15edo/Unque's compositional approach: Difference between revisions
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Added information on tritones, and significantly expanded the Functional Harmony section |
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|[[8/7]] | |[[8/7]] | ||
|[[5edo]] | |[[5edo]], [[Slendric]] | ||
|One possible choice of whole tone (see below) | |One possible choice of whole tone (see below) | ||
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|400 | |400 | ||
|[[5/4]], [[14/11]] | |[[5/4]], [[14/11]] | ||
|[[3edo]] | |[[3edo]]; [[triforce]] period | ||
|Same mapping as [[12edo]] | |Same mapping as [[12edo]] | ||
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|480 | |480 | ||
|[[33/25]], [[4/3]] | |[[33/25]], [[4/3]] | ||
|5edo | |5edo; [[blacksmith]] period | ||
|Highly contentious interpretation; see below | |Highly contentious interpretation; see below | ||
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|720 | |720 | ||
|[[3/2]], [[50/33]] | |[[3/2]], [[50/33]] | ||
|5edo | |5edo; blackwood period | ||
|Highly contentious interpretation; see below | |Highly contentious interpretation; see below | ||
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|800 | |800 | ||
|[[8/5]], [[11/7]] | |[[8/5]], [[11/7]] | ||
|3edo | |3edo; triforce period | ||
|Same mapping as 12edo | |Same mapping as 12edo | ||
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|960 | |960 | ||
|7/4 | |7/4 | ||
|5edo | |5edo, Slendric | ||
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=== 15edo as a dual-9 system === | === 15edo as a dual-9 system === | ||
The intervals 2\15 and 3\15 are both quite distant from a justly-tuned 9/8 interval; as such, some have proposed 15edo as being a "dual nines" system, in which these two intervals are both interpreted as flavors of the whole tone. This interpretation allows for a near-1:1 correspondence between the Left and Right hand versions of Nicetone (see below). | The intervals 2\15 and 3\15 are both quite distant from a justly-tuned 9/8 interval; as such, some have proposed 15edo as being a "dual nines" system, in which these two intervals are both interpreted as flavors of the whole tone. This interpretation allows for a near-1:1 correspondence between the Left- and Right-hand versions of Nicetone (see below). | ||
Where the two types of whole tone need be disambiguated, they can respectively be called the greater and lesser whole tones (after their size) or the Bayati and Slendric seconds (after the structures they generate). | |||
=== 15edo and Carlos Alpha === | === 15edo and Carlos Alpha === | ||
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[[11afdo|Mode 11]] of the Harmonic Series provides another interesting way to interpret intervals of 15edo. Notably, the intervals [0 2 5 7 8 12 13 14 15] can be interpreted as an approximation of the chord 11:12:14:15:16:19:20:21:22. The /11 logic can be extended to supersets of mode 11 to provide interpretations of other intervals, such as [[33afdo|mode 33]] providing 50/33 as an extremely accurate interpretation of 9\15, and 55/33 as an interpretation of 11\15. | [[11afdo|Mode 11]] of the Harmonic Series provides another interesting way to interpret intervals of 15edo. Notably, the intervals [0 2 5 7 8 12 13 14 15] can be interpreted as an approximation of the chord 11:12:14:15:16:19:20:21:22. The /11 logic can be extended to supersets of mode 11 to provide interpretations of other intervals, such as [[33afdo|mode 33]] providing 50/33 as an extremely accurate interpretation of 9\15, and 55/33 as an interpretation of 11\15. | ||
This interpretation may also be criticized due to a lack of accuracy, but it is notably more consistent than the Carlos Alpha interpretations as the difference between the tunings does not accrue per step. | This interpretation may also be criticized once again due to a lack of accuracy, but it is notably more consistent than the Carlos Alpha interpretations as the difference between the tunings does not accrue per step. | ||
=== 15edo's fifth === | === 15edo's fifth === | ||
The interval at 9\15 is possibly the most contentious interval in the entire xenharmonic community. Some have proposed that is represents 3/2 due to its clear function as a concordant fifth; others argue that 50/33 is more accurate and functions better alongside the other /11 intervals; still others have posited that [[97/64]] is even more accurate and simpler due to being a rooted overtone. | The interval at 9\15 is possibly the most contentious interval in the entire xenharmonic community. Some have proposed that is represents 3/2 due to its clear function as a concordant fifth; others argue that 50/33 is more accurate and functions better alongside the other /11 intervals; still others have posited that [[97/64]] is even more accurate and simpler due to being a rooted overtone. | ||
=== Dual tritones === | |||
15edo has two different [[tritone]] intervals, each about a quarter-tone away from the classic [[2edo|semioctave]] tritone. These tritones may actually be considered consonances in the context of 15edo harmony, as they approximate the 11th harmonic with only approximately 10% relative error. They are quite useful as fully diminished and half diminished fifths respectively, in chords such as the [[Ptolemismic triad|Ptolemismic Triad]]. Chords containing these tritones are often useful as dominant chords for voice leading and functional harmony (see below) | |||
== Notation == | == Notation == | ||
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Note that I will be constructing these chord progressions from back to front; this means that we will start with the resolution, then find the dominant chord, and then find a subdominant to precede it. | Note that I will be constructing these chord progressions from back to front; this means that we will start with the resolution, then find the dominant chord, and then find a subdominant to precede it. | ||
===Chord Progression in C Starfish=== | === Elements of Functional Harmony === | ||
For this example, I will use the C Starfish scale (C D Eb Fb Gb Hb A B C in Porcupine notation), and I will treat the Major (Pat 3) Triad as the tonic chord. | Just like in common-practice theory, the chords of 15edo have a tendency to rotate about the Circle of Fifths; due to the heavy damage to the 3-limit, however, this tendency is not quite as strong in 15edo as it is in more accurate tunings. Additionally, unlike in common-practice music, the Circle of Fifths in 15edo does not encompass every single note in the tuning; instead, it forms three distinct "rings" of fifths (each one forming 5edo with some kind of offset) that do not share any notes in common. | ||
Because the step size in 15edo is significantly smaller than the typical semitone, leading tones are tenser than in common-practice music. The Bayati second can be used as a useful element of voice leading, though it is not nearly as tense as the semitone. I consider voice leading to be the single most important element of 15edo harmony, because it provides a consistent sense of direction throughout melodies. | |||
Finally, it is important to notice certain tense intervals that have a tendency to voice lead by contrary motion to certain other intervals. Specifically, the Major Tritone at has a tendency to resolve inward and become a Perfect Fourth, and the Minor Tritone has a tendency to resolve outwards and become a Perfect Fifth. | |||
=== Example: Chord Progression in C Ionian === | |||
For this example, I will use the right-hand C Ionian scale (C D E F G A B C in Nicetone notation), and I will treat the Major (Pat 3) Triad as the tonic chord. | |||
Because the 3L 2M 2s scale is reminiscent of the common-practice Diatonic scale, elements of functional harmony from classical traditions may carry over into 15edo; however, it is important to note that 15edo's inaccuracy in the 5-limit may cause some amount of distortion in the functions of certain chords. Notably, the Wolf Fifth that usually occurs in the Dorian mode is no longer a fifth at all, but rather a major tritone. | |||
A recognizable Major (Pat 3) Triads occurs on the fifth degree of the scale, providing a familiar circle-of-fifths resolution as well as a leading tone from the B of the V chord into the C of the tonic chord. The subminor (Pat 1) triad on the third degree provides an interesting voice leading into the V chord if voiced correctly (with the notes E, G, and B respectively leading to D, G, and B). Finally, the major (Pat 3) triad on the fourth degree provides a leading tone from F to E and from C to B. | |||
Ultimately, our four-chord progression looks like C - F - e - G, or I - IV - iii - V. This progression prioritizes voice leading to create a coherent and flowing sound, and provides a great framework for melodies to be written over top. | |||
===Example: Chord Progression in C Starfish=== | |||
For this next example, I will use the C Starfish scale (C D Eb Fb Gb Hb A B C in Porcupine notation), and I will once again treat the Major (Pat 3) Triad as the tonic chord. | |||
First, notice that the small step occurs between D and Eb in this mode; this step is the most important place to note in the scale, as it plays a major role in voice leading. Here, neither D nor Eb | First, notice that the small step occurs between D and Eb in this mode; this step is the most important place to note in the scale, as it plays a major role in voice leading. Here, neither D nor Eb is present in the chord that we want to tonicize, which means that we won't be able to rely on it as a leading tone. Instead, we might choose to rely on the Circle of Fifths pull that can be established by moving the root by intervals of 5edo. In this case, we can use the diminished chord on Hb for dominant function. | ||
Because the Hb chord contains D, the subdominant chord that precedes it may use Eb as a leading tone. In this case, I will use the major chord rooted on B; the note B carries over from the B chord to the Hb chord, and the note Eb in the B chord leads into the D of the Hb chord. | Because the Hb chord contains D, the subdominant chord that precedes it may use Eb as a leading tone. In this case, I will use the major chord rooted on B; the note B carries over from the B chord to the Hb chord, and the note Eb in the B chord leads into the D of the Hb chord. | ||
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Finally, we can select a nondominant function that emerges from the tonic at the beginning of the progression. I will use the minor chord on A, because it sounds unresolved without sounding too tense. | Finally, we can select a nondominant function that emerges from the tonic at the beginning of the progression. I will use the minor chord on A, because it sounds unresolved without sounding too tense. | ||
[[File:I - vii - VIII - iv°.mp3|thumb|C - a - B - hb° progression]] | [[File:I - vii - VIII - iv°.mp3|thumb|C - a - B - hb° progression]] | ||
Ultimately, our four chord progression is C - a - B - hb°. This progression uses a combination of voice leading, circle of fifths movement, and tension and release to achieve a useful and functional sound, and similar principles can be applied to other scales to create similar functional progressions. | Ultimately, our four-chord progression is C - a - B - hb°, or I - vii - VIII - vi°. This progression uses a combination of voice leading, circle of fifths movement, and tension and release to achieve a useful and functional sound, and similar principles can be applied to other scales to create similar functional progressions. | ||