# twenty three x

The harmonic series can be thought, in one perspective, of as the sequence of real numbers.

1 2 3 4 5 6 7 8 9 10 11 …. ∞

Mathematically, the ratios of the frequencies between any two of these harmonics are exactly the ratio between the numbers.  For example, the ratio between the “first” harmonic (the unison) and the “second” — 2, the octave, is exactly 1:2.  The ratio between the 7th and the 11th harmonic is 7:11.  As just intonation ratios this is also notated as 11:7 or 11/7.

Powers of 2 are significant.  2 4 8 16 32 64 128… Powers of 2 are also each sequential octave (in the harmonic series, or otherwise).

This means we get these harmonic “realms”

1 2

2 3 4

4 5 6 7 8

8 9 10 11 12 13 14 15 16

16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32

I colour-coded harmonics as they show octave equivalence in the next realm.

Composite numbers such as 9 = 3*3 are both generated as a member of the harmonic series, being part of the sequence of whole numbers.  And are the combination of two or more harmonics.  In this case, the 9th harmonic to a frequency is the same as the 3rd harmonic applied twice to that frequency.

We can re-write segments of the harmonic series to be within an octave.  This is useful because it is a familiar and practical and perceptually relevant way to categorize a group of pitches.

Octave equivalence is created by multiplying or dividing any frequency by 2.  The 6th harmonic, 3*2, is an octave higher than the 3rd harmonic.

Since these harmonic realms are all in higher octaves, we can octave reduce all the frequencies by powers of 2 (some amount of octaves) to bring them all within the root octave or home octave.

1 2 just becomes 2 unisons

2 3 4 becomes 2/2, 3/2, 4/2 = 1/1, 3/2, 2/1 — this is the unison, perfect 5th, and octave

4 5 6 7 8 = 4/4, 5/4, 6/4, 7/4, 8/4 = 1/1, 5/4, 3/2, 7/4, 2/1 — this is a major triad with the harmonic 7th (a bit lower and more resonant / less beating than the minor 7th)

8 9 10 11 12 13 14 15 16 =

8/8, 9/8, 10/8, 11/8, 12/8, 13/8, 14/8, 15/8, 16/8 =

1/1, 9/8, 5/4, 11/8, 3/2, 13/8, 7/4, 15/8, 2/1

You can continue this pattern to the next harmonic series segment utilizing the denominator of 16, 32, 64, and so on.

But what about denominators that are not powers of two?  What do they mean?

Notice that the difference between the 11th harmonic and the 13th harmonic is 13/11.

So if we want to build a set related to the harmonic series, in particular the 11th harmonic, say starting from the 11th harmonic.  Lets define it at the unison 1/1.

Then the relationship between the unison (the 11th harmonic) and the 12th harmonic is 12/11.

So lets build a set

1/1, 12/11, 13/11, 14/11, 15/11, 16/11, 17/11, 18/11, 19/11, 20/11, 21/11, 22/11 = 2/1

So this set is exactly the set of harmonics from the 11th to the 22nd defined in relation to the 11th harmonic.  This is an overtone series.

We can also build a set of a constant numerator.  For example 23.  Say the 23rd harmonic.  Then 23/22 will be the distance from the 22nd harmonic to the 23rd harmonic.  And 23/11 will be the distance from the 21st harmonic to the 23rd harmonic.

Lets build a set.

1/1, 23/22, 23/21, 23/20, 23/19, 23/18, 23/17, 23/16, 23/15, 23/14, 23/13, 23/12, 2/1.

This is an undertone series of the distance between the 23rd harmonic and the harmonics from 12-22.  This is not a descending series however as 23/22, 23/21 etc are increasingly large intervals, but the description of the relationship is that of harmonic relations which are below 23.

At this set is described as 23/x (with x between 12 and 22), it can be spoken as twenty three x.

Take a listen to a recent piece composed in this 23 set 🙂

https://www.patreon.com/posts/among-dreams-23-16966657

Please consider supporting if you like the music and/or way of sharing and explaining ideas presented here 🙂 ❤

# 169/144 and 169/128

169/144 is the difference between 16/13 and 13/9.

This is 277¢, which happens to be very close to 7/6 (which is 267¢)

6/5 (minor 3rd) is the “next” higher interval (at this resolution) at 314¢

Normally the range is expanded by 8/7 (231¢) and 9/8 (204¢ major 2nd), as opposed to between 7/6 and 6/5, with which 13/11 is usually common (as 7/6 < (7+6)/(6+5) < 6/5 and (7+6)/(6+5) = 13/11.

13/11 is 289¢

and then the different between 16/13 and 13/8 is 169/128 which is 481¢, a nicely flat 4, quite sharp from the supermajor 3 of 9/7 that is 495¢

What makes this interesting more is that 16/13 and 13/8 are inverses as 16/13 * 13/8 = 2

On the Rhodes I have retuned, this 1/1 is D, and the 16/13 is the F#, the 13/9 the G# and the 13/8 the A#.   This is based of the idea that this pentatonic scale of F# (major pentatonic scape) has a very tonicizing property to it, even though the 1/1 is not played in the set.

The full note set being 1/1, 14/13, 15/13, 13/11, 16/13, 13/10, 13/9, 3/2, 13/8, 22/13, 26/15, and 13/7.

So, the F# pentatonic set of F# G# A# C# and D# are

14/13, 16/13, 13/9, 13/8, 13/7

D#,      F#,      G#,      A#,   C#

between D# and F# is 8/7

so the step sizes are, sequentially: 8/7, 169/144, 9/8, 8/7, 196/169 (between C# and D#)

196/169 is 256¢, also very much between 8/7 and 7/6

Hear this pentatonic scale at the end of the end of the 13-limit demo posted recently at https://www.patreon.com/noahdeanjordan and also the full scale, this time in “”Cminor”” at the Tribute to Phillip Glass recording (which will eventually be completed more fully with this scale)

# Instruments – Musicians – Chicken – Egg

So as the chicken was an ancestor to the dinosaur, so the egg predated the chicken, thus perhaps the vocal cords predated any intentional construction of a pitched instrument — though as natural sounds contain harmonic spectra, our control over pitch and timbre and faculty has been a growing endeavour.

We hear often natural ability on instruments, as well as the idea of music coming from the heart of the musician, and that many great musicians do not or need not “understand” the workings of music theory or the chords that they play.  Firstly, whether or not a great musician can understand an instrument as past scholars have described it, obviously they understand the workings of it deeply.  Natural ability and heart manifest and can be described in many ways – but one aspect of this particularly is what I want to discuss.

If musicians are meant to be vessels for which music can be expressed, is there an innate sense of musicality in these people – something which must transcend our European tuning system of modern history .. as of course this type of musicality transcends this time and cultural constrain.  Or is the instrument the vessel, with which a person with sufficient expressiveness or creativity, and sufficient technical competence will generate beautiful patterns from a frequency set through rhythmic space?

Evidently, the role of instrument designers has been incredible in the shaping of musics around the world.  The inter-relationship between the builders, the theorists, the musicians, and the composers is very intricate.  I am excited to these the development of this in the decades to come.  More thoughts on this eventually.

# Where does microtonality start?

In response to comments of David Dornig of Dsilton,

“…That is certainly important to explain to people who think that microtonality is only valid, if you could not approximate it in 12EDO or meantone temperaments. But than we must ask, where dose “microtonality” start. Whats the reference? Can it be objective or is it only a matter of the concept behind the music. Is a flat intonation of a blue note microtonal or not if one thinks in reference to 12 notes per octave? If those questions are settled, it would be easier to discuss.”

Where does microtonality start is a very interesting question.  Maybe lets start with a sub-question of this, which is “where is a microtone” defined.  In some of the discussions at miCROfest 2017 in Zagreb, it was said that a microtone is anything less than 100¢.  I believe that this definition is based on the visualization of 12-equal as a basis of tuning, which is subsequently a consequence of the atonal and serial conceptions of music.  This is not so much the size of the semitone, but the semitone being used as a measuring tool, as opposed to a difference tone between intervals with a small tone between them.

So there are a few places that microtonality could start.

1. Music using steps smaller than 100¢
2. Music that utilizes a greater variety of difference tones / enharmonics than are available in 12 equal music.  – this may include some historical tunings and well-temperaments
3. Music that utilizes a higher harmonic spectra or limit, for example: utilizing the 7th harmonic functionally within the music

We also have the question of note bends in reference to 12edo and if they are microtonal or not.

I think that this goes back to the idea that “microtonality” implies a subcategory or music, where as 12 tone music can be interpreted as one of many choices of tuning of music.

Therefore, I don’t really believe that microtonality actually exists outside of an atonal sense, or, to extend, outside of a harmonically functional sense.  For example, in quarter-tone music, microtonality may exist in many compositions as the quarter-tones are being used as non-harmonic or non-functional sounds in the music.  I believe, in a strict sense of nomenclature, true “microtones” must be functional only by means of symmetry or serialism.

I believe that the broader, often “xenharmonic” principles, reflect more the aesthetic of fully studying the properties of the pitch spectrum and the totality of their relations.  In the set of the pitch spectrum, 12 equal is one approach to tuning which fulfils a set of useful properties.  These properties can also be fulfils by a vast number of other tunings, and also, each tuning will not overlap with another tuning completely in all categories.  So, therefore, we must choose a number of properties that we would like to fulfil, and decide to which degree we want to accommodate each property.

1. To what degree can a major triad be “out of tune” before it is no longer a major triad, and how context dependent is this?  For example, we accommodate a 400¢ major third quite nicely (and this is made more difficult by the claim that the 81/64 pythagorean major third is in fact the functional third and not 5/4), and in 15edo we have the same third but a similarly sharp 5th (at 720¢), why is this 5th so much more undesirable than this 3rd?
2. To further this, is accuracy of tuning importance to follow order of harmonics?  ie. octaves pure, 5ths near pure, 3rds close, 7ths existent, 11ths + irrelevant.
3. How much is “out of tuneness” perceived relative to the system in place?  ie. the blue note in 12ed2 vs the 240¢ interval in 15ed2.

until next time

please check out some of the 31-tone music of Dsilton

# Rant on Optimization and microtonal misconceptions

So I have found a train of comments of “microtonal” music that have a certain tendency, that while it is coming from a good place, I believe really just shares a different aesthetic of music than the goal of many composers with interest in tuning.

I will try to explain these two points of view, and they are easiest described as the optimization path and the colour path / free path.

The optimization best often hails 12edo to be the best for most circumstances, with the sometimes allowance for 19/31/53 etc, other high accuracy/resolution tunings, so long as the music uses these new intervals in a way dictated the the classical meantone tradition.  This train of thought believes that tuning to near Just Intonation / tuning accurately to harmonic approximations is the goal of a tuning system.

There are a few problems with this.  Firstly, of all the well-tempered systems for classical era music, there is obviously no “best” system, and each has its own way of dealing with progressions, especially in the context of the music of the composer and piece.  Secondly, if a perfect system is truly desired, we have adaptive just intonation, which is exactly that; the problem being the difficulty of application in many settings.  Thirdly, any high-resolution tuning is really just a combination of the two previous problems.

On the other hand, to choose tuning systems not on an overall optimization, but on compositional preference of the notes and patterns existent in a tuning, has no need to fulfil any requirements other than to be as it is.  This method also denies the optimization is really a possibility, as essentially each system has its own parts which are optimized.  Which do you want to be optimal?  Is this always in order from lowest to highest harmonics?

Yes, many things in 15 or 17 etc can be played in 12, and in 12 is sounds debatably “better”, but only if lack of beating is what qualifies as “better”.  Each tuning has a quality of its own, and “improves” certain notes, and really always at the expense of others.  This was the whole idea of well-temperaments in the past, and why there were so many.  The ability to choose the tuning of a piece, allows the composer to choose which colours to use and how they will relate.  We no longer need to base our aesthetic judgements on tempering the 5th to the 3rd.  Of course we can, and we can expand this as we wish.  But we are asking for infinities now.

# 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

https://www.patreon.com/noahdeanjordan

# 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: https://www.patreon.com/noahdeanjordan

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.

4:5:6

and

10:12:15..

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 http://metatonalmusic.com/, or the guitar work of Fernando Perez http://www.fernandoperezguitar.com/ ).  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:

and for more videos coming every week or two please subscribe at:

https://www.patreon.com/noahdeanjordan