur without ever having read more advanced work, gotten any hands-on training, or even had a massage. It's a helpful tool but far less useful than experience of singing the actual thing.
Yes! Outside of the Palestrina-style counterpoint described by Fux's "species" -- which is to say, Modal, and eventually Tonal, counterpoint -- there are a variety of approaches. Here are a few.
These two videos give brief introductions to serialism and to serial counterpoint, repsectively.
Melharmony is a system of developing polyphonic or contrapuntal music based in music systems that are fundamentally melodic; for example, Indian Classical music.
In this answer to the question Can modal counterpoint be studied without studying harmony? Where to start? there are a number of resources listed.
Many composers have explored counterpoint in microtonal compositions, though to date, I'm unaware of a unified approach. The book Steps to the Sea: Ear Training and Composing in a Minute Equal Temperament by Julia Werntz (2014, Frog Peak Music) lays out microtonal composition exercises in a form analogous to "species" exercises.
(See also: What is rhythmic counterpoint?)
Phasing
Phasing is a technique primarily associated with composer Steve Reich. The basic concept is to create a musical line as a series of pulses, initial played in unison, but then slowly going out of phase with each other through progressive delays of each part.
A straightforward example is his Clapping Music (1972).
His piece Electric Counterpoint is a more elaborate example.
Rhythmic canons
Termed by Olivier Messaiaen, a rhythmic canon is one such that "each instrument should play the same rhythm but start at a different time."1 Further, "if the rhythmic canon is such that at every time interval exactly one instrument can be heard, then the canon is said to be tiled."2
An example of a rhythmic cannon can be heard in Messaiaen's Harawi (1945) midway though "Part VII: Adieu".
Tom Johnson works especially with tiling rhythmic canons. For example, his piece Tilework for Trombone.
Much of the writing on (tiling) rhythmic canons is highly mathematical. The seminal paper by Daniel Vuza, “Supplementary sets and regular complementary unending canons”, can be found in Perspectives of New Music.3 Moreno Andreatta has written extensively on the subject. For more musically oriented papers, Tom Johnson has published several articles. Google Scholar is a good source for papers by both Andreatta and Johnson, as well as others.
1 http://archive.bridgesmathart.org/2009/bridges2009-265.pdf, accessed 2020 Sept 19.
2 Ibid.
3 D.T. Vuza, “Supplementary sets and regular complementary unending canons”, Perspectives of New Music, Nos. 29(2) pp. 22-49; 30(1) pp.184-207; 30(2) pp.102-125; 31(1), pp. 270-305, (1991).
Heh. I just linked to this elsewhere; a marvelous book for learning species counterpoint is the Aldwell-Schachter "Harmony & Voice Leading" text. We used at the UW-Madison School of Music and it's amazing.
http://books.google.com/books/about/Harmony_Voice_Leading.html?id=-Hp1g3DWNMgC
To build on the accepted answer, the various species progress like so: First: note on note Second: two notes on one note Third: usually regarded as four notes on one note, but there are differences of opinion here Fourth: suspensions Fifth: basically "anything goes"
Compositionally speaking, these are all techniques used to put pieces together, but are never a piece in and of themselves, except, of course, when used pedagogically. For example, fourth species is often used to give a "soaring" sort of feeling due to the rhythmic offsets and harmonic blurring it employs. One good example is the second movement of Mozart's Clarinet Concerto in A, K622; that intro employ fourth species counterpoint.
Species Counterpoint is the tools by which counterpoint is composed. The five species of counterpoint make up all the pedagogical lessons needed to explain the counterpoint of composers like Palestrina.
In my experience, the term "species counterpoint" refers to only the pedagogical method by which it can be learned. Fux's Gradus was the first to break it into "species," and that's where the name came from. Other approaches exist and are often called the direct approach. This boasts being able to bring you to writing more exciting music faster by dispensing with the progression from whole notes to half notes, etc. Here is one such example:
https://www.amazon.com/Direct-Approach-Counterpoint-Century-Style/dp/B000EBPC12
