GKM Melodos Augoustou Golden Ratio by Similar Thirds mode 02 (attractions+)

GKM-Evangelos SOLDATOS tuning
This tuning of Augoustou uses only 2 types of intervals
== GEROPHONIA == a LARGE interval (A)
== Misitonia === a SMALLER interval (B)

The only exceptions are
1) in the "enharmonic fthora" passage,
and 2) in pl 2 differentiation by adding 1/72ET a) HYPHESIS on Zo and b) diesis on Ga

=========
when the RATIO between the Larger and Smaller ratios is that PHI (Apprxo. 1,6), we are dealing with a GOLDEN RATIO (or Golden Section)
(see wikipedia)

When one uses the golden ratio equation

1) [(A+B)/A] = [A/B]

as well as that of a perfect OCTAVE

2) 3A + 4B = 1200 cents


intervals A and B obtained are as follows

A = 219.29 cents [Ratio = 1.13504067434954] (Approx 13,1/72ET; 12/64ET)

B = 135.53 [Ratio = 1.08143128766047]; (Approx 13,1/72ET; 12/64ET)


Such a DIPHONOS (succession by thirds = [B-A-B-A-B-A-B]) scale presents the following charateristics
A= 219.29c [11.7/64ET; 12.43/68ET; 12.97/71ET; 13.16/72ET]
B= 135.53c [7.23/64ET; 7.68/68ET; 8.02/71ET; 8.13/72ET]
THIRD = 354.82c [18.92/64ET; 20.11/68ET; 20.99/71ET; 21.29/72ET]
FOURTH = 490.35c [26.15/64ET; 27.79/68ET; 29.01/71ET; 29.42/72ET]
FIFTH = 709.65c [37.85/64ET; 40.21/68ET; 41.99/71ET; 42.58/72ET]
==========
A: 1.13504067434954
B: 1.08143128766047
=====
Personally, I'd link the "critical" parachordi (7:57) about one GEROPHONIA (219,29c) MINUS one MISITONIA (135.53c ) i.e. 83,76c higher, which is what I did in my chanted version.
NB== the 71 ET scale works quite conveniently for the DIPHONOUS (by thirds system) using the Golden ratio, and DOES, as well, satisfy the necessity of TWO SIMILAR fourths using a tone that is EXACTLY the SAME as that which creates the DISJUNCTION between them (in other words, the DATONIC 71 ET SCALE ALSO exists for THREE different intervals = Meizon, elasson, elachistos; as well as for only TWO types of intervals A and B (gerophonia and microphonia)

Nevertheless, OTHER successions are possible:

eg ...B-A-A-B-A-B....
eg ... ...B-B-A-B-B-A....
with consecutive INVERTED thirds of the golden section

When the iverted thirds bring TWO SMALL intervals in consecutive proximity (B-B), this is NOTED using PLAGAL FOURTH interval appendices of upper Zo and upper Ni; this explains the use of the FTHORA of DIATONIC DI giving the following progression = ...B-A-A.

On the other hand, when the inverted thirds ring TWO LARGE intervals in consecutive proximity (B-B), this is NOTED using PLAGAL SECOND interval appendices of upper Zo and upper Ni; this explains the use of the FTHORA of CHROMATIC DI giving the following progression = ...A-A-B.

===========

The link between GOLDEN ratio and intervals of SECOND mode was suggested by GEORGIOS K. MICHALAKIS, who informed Evangelos SOLDATOS and Charalambos Symeonides by phone.
The two equations solved above are in the interval tables (.xls) I've put up on the server years ago.

The interval of 354 cents corresponding to the DIPHONOS system was first underlined by Charalambos Symeonides who confirmed his hypothesis with my measurements of Iakovos' intervals in "Ton Despotin"

Evangelos Soldatos connected the EQUATIONS with theoretical descriptions (gerophonia, etc) but he unfortunately continues to insist on "rounding off" values to "72ET" scale values, which is ERRONEOUS, given that a well TRAINED psaltic ear can discern as little as 0,2/72 ET.

I REPEAT HERE == ROUNDING OFF to 72 ET values is a MISTAKE == the 72 ET scale ITSELF is a mistake!!!!

The EPITROPI and KARAS theories, of course, simply REJECT this system that has come down to us ever since antiquity!
====

Equal tempered scales according to Golden RATIO




A/B = ((1+((5)^(1/2)))/2) 1.62

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