If an artifact or creature entering the battlefield causes a triggered ability of a permanent you control to trigger, that ability triggers an additional time.
Panharmonicon affects an artifact or creature’s own enters-the-battlefield triggered abilities as well as other triggered abilities that would trigger when an artifact or creature enters the battlefield. Such triggered abilities start with “when” or “whenever.”
Abilities that create replacement effects, such as a permanent entering the battlefield with counters on it, are unaffected.
Abilities that apply “as [this creature] enters the battlefield,” such as choosing a creature to copy with Clone, are unaffected.
Panharmonicon’s effect doesn’t copy the triggered ability; it just causes it to trigger twice. Any choices made as you put it onto the stack, such as modes and targets, are made separately for each instance of the ability. Any choices made on resolution, such as whether to get counters or Servos for fabricate, are also made individually.
The trigger event doesn’t have to specifically refer to “artifacts” or “creatures.” For example, Amulet of Vigor says “Whenever a permanent enters the battlefield tapped and under your control, untap it.” If a creature enters the battlefield tapped and under your control, Amulet of Vigor’s ability would trigger twice. If a land (that isn’t also an artifact or a creature) enters the battlefield tapped and under your control, Amulet of Vigor’s ability would trigger only once.
Look at the permanent as it exists on the battlefield, taking into account continuous effects, to determine whether any triggered abilities will trigger multiple times. For example, if you control Mycosynth Lattice, which says, in part, “All permanents are artifacts in addition to their other types,” an enchantment entering the battlefield will cause triggered abilities to trigger an additional time.
If you control two Panharmonicons, an artifact or creature entering the battlefield causes abilities to trigger three times, not four. A third Panharmonicon causes abilities to trigger four times, a fourth causes abilities to trigger five times, and so on.
Panharmonicon affects triggered abilities of permanents you control that trigger when an artifact or creature enters the battlefield under any player’s control, not just yours.
If Panharmonicon and an artifact or creature enter the battlefield at the same time, those permanents entering the battlefield will cause triggered abilities to trigger an additional time.
If a triggered ability is linked to a second ability, additional instances of that triggered ability are also linked to that second ability. If the second ability refers to “the exiled card,” it refers to all cards exiled by instances of the triggered ability. For example, if Isochron Scepter’s enters-the-battlefield ability triggers twice and two instant cards are exiled, both will be copied when its second ability is activated. You may cast either or both of the copies in any order.
In some cases involving linked abilities, an ability requires information about “the exiled card.” When this happens, the ability gets multiple answers. If these answers are being used to determine the value of a variable, the sum is used. For example, if Elite Arcanist’s enters-the-battlefield ability triggers twice, two cards are exiled. The value of X in the activation cost of Elite Arcanist’s other ability is the sum of the two cards’ converted mana costs. As the ability resolves, you create copies of both cards and can cast none, one, or both of the copies in any order.