This year I've decided to accompany the Easter Vigil Alleluia. Every accompaniment book I've compared has each repetition a whole step higher than the preceding alleluia. Is there some historical reason why the repetitions should be a whole step higher rather than a half step (or some other interval), or is it completely arbitrary?
That sounds tricky to accompany since it usually takes place at the door to the church and then progressively closer to the altar. Being able to see in the darkness when to give the pitches could be a challenge. We are OF and the deacon who carries the Paschal Candle just begins and the choir and congregation respond a capella. However Deacon pitches it is how it is. He always starts low enough to not be desperate by the time he gets to the 3rd repetition, the pitch does rise with each repetition and, I cannot recall for certain but, I believe it is consistently a whole step. It generally works out fine and I like the unaccompanied "Light of Christ" that we get and the response is surprisingly robust. I wouldn't call the deacon who usually does this role a real singer, but he does fine.
It makes the "Gloria" all the more glorious when it finally occurs.
It's probably a matter of keeping the pitch within a reasonable range, as going up by whole steps would make the first alleluia too low, or the subsequent chant or psalm tone too high. The rubrics only state that the priest should sing the alleluia three times, "raising his voice by a step each time" ("vocem gradatim elevando"), without specifying whether this is a whole or half step.
In the Pius X Hymnal, for example (see first attached file), the first alleluia starts on C, with the last alleluia ending on F, so that the psalm tone is at a comfortable pitch. If the alleluias went up by whole steps, the first would have to start on B-flat, which is a pretty low pitch to start on; or, if the first alleluia started on C, the reciting tone of the psalm would be on C, which is on the high side. The same would be true if the full chant verse "Confitemini Domino" were sung instead of the psalm tone (see 2nd attachment with alleluias rising by whole steps).
Easter Vigil - Alleluia before Gospel - Pius X hymnal.pdf
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Eater Vigil - Alleluia before gospel - whole steps.pdf
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