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.
It is now possible, in the developing version 2 of HE, to go to higher limits and have the scales play around that D = limit. The tones will always be in the same octave 360:720 Hz but the bricks will sequence visually and circled in the tone circle. The backdrop of the higher limit numbers (though never going far from the key three rows and five columns around D for that limit), it is now possible to fully play the Pythagorean heptatonic scale, or modern Dorian.
Pythagorean Scale a.k.a. Dorian
Comparing the Just intoned and Pythagorean Dorian gives little difference to the ear, since C and E are changed by only the syntonic comma of 81/80. However, it is important to see the Pythagorean on higher limits where it appears (such as 720 times 12 = 8640) since that was the ONLY diatonic scale before different ancient cultures record the killing of the serpent by Indra (Vritra), Marduk (Tiamat), Apollo (Python) and even Zeus (Typhon), perhaps to keep precedence.
Following on from some dialogue with Pete Dello earlier this year and some catching up of Sumerian tuning systems, I was able to satisfy my question “What role do modal scales play in the harmonic mountains and tone circles of Ernest McClain?”
I shall be publishing a number of diagrams which show where the scales are within those mountains, forming patterns like knotwork based upon the basing three vectors within the mountains, these being the Just intervals tone = 9/8, tone = 10/9 and semitone = 16/15.
To provide some audible distance learning aids for McClain and Pythagorean tuning knowledge in general, I strayed upon a crude sound system within web pages using waveform arrays, but then was informed of HTML 5 Audio components which are more reliable, efficient and easy to employ with Harmonic Explorers existing code.
I therefore have a prototype version at http://harmonicexplorer.org/app_tonal/mountain.html with the following features:
- You can click or touch bricks to make them sing. For this function stay in the limit 720.
- You can click on the seven buttons with the names of modal scales. These will play one second per note. The checkbox “ASCENDING” (default is descending) can be checked/unchecked to play a given mode up or down.
- The bricks sounding turn GREEN and the tone circle shows the usual circle for it as if you are hovering over the brick.
Two improvements have been made after years of leaving the application as was after the death of Ernest G McClain, for whom it was written. The look and feel are very similar and I expect more changes to come without the look changing.
- The control bar at the top is rationalised into buttons to increase/ decrease limiting numbers by primes or major products of 2, 3 and 5, are now grouped vertically for clarity and the layout is simpler. In the future it would be nice if scaling the graphic using zoom out for large numbers did not affect the control bar.
- The text box where limits can be entered manually also presents a list of known harmonic limits used in the ancient world, which can be clicked to see that mountain’s limit. But now, the hover text for each limit, giving brief references, now appears below the statistics for the mountain. Thus one can now see the mountain and have some inkling as to ancient meanings for that limit. In the future it would be nice to expand on this feature by clicking on a web page for that limit. I also want to put extra anaysis features such as number or rows, with numbers shown, and number of columns (on base, i.e. powers of three), and other aspects used by EGM in his own analysis and fiddly to keep counting.
Try it out here: and this version 1.12 opens with limit = 720, the Calendar Constant. To open Harmonic explorer to another limit, on launch, use a hash symbol after the URL and add your own number in the location as per harmonicexplorer.org#8640 which will resolve to http://harmonicexplorer.org/app/mountain.html#8640. Bibal group members can feedback any issues, and other can use the contact form at http://richardheath.info.