Angel (noun)
About 1300 angel, aungel (with a hard g as in gem),
replacing earlier angels, angles (with a soft g as in ghost),
​a fusion of Old English engel and Latin angelus.

The latter Middle English angel, aungel are a fusion of Old English engel & Old French angele, angel, aungel, from Latin angelus, from Greek ángelos, originally meaning 'messenger'.

The Greek word was a loan translation in the Septuagint of Hebrew mal'akh 'messenger, angel' 
& ángaros 'royal mounted courier'.

The Old English engel was a borrowing of Latin angelus. Other Germanic languages made a similar borrowing of the Latin: Old Frisian angel, engel, Old Germanic angil, engil, Old Icelandic engill.

Angelic (adjective)
About 1385, borrowed from Old French angélique,
from Latin angelicus, from Greek angelikós,
​from ángelos 'angel'.

...The original pronunciation was a hard g & shared understanding with the word angle. In about 1300AD the pronunciation of the word angel (hard g) changed to angel (a soft g). The meanings of messenger/royal mounted courier remained attached.

Angle (noun)
A space between 2 lines that meet.

About 1380, 'corner', borrowed through Old French angle, or directly from Latin angulus 'corner'. Cognate with Greek ánkos 'bend', 'valley'.

Angle (verb)
Fishhook & Line.
About 1450, verb use of angle (n. fishhook);
developed from Old English angel (before 899);
​related to anga hook.

Old English angel is cognate with Old High Germanic angul fishhook, ango hook, Old Icelandic ongull fishhook.

Angles referred originally to the people of Angul,
a region in Germany that was named after it's 'hook-like' shape.
A Germanic tribe accompanied by Saxons, Jutes and Frisians
that crossed into Britain in the 400's & 500's AD.

...In the Ethos of Music, Angels & Angles are analogous to each other. The Hierarchy of Archangels are a direct representation of the arc angles of light that Sir Isaac Newton defines in his book Principia.

Ankle (noun)
About 1350 ancle;
probably before 1300 anclowe;
before 1150 ancleowe;
developed from Old English onclēow (before 800)
cognate with Old High German anchal, enchil which means ankle (modern German enkel),
Old Icelandic okkla (from proto ANKULAN)
& Latin angulus meaning 'corner' and 'angle'.

Old English had another form anclēow cognate with Old High Germanic anchlāo,
apparently influenced by Proto-Germanic klāwa meaning claw,
​but eventually this was supplanted by the simpler ancle.

...When applied to the 12 key signatures & signs of the zodiac,
Jacob (patriarch of the 12 tribes of Israel) takes on the meaning 'heel-catcher'
which is descriptive to the development of human physiology.
A connection of 2 systems;
A ladder of knowledge represented (clearly) through
the angles of angels that are eternally watching from the skies above;
shining down from space, evolving our faculties & guiding us through time.
Hidden behind a tactile mirror of the alphabetic letter G.

The purpose of this exercise focuses on the finer elements involved in molding a melody. Specifically turning something unmusical into something musical. This technique is accomplished on 2 levels. First by physically changing the alphabet of literature into music notes & secondly by turning that result into something musical through a specifically limited set of compositional tools.

The text used in this example:
"Have mercy upon me, for I am weak.
Heal me for my bones are vexed.
My soul is also sore vexed.
Deliver my soul: Oh save me for thy mercies sake.
For in death there is no remembrance of thee."
​-Psalm 6:2-5

The result is a string of notes replacing the paragraph of words. Barlines are inserted so that each word occupies one measure: (fig.2) 

This substitution method provides a boundary, a playground for students,
where creativity is focused on phrase marking,
employment of dynamics,
register choice as well as
utilization of tempo & rests.
There are no time signatures.
The meter is provided by the original text. 
The tempo is maintained by counting 16th notes.
The goal is to reform what is provided without changing the order of the notes.

​This style of abandoning the time signature was used extensively by composer Olivier Messiaen during the first half of the twentieth century. For the purpose of this exercise abandoning time signatures allows for greater aesthetic flexibility and it strengthening the bond to the original source material. 

What may seem formulaic is actually a technique of encoding. Similar to how a doctor injects dye into a patient in order to highlight an area of interest before performing an X-ray. For this exercise, the dye that's injected into the text is pitch and duration & it highlites the need for dynamics, articulations & tempo. This is an exercise for very specific, invaluable aspect melody. The final step is to make it all musical by using only what has been mentioned.
​

The completed exercise may be useful to study from a linguistic perspective as well
through concepts of Transformational Grammar.
The composition as a whole resembles layers of passive transformations
that cycle together forming even larger layers
of more cycling passive transformations.

1. A Note and a Duration is analogous to a Phoneme (fig.3 & 5)

2. A Motive Particle is analogous to a Morpheme (fig.3 & 4)

3. A Phrase is analogous to a Word (or a group of motivic particles) (fig.5)
​3a. A Phrase can also connect Words. (fig.4)

4. A musical Section can be analogous to a Clause: "For I am weak."
​4a. A Section can also be analogous to a Word (fig.4 & 5)

5. A musical Statement is analogous to a Sentence: "Have mercy upon me for I am weak."
5a. A musical Statement can also be analogous a Clause (fig.3)
5b. As well as a Word (fig.3)

​6. A complete Piece can be analogous to a Piece.

​

In terms of music, a Comma is a word used when
referring to the discrepancies of distance between intervals
​inside of any given system.

* There is no single unit that represents a comma but there are
​2 main numeric values in widespread historical use:

The Pythagorean Comma
(also known as the Ditonic comma)
is the discrepancy between twelve pure 5th's
​& seven pure octaves.
​It's the comma used in constructing the Equal Temperament system
& is defined by the ratio 531441:524288
​and the decimal 1.0136433.

​The Syntonic Comma
(also known as the Diatonic comma) 
is the discrepancy between four pure 5th's
& two octaves plus a Major 3rd.
​It's equivalent to the ratio 81:80
​and the decimal 1.0125.

A Diesis is the difference between
a major semitone & a minor semitone.

​The distance of a diesis varies depending on
the amount of tempering used in constructing the 5ths.
The greater the tempering of the 5th's, ​the greater the difference between
​major & minor semitones (the diesis).
​
​
​For example:

* Quarter-Comma Meantone tuning has a larger diesis than the 6th-Comma Meantone tuning system.. A quarter (1/4) is larger than a sixth (1/6).

Just Intonation requires that all notes in a scale have the flexibility to
vary in pitch according to the needs of the harmony at any given moment.
​In other words, it's a musical system that calls for harmonically pure intervals at all times.

For that reason alone it's difficult for most modern instruments to play in Just Intonation,
especially instruments that are pre-tuned or fretted (Piano, Harp, Guitar, Bass, Flute, Trumpet).
It is possible for unfretted instruments like Strings (Violin, Viola, Cello)
& the Human Voice to perform in Just Intonation because
they can slide into pitch but it is still difficult (no matter the instrument)
​and takes a very refined ear to confidently recognize & execute the correct pitch.

*It would also be useful to think of Just Intonation as a 'divisive system‘
... time cuts space into different parts.
This is how we get (& why we use) calendars to coordinate time.

Equal Temperament (ET) is a system that gives equal distance & balance to every interval inside of its respective octave. ET is globally attractive because it's relateable to all 12 keys signatures at once.

ET tuning scheme is tempered,
meaning that the intervals are altered in order to keep
equal distance from octave to octave.
In numeric definition this means that​ 1+1=2 (c1),
2+2=4 (c2 another octave),
4+4=8 (c3 another octave),
8+8=16 (c4) and so on..
​This is considered as the Law Of Octaves.

​1+1=2 is conceptually the same as:

​Using the note name of C vibrating at 128Hz
​the Law Of Octaves continues as:

​After 12 repetitions of this cycle (12 octaves) a drift in tuning occurs between the original tone & the final tone. The numeric representations of these types of discrepancies are known as Commas & Diesis.
​
​The extra distance after 12 Octaves is equal to the Pythagorean Comma (PC). In order to rectify this extra space, ET tempers each octave by 1/12 of the PC, the extras are compensated for.
In other words,
ET folds the extra space equally
& then disperses it back into itself.

The tuning scheme of Equal Temperament allows for the 'mechanics & reasoning' that govern the natural motions of harmony to become self-evident. It's a harmony purely for the sake of harmony itself. This makes the ET system 'out of tune' (or artificial) in comparison to the vibrational functions of nature that Just & Pythagorean systems both outline. ET is unnatural.

Here is a chart that shows the differences among the 3 systems:

Pythagorean Tuning is a tuning scheme
that is constructed by co-relating the ratios
obtained from a chain of perfect 5ths.
In other words, the notes in 1 octave are determined
by their perfect relative 5ths
​from other octaves.

It's considered an un-tempered scale,
a natural system,
based on the interval of the Perfect 5th (or ratio 3:2).
After the span of 7 pure octaves & 12 pure 5ths a discrepancy develops,
​known as the Pythagorean Comma.

The Pythagorean Comma (PC) represents a measurement of 'left-over space'.
It is also known as the Diatonic Comma,
​as well as the ratio 531441:524288,
which is the numeric value of 1.0136433.

The PC is closely associated in the construction of today's Equal Temperment (ET) Tuning System.

* Pythagorean Tuning may be thought of as inclusive
because it bridges numeric gaps that the
​divisive Just Intonation system does not blanket.

Thousands of years ago music was understood as a force of nature, represented through numbers and ratios instead of notes on paper. It wasn't a form of entertainment, instead it was used as a tool to bring harmony back into that which was in dis-ease. History provides us with the mathematical language of music, all that we need to do is to re-insert those numbers and harmonic concepts back into the science of today. By putting the concepts that govern harmony back into the scientific process & applying the results into the needs of daily life, as a society, we end up moving along With the flow of nature's current as opposed to against that current. This current of energy that effects every cell in our bodies & continually flows through the natural world is known as the frequency spectrum. This energy can be fashioned & directed, through the study of Music Theory and Harmony.

Musical relationships based on Astrological principles:

If you have no prior knowledge of music theory this is going to make absolutely no sense, I'm sorry. Those who have studied music will notice that some fundamental rules, that are taught in schools, are different. Academically altered for the practicality of musical performance ..in short, this type of music theory does not describe the laws of 'listenable music' but instead the motion of the natural force of electromagnetic & electroacoustic energy. The first relation of electromagnetic frequency as music theory is to be made from our solar system, sun, moon & stars.

Aries is traditionally the key signature of Db Major (Bb minor), Taurus is Eb Major (C minor), Gemini F# Major (Eb minor) and Cancer is Ab Major (F minor). The key signature of Bb Major (G minor) has historically been associated with the constellation of Leo the Lion, C Major (A minor) with Virgo, D Major (B minor) associates to Libra and E Major (C# minor) to Scorpio. The key signature of Sagitarius is F Major (D minor), Capricorn is G Major (E minor), Aquarius is A Major (F# minor) and Pisces is B Major (Ab minor).

*It's interesting to note that the scales of C Major/A minor are associated with the constellation Virgo. Virgo has been long associated with the notion of purity (Venus/Ishthar/Virgin Mother Mary) & historically the corresponding scales are constructed of pure intervals, lacking accidentals of sharps or flats.
The zodiacal key signatures are not the familiar key signatures of today's modern music theory. Much of the mythology remained intact while the actual logic & reasoning, which pushed the applicational development of modern music theory over time, subconsciously changed.

The principles that guide the naming of notes and scales used in music are quiet different from the natural tuning system that the universe offers us, but the construction of those scales is no different, it's all based on same mathematics, with only slightly altered algorithms. The universe is constructed upon principles of both the Just & Perfect intervals. The generation of Pythagorean tuning brings the concept of tones & scales closer to the functioning faculties of our biology.

In the quest to share written music around the world, and for the performers to reproduce those written notes in a musically pleasing way, the tuning system of Equal Tempernment (or equal distance) was adopted around the world & the piano keyboard became the main vehicle for musicians. This switch to piano was good for the concept of consistency of performance but bad in the fact that the natural meaning (& exactness of the desired pitch) within music was approximated and sacrificed ..Like the Tower of Babel, in trying to capture communication the original meaning behind the language was destroyed.

So the idea of pure intervals is not limited to just white or black keys on a keyboard but instead: pure intervals refer to numbers - specifically, whole numbers that represent distances involved in the construction of musical intervals. When combined together those intervals make scales & key signatures. Thus what we know & recognize today as C Major (c-d-e-f-g-a-b-c) in Equal Temperment is both an approxiamation to the natural frequency and inaccurate according to the laws of nature and the universe. The scales of music theory were fashioned over time for our aesthetic sensibilities as opposed to natural laws.

The notes representing the scale of Virgo (C Major/A minor) are naturally approximated by its particular function. For example: The sign of Virgo is an earth sign and alludes to the geometry of the triangle. The mathematical proportions of the triangle point to the interval of the Major/minor 3rd & 6th. The quality of Virgo is known as mutable (or adaptable) and speaks to the geometry of the square. The square is mathematically descriptive of the intervals of the 4th & 5th. From these geometric intervals it's possible to create Tetrachords which act as 'functions of the harmonic elements that radiate from the constellation Virgo'. Keeping in mind that harmony has a fluid-like motion & constantly fluctuates within itself, it's extremely plausible to map the route of that motion by combining the geometric distances to the corresponding Enharmonic spellings. This a very non-exact representation of the Virgo scale (based on note name instead of frequency): C-D-Eb-F-F#-G-A-B-C/C# ..Very different than the C Major scale I learned in music school.