* Audio examples are best demonstrated using earphones or earbuds.

In western harmony scales are defined from Tonic to Tonic (from C-C). The Tonal definition exists around the outside, containing the harmony like skin around a body or endcaps on a bookshelf. It seems counterintuitive & I believe that it is.

Up until the 1200-1300's AD the Tonal center of a particular harmony was expressed harmonically through the center by the modal system passed down from the Greeks. The move towards a definitive 'endcap' practice was the result of a developing of chromaticism & the confusion in trying to contain it. This led to the birth of 12 different key signatures, all with their own tonic point of view. This gave harmony the space it needed to chromatically grow & the structure for composers to understand it. After 700 years Harmony still has new space to grow into & we can still learn from what it can shows us. Let's start putting harmony it back into itself...

C Major has the note C as its tonic. Thinking of C as a tonal center would place the note C in the center of the scale, theoretically starting on the 4th or 5th of C (G A B C D E F G, or F G A B C D E F) & the same would apply with each of the 12 separate key signatures. Essentially it would be applying the system of modes into each keysignature. Nothing new or drastic & really not even a big deal. The only difference is the definitions within the new foundation, a little change in perspective making a big change in direction.

A scale attached to the English alphabet, 26 letters attached to 26 intervals.
Since C will be the Tonal center of this scale, start with the letter M as the note C and continue diatonically, N as the note D etc.. & the same in the opposite direction, L as the note B, K as the note A, etc..
Repeat the same C Tonal center scale, overlapped chromatically. Start with the letter M as the note C, N as the note C#, etc.. & the same in the opposite direction, L as the note B, the letter K as the note Bb:

The result is extremely different than any traditional scale. The result is a chromatic scale that has a length of slightly more than 2 Octaves (from bass stave C, passed middle C, up to C# treble staff) that's overlapped onto a diatonic scale that has a span of 3 Octaves plus a 5th, all balanced around a C Tonal center. Two natural scales radiating out of the center note C, like the solar system, or a galaxy contained by phonics.

When provided a structure to contain itself, the Tonal center provides it's own natural self-space that can be defined through intervallic distance. This particular example, using the alphabet, provides an outer circumference of 26. Below is another example using the alphabet structure with a Tonal center focusing around the note A:

The intervals present diminished 4th's (#19) diminished 5th's (#22), demonstrates symmetry, as well as reciprocal motions (#7 & 19, #4 & 22) that match its overall macro form.

An exercise in producing all of the 12 key signatures (& the modes of those key signatures) ends up painting a tonal palette, revealing distances of augmented unisons & double diminished intervals. A key note tonal center is reminiscent of a Pythagorean system, a natural motion within.

This key note perspective contains a sense of overall definition and symmetry but it lacks movement, it has no rhythm within itself. I imposed the idea of assigning the English alphabet onto 12 rhythmic durations (from an earlier post). That chart can be superimposed onto this process in 2 different ways by corresponding the letter to a rest, or a keyscale note with a rest. Both produce a motion that's reminiscent of Wagners Nature Leitmotiv from his epic opera cycle The Ring Of The Nibelung. Choosing rhythmic duration based on alphabet letter produces a 'block chord' rythmic motion. Corresponding the rhythm through note name produces a melodic/polyphonic based motion.

Here's a short composition where all of the notes were derived from the letters in the name 'Joseph'.
​The scale used was a C KeyNote superimposed onto the English alphabet:

Seeing new extra-musical relationships, mixing principles & springboarding new ideas, expand areas in order to refine & redefine boundaries, demonstrating points of natural change, understand the sense of space.

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: