St Joe and 2

I’m going to be posting today a cover of the song Saint Joe on the School Bus by Marcy Playground.

The song’s verses have a simple F#m A E progression.  But when playing it in 15 you have to decide which F# to use (if you visualize it like me and take the open E as a reference point).  So in this tuning there are 3 types of 2nd, and essentially two types of major 2nd, one at 160¢ and one at 240¢, conveniently located exactly opposite the 200¢ of 12edo.

And actually, in the case of this song, the small (140¢) major second between the E and the F# is the choice I use because it shares the C# chord tone with the A.

What is more interesting also is that in just intonation, the big and small major seconds are 9/8 and 10/9, and there are combined to make the just major 3rd of 5/4  – two unequal step sizes.  This is in fact a property of 15edo: that you need a 240¢ and a 160¢ to get your 400¢ major 3rd.  In this case, it is quite exaggerated from the 9/8 and the 10/9 (~204¢ and 182¢ respectively).

In 15edo as in the just system, the bigger whole tone (240¢ and 9/8 in these cases) is the difference between the unison and the major 2nd, and the distance between the fourth and the fifth, and the distance between the major sixth and major 7th.  The small tone (160¢ and 10:9 in these cases) is the distance between the major 2nd and major 3rd, and the fifth and the major 6th.

Here is the song.  For more please see the Patreon page

Some thoughts on resolution

There are 3 types of 7th in 15edo, and as such, with the major 3rd and the lower 2 of these 7ths, we have 2 types of tritone. Both of these 7ths can resolve down 1 or 2 steps to a major or minor 3rd, as in 12 tone music.

Harmonically, when the neutral / middle 7th resolves down one step, the new root is 1 step higher than the previous. This means that the V is going to the augI instead of the I. This is balanced a bit by the fact that the V in this tuning is quite sharp.

I have made a video demo of this accessible from my page:

If all this is agreeable to you , we can then suggest that some of the following may be true:
1. the step size from the 7 of the V to the 3 of the I can be of a variable size, whether it resolves to a i min or a I maj.

2. the V chord could resolve to places other than the I or the vi

3. different tritones may have different tendencies

4. the just harmonic approximation of the tritone or the 7th may determine in part the quality of the resolution

– this last part begs a question:

In 15edo, the lower tritone is a 7/5 or 11/8 perhaps,

and the higher tritone a 16/11 or a 10/7, the undertone versions.

That means that in this tuning, the distinction between these tritone is not 7th harmonic vs 11th harmonic but overtone vs undertone.

On the other hand, the lowest of the 7ths in 15edo is a near 7/4, and the middle 7th a 11/6 or a 9/5 perhaps. And in any event, these are not inverses of each other, but definitely of different harmonic qualities.

So the questions are: does the harmonic limit of an interval affect the quality of the resolution? will the over/undertone nature of an interval affect its quality or functionality? how are these related? particularly if the tritone and the 7th within a resolution are of different harmonic limits. — furthermore, in this 15edo tuning we have a few possibilities for the interpretation of these senses:

1. In the 4:5:6:7 chord (major triad with the lowest 7th (7:4)) – we can interpret it as a 7:5 and a 7:4 or as a 11:8 and a 7:4… will this 7:4 predominance force us to hear the lower tritone as a 7:5? — YES, if this is a 4:5:6:7 chord. But if we don’t analyze this as a full form (because we are not in just intonation), the other approach could be possible.

side note:

to read 4:5:6:7, we are looking at a series of ratios

the ratio between the first two notes, the root and the major 3rd is 4:5 (5:4… 5/4)

the ratio between the 2nd two notes, the M3 and the fifth is 5:6 (6:5, min 3), and the ratio between the root and the fifth is 4:6 or 2:3 (3/2, P5)

the ratio between the final two is 6:7, between the 2nd and the final is 5:7 (7:5, the tritone in question),

and the ratio between the root and the final is 4:7, or 7:4 -> the note we define.
2. In the 12:15:18:22 chord with the neutral 7 (as a 11:6) on the major chord, we get the tritone defined in this sense as 15:22. If we want the large tritone to be an undertone, any of out close approximations of 10:7 or 16:11, we cannot fully analyze the chord this way.  The combination of an overtone and an undertone being analyzed in the same chord has difficulties.

This difficulty is shown in the different of complexity between writing a major and a minor triad.




what happens is you want to write 5:6 (for the minor third), but must make the 2:3 as well, and as 5 is not a multiple of 2, we must multiply 2 to all.


More thoughts into this to come..

A beast

In Zulu, harmony is called Isigubudu, which means “a beast with converging horns so that the horns touch the skin of the animal”.

It is said that harmony in music is of particularly emotional characteristic, it evokes pain or sorrow; it invokes the heart of performer and listener alike.

Melody in Zulu is called Indlela, which means a path.  This is something more linear or directional, or even solitary, small modifications to add colour to a journey.  When a second or third voice is added, these small modifications may add tension exponentially.

See: A study in Nguni and Western Musical Syncretism by Bongani Mthethwa, for more

In my opinion, it is particularly the harmony of the 5th harmonic, the major and minor 3rds and 6th, that add this emotive character to harmonious music.  I believe the higher harmonics, when used functionally and contextually suited for their roles, evoke another state of interaction with consciousness.  While it may still be emotive, the 7th harmonic does not pull the heart-strings in the same way at the 5th, is seems to me more of the realm of acceptance, of chaos passing through, similar to meditations on non-thinking, or non-interaction with thoughts as they pass through.

Of course, if the 5th and 7th harmonics are used together, this synthesis takes the relation to an even greater level.

And the 11th harmonic seems to transcend, and the 13th to hold a great mystery, the 17th and 19th to be like cogs or elementary particles holding the low structures intact.

“That one has no name, it has no use” – referring to a plant





Tokenism in Music

The idea of microtonal music is not necessarily to add some special frill with which to differentiate it with other music.  The heart of it is to open musical tuning from having one standard system to having an infinite amount of possibilities and musical systems.

To use an oud, for example, in a group with standard guitar and/or piano often will call for the oud to adapt to the tradition of the guitar or piano and thus lose the dynamic of the expansive range of scales and intonational dialogue that has developed over centuries of use of the instrument.  This is an example of what I feel to be tokenism in music, the use of a foreign instrument in music to add a foreign flavour too it, where as this method forced conformity rather than synergy.

This is or course partially related to the challenge of fixed-pitch instruments in performance.  A piano can not realistically be tuned in between performances, nor can a guitar change its fretting.  This is of course within current technical specifications.  Guitars with interchangeable fretboards already exist (see: Ron Sword’s Metatonal music, or the guitar work of Fernando Perez ).  It is also plausible to consider a piano with an auto-tuning mechanism, and of course synthesizers can easily be developed to be retunable.  Harps and zithers, whether modern or traditional, are fixed-pitch instruments which have been developed to easily change tunings between or within performances.

Jazz is an interesting example of a synergy in which a set of instruments were adapted to play a new style of music in which they were not used for previously.  The same could be said for string quartets playing rock music, or guitar adaptations of foreign or folk melodies.

It is not my objective to state that some type of musical tokenism is inherently wrong.  But more so to show that to reopen the dialogue of musical tuning can open many intercultural musical synergies which can create a greater amount of interplay and musical possibilities, both of the preservation of past traditions and of the creation of new musical styles.

Music often has very strong cultural connections and significances.  As the world becomes more globalized, it is important to consider certain effects of globalization and of multiculturalism.  Can a truly multicultural society still have a dominant culture, or a cultural hierarchy, or must deeper levels of integration be considered?

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.

For the video to hear this please check out:


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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 (, in 18-edo or 1/3 tone (Sonido 13), and The Devil (  I have been releasing one of the improvisations each week on my new Patreon page (, 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.

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:

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.


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