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As I understand cantus firmi means that fixed songs and it is used in counterpoint melodies but I could not understand why is it called cantus firmi and its function. Does anybody know why is it called fixed songs but it is not fixed at all and its funcion?

Like it is given an example below enter image description here

But how does one determine which one is c.f?

Edit: I read that it is used for music education but still could not understand its function at all!

Edit2: Found some information here

The following characteristics are typical of all well formed cantus firmi:

length of about 8–16 notes
arhythmic (all whole notes; no long or short notes)
begin and end on do
approach final tonic by step (usually re–do, sometimes ti–do)
all note-to-note progressions are melodic consonances
range (interval between lowest and highest notes) of no more than a tenth, usually less than an octave
a single climax (high point) that appears only once in the melody
clear logical connection and smooth shape from beginning to climax to ending
mostly stepwise motion, but with some leaps (mostly small leaps)
no repetition of “motives” or “licks”
any large leaps (fourth or larger) are followed by step in opposite direction
no more than two leaps in a row; no consecutive leaps in the same direction (Fux’s F-major cantus is an exception, where the back-to-back descending leaps outline a consonant triad.)
the leading tone progresses to the tonic
in minor, the leading tone only appears in the penultimate bar; the raised submediant is only used when progressing to that leading tone

Cantus firmus composition gives us the opportunity to engage the following fundamental musical traits:

smoothness
independence and integrity or melodic lines
variety
motion (towards a goal)
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  • Canto fermo...? (Italian, plural - canti fermi_.
    – Tim
    Aug 26, 2020 at 12:11
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    I think Canto fermo and cantus firmus are same words but cantus firmus is latin.
    – Nabla
    Aug 26, 2020 at 12:13
  • This is all covered by an encyclopedia reading. Aug 26, 2020 at 12:31
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    At first I had difficulties about why is there such thing but now I understand
    – Nabla
    Aug 26, 2020 at 12:32
  • @MichaelCurtis And that shouldn't be an issue. Please refer to meta.stackexchange.com/q/5280/247309 and meta.stackexchange.com/q/8724/247309
    – NPN328
    Aug 27, 2020 at 0:28

2 Answers 2

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As Michael Curtis mentioned in comments, information on this can be found in an encyclopedia article on the topic.

Briefly, they are called cantus firmi or "fixed songs" because in the renaissance and earlier times, they were derived from pre-existing melodies. Most commonly, they were Gregorian chant melodies, but sometimes other popular melodies (like popular songs) could be used. In the religious tradition of medieval times and the renaissance, the most popular way of composing religious music for church services was to base it on a pre-existing chant melody. Hence, these melodies ("songs") were "fixed" and couldn't be altered, whereas the other parts created around these melodies were flexible.

In the baroque period, composing around a cantus firmus became less common, but students were still frequently taught how to compose in a traditional style by being given a fixed melody (a cantus firmus, which may or may not have anything to do with chant). Students were then asked to write other parts around this cantus firmus. The idea is that the C.F. forces some structure on the student, rather than letting the student compose freely without any constraints. (Most beginning students wouldn't know how to create a good C.F. easily.) Such exercises remain a common pedagogical tradition in counterpoint even in some classes today.

You know which melody is the cantus firmus when it is labeled as such (generally with "C.F."). Or, in many cases, the C.F. is the only melody given, and the student is expected to write other parts to go with it.

As the edited question notes, most cantus firmi that are used in classes today tend to have certain kinds of melodic characteristics, themselves often derived from patterns common to the chant melodies which used to be used for actual composition of renaissance pieces (whether or not they are actual phrases from pre-existing chant melodies.)

(Note: As it came up in comments, cantus firmus is the singular form in Latin, the traditional language used to reference these melodies. As cantus is fourth declension in Latin, its plural is cantus with a long u, whereas the Latin plural of the adjective firmus is 1st/2nd declension firmi. Hence the plural cantus firmi. Some sources use the Italian name instead, which is singular canto fermo, plural canti firmi.)

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    +1 and nice detailed explanation of the Latin. You have actually spared me a dictionary search to confirm whether cantus really was the 4th declension... :—).
    – Ramillies
    Aug 27, 2020 at 0:23
  • Regarding the "encyclopedia article" part, please see meta.stackexchange.com/q/5280/247309 and meta.stackexchange.com/q/8724/247309
    – NPN328
    Aug 27, 2020 at 0:28
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    @Ramillies - in late Latin sources like the time of the renaissance, cantus is actually sometimes treated as 4th and sometimes as 2nd declension, actually. But this sort of inconsistency in medieval sources isn't unusual. To make things even more confusing, cantus is treated as an adjective (usually meaning "sung") at times as well, taking the 1st/2nd declension endings. But yes, the noun is usually treated as 4th.
    – Athanasius
    Aug 28, 2020 at 16:19
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    @VonHuffman - thank you for the reminder, though I'd note my intent was not to shame or belittle the question, but rather to note that more information is readily available on this topic by reading an article in many standard references with the title "cantus firmus." I tried to address the specific aspects of the question in my answer, but it seemed like the questioner in general wants to know the meaning of a very broad concept. I was trying to signal that my answer was not intended to be an "encyclopedic" answer to the broad question, but rather addressing some specific points raised.
    – Athanasius
    Aug 28, 2020 at 19:02
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This is a partial answer to supplement Athanasius's, which is excellent as usual.

There are several ways one might identify a cantus firmus. Two have already been mentioned: it may be marked as such, or, if you are composing the piece yourself, you will just choose it (or have it assigned to you by your instructor).

If you are looking at an actual composition from music history, the second method won't apply, and the first method may let you down because the cantus firmus often isn't marked as such. The composer will likely have assumed that everyone would know which part has the cantus firmus. In such cases, one or more of these points will help:

  • The cantus firmus is usually given in long notes, while the other parts have shorter notes. The example in the question is a pedagogical exercise; settings like that are not common "in the wild." This practice continued through the baroque and classical periods into the romantic period.
  • In the renaissance, the cantus firmus is usually, but not always, in the tenor voice. (As time passed, it became more common to give the cantus firmus to the top voice.)
  • In renaissance masses, the title of the mass will typically be the name of the tune used for the cantus firmus. (A famous example is Missa "L'homme armé," because almost everyone wrote such a mass during the late 15th century.) You can then look up the tune and compare it to each part to see which one matches. (But some masses, called parody masses, are based on other polyphonic compositions, which then lend their name to the mass, so this can be misleading, because these masses may not have a cantus firmus.)
  • In some cases, the cantus firmus appears with the original words. If you see a piece where most parts are singing one text and one part is singing a different text, it is likely that the one part has the cantus firmus.

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