figure 1	figure 1a

consonant ligatures	(bad) consecutive 5ths

figure 2

dissonant ligatures, suspension & resolution
figure 3

(bad) consecutive unisons

Fourth Species

FUX says that this species is called ligature or syncopation. The rules for fourth species are:

There are two minims for each note of the CF.
In every bar the second minim is tied to the first minim of the following bar. (The final semibreve is not tied to the previous minim.)
The second minim in each bar must always be a consonance.
If the tied note on the first beat of the bar is consonant with the CF Fux calls this a consonant ligature. Movement from a consonant ligature may be by step, or by leap (as in figure 1). [Note that if we shift the counterpoint in this example onto the first beat (figure 1a) consecutive fifths are apparent; recall that direct motion to a perfect consonance is forbidden. However, Fux allows figure 1, maintaining that the syncopation mitigates the effect of parallel fifths.]
If the tied note on the first beat of the bar is dissonant it must be resolved downwards by step on the following minim (figure 2). In this example the tied D is a dissonant 7th above the E, so the next minim resolves down a step to a consonant 6th on C. We can also say that the D is a suspension from the previous bar (in which the preparation occurred), and that the resolution occurs on C.
Successive dissonant ligatures resolving to either an 8ve or a unison are not allowed (figure 3). Although the consecutives here appear to be similar to the example with 5ths in figure 1, Fux explains that 8ves and (especially) unisons are too lacking in harmonic colour for the effect to be acceptable.
In the penultimate bar the first minim must be a seventh (tied over from the previous bar), which resolves on the second minim to a major sixth above the CF.

Although it is permissible to use consonant minims without ties occassionally, this should be avoided if possible. In the rare cases where it is not possible to use ligatures and untied minims must be used, ties should be reintroduced at the first opportunity.

An example from Gradus

Note how Fux writes consecutive octaves in bars 9 & 10, despite his earlier warning. He does this in another example, too; it may be significant that in both cases a leap of a 3rd (consonant ligatures) is involved, rather than a step.
Also note that the closing formula requires preparation in the bar preceeding the penultimate one.

How to use the applet

The basic procedure is the same as in the preceeding applets, though the introduction of ties causes some slight changes. When the user clicks in the second half of a measure, a minim is entered there and tied to another minim which is simultaneously entered on the first beat of the next bar. When the user clicks in the first half of a measure, an untied minim is entered on the first beat of that bar; if a tied minim was previously there, the tie is removed from the minim in the preceeding bar.

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