15-edo Blackwood [10]

In 15 tone equal temperament, as well as all equal divisions of the octave that are multiples of 5, have a 5 tone equal scale.  Notation: 5n-ed2s. – which means 5 times some number n – equal divisions of 2 times the base frequency.

This 5 town equal scale works well as a standalone pentatonic scale, and every note sounds good with every other note.  It doesn’t have the polished harmony of western romantic music, which is characteristic of near-perfect tuned 3rd harmonics (fifths and fourths) and well tuned 5th harmonics (major and minor thirds).  It does however place its characteristic interval between 7:6 and 8:7, making it essentially an embodiment of the 7th harmonic.

Side question: since 15:11 rests between 7:6 and 8:7 (as (7+8)/(6+7) is (15/11), why in this context does this 240¢ interval seems to hold the characteristics of the 7 instead of the 15 or 11?

The Blackwood [10] scale is a combination of two of these 5-equal scales.  In 15-edo, there are two modes, a major and a minor mode.  The major being a LsLs… symmetrical scale structure, and the minor being a sLsL.. scale structure.

In the major version, you get the leading tones below each of the notes in 5-equal, therefore adding the major 3rd, major 6, and major 7th to the scale set, as well as the large tritone and the neutral 2nd.

In the minor version, the opposite it true, adding the minor 3rd, 6th and 7th, as well as the small semitone and the small tritone.

The 5-equal scale base set can be described as having the large tone (septimal), the 4th, 5th, and harmonic 7th.

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Travel Music

Greetings fellow humans,

I’ve finally made it to a beach where I can relax and go over some recordings and work from the past month or so.  I’ve been working with my 15-tone guitar, recording improvisations each day with the goal of learning in depth the sounds and patterns of this instrument.  This was the same method I used for recording my two solo piano albums The Moon (https://noahdeanjordan.bandcamp.com/album/the-moon), in 18-edo or 1/3 tone (Sonido 13), and The Devil (https://noahdeanjordan.bandcamp.com/album/the-devil).  I have been releasing one of the improvisations each week on my new Patreon page (https://www.patreon.com/noahdeanjordan), which you should definitely check out if you are interested in this progress.  There are also live videos I am releasing from places where I can record audio and video, so far this has been in Swakopmund, Namibia,, Klein Brak Rivier, South Africa,, Joza, South Africa,, and most recently: Maputo, Mozambique where I recorded the Bad Canada song: SWOUP.

Two days ago I was at the Timbila Music Festival in Zavala, Mozambique.  I was able to spend the day watching great performances of music and dance with abundant cervejas and the most delicious pork skewers.

The timbila is a mallet percussion instrument, similar to the xylophone or marimba but with a very unique and wonderful tuning.  The tuning is heptatonic (7 tones per octave) and they are somewhat evenly spaced, the first steps actually being very close to the steps of my 15-tone guitar (160¢).  There are 3 sizes of timbila, with the lowest generally having only 3 or 4 keys.  The timbila music is very very lively but with a soft sound of the mallets on the wood.  There are almost always many layers of rhythms and the instruments blend together to make a hypnotic texture of layers upon layers of melodies and rhythms.

I believe that this tuning, being a near 7-equal tuning, is well suited for this type of music.  Because of the many layers of the music and the symmetries, it creates a diatonic-type texture that is familiar, but with a neutral second as the interval that is characteristic to the music and heard by the musicians as quite fundamental.  In a sense this is a music characterized by the 11th harmonic.  The perpetual beating I believe helps to push forth the energy of the music, and allows all the instruments to blend together to create a very rich-textured harmony but with simple components, allowing melodies to flow in and other of each other, with every player playing a unique part which was a piece of the whole, the music completely acoustic but with great volume when desired. This festival occurs every year in August in Zavala, and has for the past 23 years.

It is important to consider attributes of tuning in different traditions of music, as this same effect could not be obtained by any subset of 12-tone equal temperament, or with any just intonation 5-limit diatonic scale.

Some questions I will put forth about this:

What is the function of the 11th harmonic?

Can 7-edo be legitimately described as a tuning representing the 11th harmonic?

In my opinion, 5-edo is in the same way a tuning representing the 7th harmonic.  Can the 720¢ fifth be described in some what as a fifth of the 7th harmonic?  Even though it does not actually approximate this harmonic in any way.

Stay tuned for the next part where I will speak about the Blackwood [10] symmetrical scales in 15-edo (and other tunings)

Seventh chords in 15ed2

Hi everyone,

In 15 tone equal temperament or 15ed2 / 15edo (15 equal divisions of the 2nd harmonic / the octave), there are three types of 7th, three types of 2nd, and two tritones.  Thus, the three “extra” notes.  Of course, a majority of the other notes have changed position from 12 tone equal temperament (or 12edo / 12ed2), aside from the root position augmented triad.  This is because both 12ed2 and 15ed2 divide the octave into 3 equal parts (or 400¢) as 12 and 15 are both divisible by 3, and thus the augmented triad.

The lowest of these 7ths is a 960¢ interval, very close to the ~967¢ 7:4 harmonic 7th, which is an octave reduced 7th harmonic, thus the name.  Because of its close proximity to a low harmonic, and perhaps the simplest ratio describing a 7th, there is a low amount of beating, and thus paired with a major third, this 7th can function as a root position chord as well as a dominant function chord.  It can function as a dominant as well due to it still being a tritone (in this case a lesser tritone) from the major third.

It is debatable whether 7:4 or 9:5 is in fact the simpler ratio, due to the first being the 4th prime number (7) octave reduced (over the 1st prime / 2nd harmonic (2) twice), or the 2nd prime twice (3rd harmonic / 3) over the 3rd prime (5th harmonic / 5).  In fact, simpler may not be an accurate term to describe the distinction, as both functional and harmonic relations are existing here.

The middle 7th in 15ed2 is a 1040¢ interval, quite close to the 11:6 neutral 7th ratio.  Being a 4th (flat 4th) away from the minor 3rd, it can function as a minor 7th.  And being a greater tritone, away from the major third, can also make a dominant function.  As a melodic interval, its function as a part of the 11th harmonic is more prevalent.

The high 7th in 15ed2 is tuned to 1120¢ and essential functions normally as the major 7th.

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PS: naming is a challenge currently, which tuning systems often having multiple names.  I prefer 15ed2 as the most technical, accurate, and general name.  However, 15 tone equal temperament is most widely understood, followed by 15edo.

The “tritone” interval was named such as the sum of three whole tones.  In 15ed2, the small whole tone (of 2 steps) used 3 times is the 4th (of 6 steps), and the large whole tone (of 3 steps) used 3 times is the 5th (of 9 steps).  Hence the tritone name is erroneous here in this context.  Unless, of course, you accept two different sized tones in combination to generate the tritone, in which case, both 15ed2 tritones may be generated.  Such is an argument for the role of context in music, and particularly the large amounts of compositional uses of this type of contextual structure in microtonal / xenharmonic music.

Bad Canada Review

My Bad Canada project – psychedelic indie canadiana folk – got a really great 5/5 from Canadian blog: Ride the Tempo.

http://ridethetempo.com/2017/08/06/album-review-bad-canada-bad-canada/

This album was recorded primarily in San Luis Potosí, México with the Sonido 13 pianos of Julián Carrillo.

There is new Bad Canada material in the works and some will be played at miCROfest in Zagreb, Croatia at the end of October this 2017.

If you are interested to follow the process of this new work, and hear some stuff along the way, please subscribe to my Patreon at:

https://www.patreon.com/noahdeanjordan

Something about Timbilia music in Mozambique to come soon 🙂

South Africa

Traveling around South Africa has been pretty amazing.  From Cape Town (after flying from the Namibian coast) driving up to thus far to Grahamstown where the ILAM (International Library of African Music) is.

The ILAM has a collection of instruments and writing and recordings started by Hugh Tracy in the early 20th century.  There is a huge amount of documentation there of musical traditions and tunings ( 🙂 ) of which many have changed or disappeared since then.

In the week I have been year I have been talking with ILAM director Dr. Lee Watkins, and the Tracy family about African music, tuning, and the past and the future.

When speaking about microtonality today, the point emerged about “alternate tonality” as a more tonal based way of describing microtonality, at least as I see microtonality.  The word microtonality, to me, brings up ideas of 20th century serialism, atonalism, and dissonant experimentalism.  Which, while great and interesting, does not really cover the usage of near 7 equal scales used by the Chopi timbila musicians of south-eastern Mozambique (who I will be visiting soon).  Standardization has been a very devastating occurrence to many musics of the world, and a microtonal/xenharmonic/free-pitch view point could help to once again free this part of the artistry (begin dialogue of if tuning can be an art in itself) and help revitalize traditional tunings which are tuned uniquely and be memory of a tradition.  A unique tuning, which are theoretically infinite, can help define a community and a tradition.

We will see in these travels to come, as I make my way north to Egypt, if I can generate a case for microtonal music for its use of both preserving musical traditions of the past, and generating new musics for the future, as well as the synthesis of this as cultural traditions fuse with each other and share and morph defining characteristics with each other.

Also, there is much new music in the works from me.  I am starting a crowdfunding platform with which to share my own playing and composition and improvisation during these travels, and afterwards.  Please consider supporting for $3/ month for 2 songs per week.

https://www.patreon.com/noahdeanjordan

Mahalo

How to make a 5th

In 15 tone equal temperament (15edo / 15-ed2).  The “perfect” 5th is tuned to 720¢.   720¢ is almost exactly 47:31, or decreasingly, 44:29, 41:27, 38:25, 35:23, 32:21, 29:19, 26:17, 23:15, 20:13, 17:11, 14:9, 11:7, 8:5, 5:3, 2:1.

In between any two intervals described in Just Intonation, an interval between them can be found by adding the numerators and the denominators.  For example: between 3:2 and 4:3, we have (3+4):(2+3) or 7:5.  In the first pattern we are finding intervals closer and closer to the 3:2 starting from the octave.

The first are familiar: the 2/1 octave, the 5/3 major sixth, the 8/5 minor sixth.  Then they become more abstract: we could suggest 11:7, 14:9 and 17:11 to be subminor sixths, but this is challenging territory; what is 11:7?  It describes an interval generated by the 11th harmonic up and the 7th harmonic down.  Is some amount of the characteristic of the subminor 6th to depend on a relation between the 7th and the 11th harmonic?  Or maybe 14:9 is more characteristic of the sound?

I pose this as an open question?  Which is more consonant and in which contexts (for context may lead to many answers), 14:9 or 11:7?  And further, is it solely larger number that makes more dissonant or complex intervals?  Or is the composition (and prime composition) a large factor as well?  And do these have functional relations?

In addition, 32:21 may be described as a diminished 6th, a minor sixth lowered by a chromatic semitone.  In this case, 8:5 lowered by 21:20, creating 32:21.

Back to the idea of the 5th.

It is evident that by analysis of Just Intonation from this standpoint that a 5th must be tuned quite sharp for it to appear to be something other than a fifth.  Where is this bound and how is it defined?

**this is coming from the many debates on xen alliance II fb group of whether or not the 15-ed2 fifth is good/usable/etc or not

 

 

tones and context

To whom it may concern,

If you are here it may be by mistake or it may be though the so called “microtonal” music or through some other means.  I would like to make some comments about the content here and what to expect in the future.

Firstly, the term “microtonal music” is misleading at best.  It is to imply that the tones used are smaller than usual.   This is only meaningful if it is in reference to another arbitrary system of tuning.  For even if I choose to use a step of 80¢ instead of 100¢, is my tone of 160¢ bigger than something or smaller than something.  The “micro”-tone does not apply in a sense.

I do however intend to explore a multitude of tuning systems and approaches to tuning in my music here, as well as hopefully open some dialogues about tuning, in its practicality, aesthetic, notation, and possibilities.

Some terminology:

edx: equal divisions of x

ed2: equal divisions of the octave (2) .. this is due to the octave being defined as a doubling of a frequency

15-ed2: 15 equal divisions of the octave

just intonation: a tuning system based on intervals which can be described as ratios of whole numbers.   3/2 as the perfect 5th, 5/4 as the major third etc

temperament: a tuning system which equates certain intervals in just intonation so as to manifest different properties.  ie: in meantone temperament the pythagorean major third 81/64 and the just major third 5/4 are equated, removing the syntonic comma of 81:80

from Scott Dakota of the XAII:

“Starting from the beginning with what “temperament” means:

1) We assume pure whole-number frequency ratios as a default or beginning state for our musical intervals. A classic example is “the 5th” in common western musical terms, which is a pure 2:3 frequency ratio in just intonation terms. One can easily tune pure (beatless) 2:3 by ear, and then tune them in chains as long as one likes.

2) With the pure 2:3 ratio, if you tune a chain of those out to 12 notes, the 12th note is *near* being the same pitch as the starting note, but not quite. So one does not actually get the closed cycle of 5ths that people are used to with common 12-equal-temperament, when using pure 2:3 intervals.

3) But if one makes all of the 2:3 intervals slightly flat of pure (about 2 cents flat, a 50th of a common half step), then the 12th note does indeed come back around in the cycle, and is the same as the starting note.

4) To choose to alter a pure frequency ratio beatless musical interval (just intonation) a bit flat or sharp so as to reach some other pitch target, like closing a cycle of 5ths, this alteration is “tempering” pure ratios in service of some overall goal.

5) To give a complimentary example, if we make each 2:3 5th around 3 cents sharp each, we get a different cycle that closes and meets itself at 17 notes. And that is 17-equal-temperament.” – Scott Dakota

 

comma: the difference between two just ratios

for more please visit: http://lumma.org/tuning/faq/

 

Some questions I hope to answer:

  1. Can small edx visualizations and ear-training allow for performers and composers to more easily use complex microtonal tunings?   /-/  We already utilize 2-ed2, 3-ed2, 4-ed2, and 6-ed2 as symmetries in our playing.  Would extending this to 5-ed2 and 7-ed2, or a bit further give us potential to divide the octave and other tones in a way that time signatures based in 5 or 7, or polyrhythms as such are common.
  2. Can the study of small equal divisions and regular temperaments lead to a more precise description of functional harmony?  
  3. Does functional harmony exist in the sound continuum?  Is functional harmony related to just intonation or to temperament (or both, or an interaction of both)?  Is functional harmony related to the prime-composition of a rational interval (or an approximation to this)? 
  4. How do approximations/temperings/sense play a role?

Essentially, we would like to: create a notational system which can incorporate any possible system of tuning and be flexible to include further developments as they occur.  We would like this system to be intuitive to sight-singing and to pattern recognition.  We would like this system to have some semblance if possible to conventional notation systems.  We would like this system to be precise and flexible with minimal accidental, but maximal functional properties.

I believe this can be solved through a modulatory perspective.  To state a piece is initially in 15-ed2 with a tonic at A440hz could for example imply a specific notation to a diatonic scale… or to a Porcupine (http://xenharmonic.wikispaces.com/Porcupine) or Blackwood (http://xenharmonic.wikispaces.com/15edo) scale, or possible a 5-ed2 scale, for these are the most fundamental bases of 15-ed2.  With these we could also specify a degree of accuracy in which tuning would be related to just intonation, for example, in a 15-ed2 piece in Porcupine, should perfect fourths be tuned closer to the just 4/3 when possible, or should they maintain strict 480¢.  It is my believe that the understanding of these relations is ultimately much more simple and fulfilling for both performer and composer than a system of an incredible amount of accidentals or perpetual pitch adjustments notated above known pitches.  Those systems are only momentarily utilitarian, but eliminate the performers possibility to play based on contextual relations and knowledge and are forced to be arbitrarily and often inaccurately precise.

Comments and discussion more than welcome

mahalo

n