The essence of what Harmonic Explorer presents is a tool for exploring musical invariance as it is today and as it probably emerged, in an age when numeracy was about the properties of number rather than, what you could calculate using them. If the modal scales had been discovered in the past by innovating the mixing of powers of three and five (i.e. Fifths and Thirds) then surely the scales upon the mountains of flood heros and others should be considered as informing the cultures having such tales, and their search for a practical yet often sacred music.
One simple observation is that symmetrical twin scales either rise up to the upper register by the major third or descend into the minor thirds. It is this that led equal tempered music to relegate the significance of modal scale names, since the scales could now be played in any key and their defining character was chosen to designate via the key followed by major or minor.
- Dorian can be seen to both rise and fall in its sequencing because Palendromic – self symmetrical.
- Before there were modes, the modern Dorian was the Pythagorean Diatonic (864), which required another power of three to the opening Calendar Constant (720), and includes the Tyrant number (729).
- The calendar (720) can only express five modes: self-symmetric Dorian, Mixolydian-Aeolian, Ionian-Phrygian. Doubling to 1440 generates 729 but has a cornerstone of 1024 but no increase in scales.
- Doubling to 2880 generates twin peak 1875 and also adds the Lydian-Locrian twin scales, the scales that require the 12th tone of a-flat/g-sharp to be symmetric – a condition where the whetted area is a rhomboid, called by McClain the Bed of Ishtar.