Pass formula
A pass formula (also paramodal formula, hypermodal formula, hypermodal key) is a parameter in hypermodal systems that determines the arrangement of paramodes. It's derived from the paramode of the first mode - if it's, for instance, the 3rd in it's own modal system, the pass formula of the hypermodal system would be I:III, or III2:I1.
Pass formulae are expressed through a ratio of roman numerals which are the orders of both scales in their own modal system. The first numeral is usually I. If it's not, an arabic numeral is added after both roman numerals to distinguish the original modal system (1) and the paramodal system (2). It doesn't matter at all though, as anyone can understand even with minor incorrections.
The paramodal system is always upside down (ascending from the top to the first mode), which is why it's called a "pass" formula - the modes are passing by in parallels.
I:V pass formula
This pass formula can be seen on the image above. It's the most common pass formula in terms of usage - the Harmonic Ioni-Aeolian and the Harmonic Phrygi-Lydian hypermodes use it.
I:VII - Perfect pass formula
A perfect pass formula is a formula with a ratio of I1:VII2, which means the paramodal system is perfectly reversed next to the original modal system. It is therefore the easiest to derive and to find in a reversed context. An example of a hypermodal system with the perfect pass formula can be the ZIPH hypermodes. In this case, the scales Clordian and Parafizzardic have their modes named just to fit their paramodal counterparts. For example, the paramode of Clordian is Flordian, and the paramode of Phononic is Freenonic.