While Really Wild West  spellcasters can obviously use standard Starfinder spells, the flavor of spells can strongly impact the feeling of a campaign.

So, we’ll want some Western/Victorian themes spells. Here is one each at 0 and 1st-level.

(art by storm)

Phantom Pocketwatch [Mystic 0, Technomancer 0]
School enchantment (mind-affecting)
Casting Time 1 standard action
Range touch
Targets Up to 5 willing creatures, +1 creature per level
Duration 24 hours
The technomancer version of this spell literally summons a phantom pocketwatch. The mystic version creates a small tattoo of a sundial.
Creatures effected by this spell know what time it is, including the date and year, day of the week, phase of the moon, time zone, and time down to the second. They can also have it track how much time has passed since a specific moment, and inform them when a specific time has arrived.
The most common version of the spell in the US use its official time zones, the Gregorian calendar, and so on. Variations of this spell from other cultures use other calendars, an absolute time based on dawn, and other standards. If the spell is cast while touching some timepiece or calendar, it can use that as its standard rather than its normal units.

Ask The Wind Their Name [Mystic 1]
School divination
Casting Time 1 full round
Range personal
Duration instantaneous
This spell can only be cast while outside. You learn about one creature you can currently see, or that you have previously seen and know the name of. You receive information about the creature as if you had used the identify creature skill task with a result equal to 20 + your Mysticism bonus. You do not learn any information about their allegiances or true name or identity, but if you cast it using their name you do learn if that is a name they are commonly known by.

PATREON
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