My Theme from "Invisible Haircut" is simply a 6-measure phrase, with the bass repeated in the manner of a passacaglia.

In 1993 I retuned it to just-intonation, without any microtonal instrument available, doing all the work "theoretically" on paper. That year, my friend Jeff Morris was producing and directing an off-off-Broadway play he wrote, called Invisible Haircut. He commissioned me to compose incidental music for the play, and so I adapted this piece, in its JI version, using this as the Main Theme, and other related pieces for other parts of the play. It was performed in New York in December 1993. Here is a YouTube video of Invisible Haircut in 19-limit JI, rendered with my Tonescape software.

In June 2023 I made a new version of this piece, retuned to 41-edo, which gives excellent approximations to the primes I used in this JI version, and provides an excellent example HEWM notation adapted to 41edo. Here is a YouTube video of Invisible Haircut in 41edo-HEWM.

There is a brief passing-tone chord near the end of measure 2, which I left in 12-Eq on this sequence. It lasts such a short time that the intonation isn't really noticeable.

JustMusic analysis (m = measure):

[In this analytical method, utonalities have the numerary nexus above the dividing-line with the udentities below, and vice-versa for the otonalities.]




m 1	m 2	m 3	m 4	m 5	m 6
C n0 -----	F 3-1 ------ [12-Eq E] -----------	Eb 33 5-1 71 ---------------		F 33 5-1 ----------
1 5 3 15 3 9	5 3 1 5 15 9 5 3 1 5 15 9	5 3 1 5 15 9	11 19 7 5 3 1 19 13 5 7 1 1	3 1 5 15 9	9 5 13 1 5 3 7 5 1 1
			-------------------- F 32 5-1 71 19-1 ----------- Bb 32 5-1		----------------- Ab 5-1

In Partch's terminology, the letters-with- numbers and exponents below or above the line give the 1-identity of the Otonality or Utonality, respectively. He would call these "roots" as follows:

m 1  C    1/1  -Utonality
m 2  F    4/3  -Utonality
m 3  Eb 189/160-Utonality
m 4  F  126/95 -Otonality
     Bb   9/5  -Otonality
m 5  F   27/20 -Utonality
m 6  Ab   8/5  -Otonality

The stack of numbers alone, on the opposite side of the line, are the identities present in the chord. I find my notation much simpler than Partch's. If you understand what I wrote above, then you could easily reproduce the tune. It's also easy to visualize the pitches on a lattice with my notation.

I wrote it originally in 12-Eq, then figured out common-tones on paper, which is how I got those strange F and Bb chords in measure 4. I've tried this kind of thing for other people's (older) music and it didn't work, because the high-prime common-tones throw the chord-roots off into an odd-sounding high-prime key, but surprisingly here, it sounds great!

Part of the reason why I left that passing-chord in measure 2 in 12-Eq is because, by use of all the common-tone relationships in the following chords, I had to find a "break" in the tonal fabric somewhere, and I decided to do it with the parallel descending chords going from m 2 into m 3. So the chord in m 3 is the only significant chord in the 6-measure phrase that's not closely related to the preceding chord. Making the passing chord also unrelated (by leaving it in 12-Eq) helps to mask the "break".

There are some pretty "far out" chord changes in there, and with the employment of the 5-limit xenharmonic bridges ("unison vectors" as Fokker called them) they could be well represented in a simple 5-limit tuning!

Whether in 19- or in 5-limit, it sounds MUCH better in JI than in 12-Eq. The JI versions have a richness that is entirely lacking in the 12-Eq version. When you hear the JI first, the 12-Eq simply falls flat, so to speak.

graph of pitches

(Chords from Piano part only - no bass line)



---

Joe Monzo: Invisible Haircut
lattice diagrams

Showing each of the main chords in red.

Click here to open a window with an animated applet of these lattices.



Click here to open a window with an animated applet of these lattices.

---

Some further observations on this piece:

• Me, Mills College TD 1415.17, Thu, 14 May 1998 16:33:41 -0400

  ET vs JI
  monz@xxxx.com (Joseph L Monzo)
  
  5/14/1998 1:33:41 PMCarl Lumma wrote:
  
  > I've always viewed these "commas" as making modulation
  > more interesting. But many disagree. Partch's chapter in
  > Genesis of a Music is really great about this point (the one
  > with the letter from Fox-Strangeways). Boy that chapter
  > is really a thrill!
  
  > Basically, what I got from this chapter is that modulation is
  > best defined simply as switching the 1/1, and that common
  > tones, while playing, of all things, perhaps the most important
  > role in the use of modulation, are not necessary in its definition
  > or execution. And perhaps, that a theory of modulation may
  > be constructed where tones separated by a comma can still be
  > considered "common"!
  
  I've written about this same passage in my book. Partch describes
  three ways of modulating, assuming two chords which possess
  a "common tone":
  
  1) Making the "common tone" consonant with the first chord and
  dissonant with the second.
  
  2) Making the "common tone" dissonant with the first chord and
  consonant with the second.
  
  3) Making the "common tone" actually two different tones which
  are close by in frequency (differing by said "comma") and
  which are each consonant with their respective chords.
  
  His conclusion, with which I agree, is that all three methods can
  be used to effect a modulation in just-intonation, giving a richness
  and subtlety of expression which is _utterly non-existent_ when
  utilizing the 12-Eq scale to present the same musical passage.
  
  I have some interesting observations of my own in this regard,
  involving not full-fledged modulation, but rather short-term
  tonicization:
  
  1) I have tried sequencing Mozart's 40th Symphony in several
  different versions, using ratios that were 5-limit, 7-Limit and
  19-Limit. 5-Limit sounded best, 7- and 19- were both OK, but
  when tonicization was effected by a common-tone related by
  7, it didn't sound right. It sounded to me like the tonality veered
  off into a weird key that was microtonally "off". This argued
  against the applicability of 7 in Mozart's music.
  
  2) In my own "Incidental music to 'Invisible Haircut'", I use
  tonicizations
  which have 19 as a factor in the common-tone. This piece has
  a jazz/ blues flavor, and I find that the 19-limit tonicizations work
  well (they certainly provide a richness that the bland 12-Eq version
  lacks completely). This may, however, be because 19/16 is so
  close to the 12-equal "minor 3rd" that the _interval of modulation_
  is not so strange to my ears. I'll grant that it's possible that, had
  I
  used it in the tonicization, 7 may have sounded just as strange in
  this piece as in Mozart.
  
  3) I sequenced some of Satie's "Sarabande No. 1" in just-intonation,
  and found that tonicizations involving 7 sounded "off" here too,
  tending to corroborate what I said in #2.
  
  My original point was that no matter how well any equal temperament
  represents whatever ratios, there's no substitute for the richness
  and subtlety of expression which is possible when using ratios
  themselves. Numbers can be compared in all sorts of different
  patterns and combinations, and when these numbers represent
  _easy-to-hear_ musical relationships, the variety of musical
  relationships is correspondingly expansive. I'll grant that equal
  temperaments are easy to hear in melodic terms, but, aside from
  the ratios they imply well or badly, they don't have much
  significance from a _harmonic_ standpoint.
  
  Part of the problem I have with 12-Eq serial music is that I just
  can't hear many of the supposed relationships in the music that
  have been pointed out by theorists. I will admit the possibility that
  things that are happening in music that are numerically related
  but are not consciously audible may still have some kind of effect
  on our nervous system, but at our present state of theoretical
  knowledge, I think it's best if we deal in terms of what can
  demonstrably be _heard_.
  
  I think a large part of the reason Schoenberg stuck with the
  12-Eq scale was because he realized intuitively that within the
  vastly expanded resources of his implied 13-Limit, were he to
  work in just-intonation, the number-play involved could quickly
  become a bottomless pit from which his musical inspiration would
  never again emerge into the light of day. Then again, I also
  think he wanted to make use of the ambiguities made possible
  by a comparison of so many close ratios on the one hand and
  their nearby 12-Eq equivalents on the other.
  
  My whole idea of primes having distinct qualities is useful
  compositionally when, for example, using a precisely-tuned
  81/64 "Pythagorean major 3rd" or 9/7 "septimal major 3rd"
  instead of the usual 5/4 "just major 3rd", or when sliding
  around between them, as good blues singers do, to create
  a specific effect that 5/4 just doesn't give.
  
  I refuse to accept an equal-temperament because it makes
  modulation "easy" or (excuse me while I laugh) "possible".
  Part of the reason JI composers use JI is because, within
  a restricted JI scale, modulation to a new key brings a
  whole new set of intervallic relationships into play, giving
  each key its own distinct "flavor", far more pronounced
  in their differences than anything a well-temperament can
  do.
  
  The only reasons I can see for accepting any equal-temperament
  is that it is easier to play on most instruments, and, as I stated
  in another post to this issue, the study of the interplay between
  JI and ET in the same piece is becoming more and more interesting
  to me.
  
  Joseph L. Monzo
  

  ==================================================================
• Erlich, http://groups.yahoo.com/group/tuning/message/1788 Wed Mar 17, 1999 2:07 am

  authorName: Paul H. Erlich"
  from: Paul H. Erlich; PErlich@xxxxxxxxxxxxx.xxxx",
  subject: Re: webpage of my JI tune
  msgId: 1788,
  
  Joe,
  
  I really like this tune! It is magical how JI transcription happened to
  "work" on this one. It would be nice to see a post on your thought
  processes on figuring out the common-tones, etc. This type of
  transcription, with 11 and 13 and 19 identities, leads to a pronounced
  "periodicity buzz" in the otonalities that is radically different from
  the jazz aesthetic but has a flavor that can certainly grow on you (Kami
  Rousseau had some similar transcriptions on his web page, Kami, are you
  still with us?)
  
  Also, the progression from the 126/95-Otonality to the 9/5-Otonality is
  quite nice as a V-I, even though the root rises by a fourth of 529 cents
  (or drops by a fifth of 671 cents). This underscores what I was saying
  to Dan Stearns, that a fourth of 533 cents can be acceptable
  melodically, and that applies at least as well when the "melody" is a
  bassline.
  
  At 2am, it really is nicer to hear music than to look at theories! Joe,
  can you help me MIDI-ize a 22-tET piece of mine (the one I keep talking
  about), and maybe we can create a JI rendition too? Peter Blasser, if
  you're there, I successfully got csound as you directed, but couldn't
  get Rocky to work. My home e-mail doesn't seem to be working, so can you
  reply to me here? Thanks!
  
  -Paul
  
  ===================================================================
  

• Me, http://groups.yahoo.com/group/tuning/message/1798 Wed Mar 17, 1999 1:02am

  Yahoo! tuning list
  msgId: 1798
  
  Wed, 17 Mar 1999 04:02:13 -0500
  From: Joseph L Monzo  I really like this tune!
  
  Thanks, Paul.  Glad you like it so much.
  
  It's been so long since I wrote the tune,
  I really don't remember what guided me in
  choosing the JI pitches and common-tones
  I chose.  I'll try to reconstruct if I can.
  
  First of all, it's very perceptive that you
  call it a transcription - I don't recall saying
  that anywhere on the webpage.  But it's true,
  I wrote the tune on a little battery-operated
  Casio keyboard, in 12-Eq of course, then calculated
  on paper how I wanted it tuned.
  
  This was before I had formulated my lattice theories,
  so I was just multiplying fractions Partch style.
  Having 19 in there made me do a little work.
  
  To start with, I had what were clearly "minor 9th"
  chords in measures 1, 2, 3, and 5.  I pretty much
  trivially decided to make these 5-limit chords, of
  the "minor 7th" type we've been discussing here recently,
  with the 9th added:  10:12:15:18:45.
  
  I knew that for measures 4 and 6 I wanted to use
  otonal "dominant type" chords, and that I wanted
  11-, 13-, and 19-identities in those chords.
  
  I mention on the webpage that there is a "break"
  between measures 2 and 3, where a descending passage
  occurs too fast for a tuning to be recognized.  I took
  advantage of this because I assumed before I started
  that in JI, with "interesting" chords and lots of
  unusual microtonal intervals, I'd probably end up
  modulating a comma or two off by the end of the phrase,
  and it repeats, so I had to work around that.
  
  So I started at the end, the 8/5-Otonality, and worked
  backwards from there.  It was mostly just trial and
  error.  I remember writing out at least 4 different
  tunings, probably up to 8 for some chords.  I just
  tried using all the different common-tones I could,
  and played around with the numbers until everything fit.
  
  I say there was a break between measures 2 and 3,
  meaning that there was no common tone as it is a
  descending parallel chord passage.  So I left the
  quick passing chord in 12-Eq, and everything worked
  out.
  
  I didn't actually hear it until a few years after I
  wrote it, when I got my [Yamaha] TG-77.  I've always liked
  the JI version much more than the 12-Eq.  The 12-Eq
  sounds totally bland by comparison.
  
  Also, when the Grand Piano patch plays, the high-limit
  ratios blend in such a complex interesting way . . .
  I suppose that's the "periodicity buzz" you mentioned.
  
  This tune was sequenced in Cakewalk 2.0, the tedious
  way - by putting the proper amount of pitch-bend on
  every note.  I should remove all the pitch-bend and
  put the 12-Eq version on the webpage too, for comparison.
  Hmmm, maybe versions in several different tunings...
  
  I'd definitely like to help you do your 22-Eq song.
  Send it over.  I'm trying to get my copy of Rocky
  working too.
  
  -monz
  
  =====================================================================
  

• Lumma, http://groups.yahoo.com/group/tuning/message/1804 Wed Mar 17, 1999 6:11am, Onelist TD 108.5

  Yahoo! tuning list
  msgId: 1804
  
  Date: Wed, 17 Mar 1999 09:11:40 -0500
  From: Carl Lumma I really like this tune!
  
  Isn't it killer?  Have you heard his 24Tune?  It's on his webpage under
  list of works.  I actually like it better than invisible haircut.
  
  
  >It is magical how JI transcription happened to "work" on this one.
  
  Ackg!  JI can always work this well, no magic.
  
  
  >This type of transcription, with 11 and 13 and 19 identities, leads to a
  >pronounced "periodicity buzz" in the otonalities
  
  This is the most pleasant thing, isn't it?  It is what I call "florescent
  lightingness" at the 7-limit.  It is the first thing to leave a just chord
  with detuning.  Only 34 and 53, of the ET's I considered, have it for the
  4:5:6 chord.
  
  =======================================================================
  

• Me, http://groups.yahoo.com/group/tuning/message/1842 Thu Mar 18, 1999 6:45am, Onelist TD 110.9

  Yahoo! tuning list
  msgId: 1842
  
  Date: Thu, 18 Mar 1999 09:45:13 -0500
  From: Joseph L Monzo  Joe Monzo's Invisible Haircut
  
  [Erlich:]
  > I really like this tune!
  
  [Lumma:]
  > Isn't it killer?
  
  Thanks, guys.  I really appreciate the kind words.
  
  We've been slammed for "so many numbers, so little music",
  and I feel that to a large extent I'm dropped into that
  category, so I figured it was time I had better get some
  good microtonal music "out there".
  
  [Lumma:]
  > Have you heard his 24Tune?
  > It's on his webpage under list of works.
  
  > I actually like it better than invisible haircut.
  
  You told me you like the "24-Eq Tune", Carl, but
  I'm really surprised that you don't like "Invisible
  Haircut" more.  Not that I'm trying to slight the
  other tune, but I wrote that mainly because I felt
  24-Eq had been criticized "unjustly", and I took
  it as a challenge to write something that sounded
  good in that tuning.
  
  "Invisible Haircut", on the other hand, was really
  an inspiration.  It just popped into my head and/or
  fingers one day, and immediately, as soon as I'd
  played it, I could imagine all the different just
  ratios swirling about.
  
  [Erlich:]
  > This type of transcription, with 11 and 13 and 19
  > identities, leads to a pronounced "periodicity buzz"
  > in the otonalities that is radically different from
  > the jazz aesthetic but has a flavor that can certainly
  > grow on you (Kami Rousseau had some similar transcriptions
  > on his web page . . .
  
  I'm quite happy with the way my "justification" of it
  turned out.  The "periodicity buzz" in that V/V - V - I
  \progression really does it for *me*!
  
   Also, the progression from the 126/95-Otonality to
  > the 9/5-Otonality is quite nice as a V-I, even though
  > the root rises by a fourth of 529 cents (or drops by
  > a fifth of 671 cents).
  
  These are the two chords that really display
  that "buzz".  I thought I'd go into a little
  more detail about Paul's observation about the
  size of the intervals.
  
  I'm adding the following "minor" chord into this
  description.  Altho my harmonic analysis (on the
  webpage) of the "Eb minor" chord is as a 27/20-Utonality,
  it can also be thought of in the traditional sense
  as a "minor" chord built upwards on "Eb" 6/5.
  
  Here's a diagram, in cents (rounded to 2 decimal
  places), of the "root" movement of these three chords,
  and the intervals of both the V-I skips, and the
  stepwise movement in the (Schenkerian) "prolongation"
  from the 126/95 to the 6/5:
  
  
  126/95        9/5          6/5
  488.91      1017.59       315.64
    | \        /     \        /  |
    |   671.31         498.04    |
     \________ 173.27 __________/
  
  
  I believe the reason this works so well
  is because there is a "xenharmonic bridge"
  at play here.  The difference between
  126/95 and 4/3 is the interval 190/189
  [= 9.13+ cents].  If we analyze the intervals
  in the above progression with 4/3-Otonality
  substituted for the 126/95-Otonality, we get:
  
  
  4/3           9/5            6/5
  498.04      1017.59        315.64
    | \        /     \        /  |
    |   680.44         498.04    |
     \________ 182.40 __________/
  
  
  Although it's not your usual Pythagorean [3-limit]
  bass-line, this is a "root"-movement that would be
  familiar to most people, since it occurs in the
  (implied) JI/meantone diatonic scale.
  
  The II (supertonic - "minor") is *the* problem chord
  in (implied) 5-limit diatonic music.  Sometimes the
  II-degree has to be tuned to 9/8 to be consonant with
  the Dominant (V), sometimes tuned to 10/9 to be consonant
  with the Subdominant (IV).
  
  Notwithstanding Paul's observation about how my chords
  here are "radically different from the jazz aesthetic",
  jazz harmony was very much how I was thinking when I
  wrote this tune.
  
  In jazz , the standard chord progressions are strings
  of V-I's, and that's what I did in this little section.
  Thus, we can assume a local tonicization of the "Eb" 6/5
  in the above progression.
  
  In the "key of Eb 6/5", the "I" is 6/5, the "V" is 9/5,
  and the "II" (the "V of V") is, in this case, 4/3.
  
  So I think that it's entirely possible that we're
  "hearing" 4/3 for 126/95, as well as perceiving a
  more familiar 40/27 skip for the 28/19 actually heard
  between "F" and "Bb", and a 10/9 step for the 21/19
  actually heard between the "F" and "Eb", by making
  use of the 9-cent xenharmonic bridge from the 3- to
  the 19-limit.
  
  BTW, I didn't plan any of this - it "just" worked
  out that way.  I figured out the JI "translation" years
  ago by writing out pages full of common-tone chord
  progressions.  It was only much later that I
  "saw that confounded bridge".
  [pun intended, for Led Zeppelin fans]
  
  I wish I could draw a lattice of it here to simplify
  the explanation (and because it's *beautiful*!),
  but 19-limit is too complex (and too big) [to draw in ASCII].
  
  Other small 3--19 bridges were implied in the
  Chromatic and Enharmonic genera of the ancient
  Greek Eratosthenes, c. 200 BC, and in the Chromatic
  genus of Boethius, c. 505 AD, and have also popped
  up in my explanation of Marchetto's "fifth-tone"
  theories, c. 1318.  (c. - I mean see, my website)
  
  I'd like to point out that, tho I already admitted
  this was a transcription (since it was first composed
  on a 12-Eq keyboard), the sound of the JI version you
  hear was very much in my head from the moment I wrote
  it.  This is the case for a lot of 12-Eq music I've
  written over the last ten years.
  
  The entire reason I developed my lattice theory is
  because I despaired of having an instrument that could
  give me all the pitches I wanted, and the lattice
  diagram model was the best way for me to understand
  and internalize the musical relationships.
  
  Associating the visual mapping with the sound of
  the intervals was the only way I could make sense
  of the vast array(s) of ratios in the musical fabric.
  So even as I write or play in 12-Eq, I'm hearing
  (in my head) ratios that I can visualize on the lattice.
  
  Having instruments available now in 17-, 19-,
  22-, and 31-Eq helps a bit, because at least it
  gives me some more of the sounds, but I still think
  in JI and sure wish I could afford a Microzone!
  (or even a Clavette!)  It would take a lot of the
  work out of what I'm doing . . . 
  
  ("justified" another of my old tunes last night)
  
  -Monzo
  http://www.ixpres.com/interval/monzo/homepage.html
  
  Listen to "Invisible Haircut" at:
  http://www-math.cudenver.edu/~jstarret/haircut.html
  
  ===================================================================
  

• Erlich, Fri Mar 19, 1999 1:08pm

  Yahoo! tuning list
  msgId: 1870
  
  Date: Fri, 19 Mar 1999 16:08:02 -0500
  From: "Paul H. Erlich" >It is magical how JI transcription happened to "work" on this one.
  
  Carl Lumma wrote,
  
  >Ackg!  JI can always work this well, no magic.
  
  If you read the "Invisible Haircut" page itself, you'll see that Joe
  says he could not apply this kind of JI trascription to works of other
  composers that he tried it on.
  
  ======================================================================
  

• Me, http://groups.yahoo.com/group/tuning/message/1888 Fri Mar 19, 1999 6:00pm, Onelist TD 112.12

  msgId= 1888
  
  Date: Fri, 19 Mar 1999 21:00:28 -0500
  From: Joseph L Monzo  It is magical how JI transcription happened
  > to "work" on this one.
  
  [Lumma:]
  > Ackg!  JI can always work this well, no magic.
  
  [Erlich:]
  > If you read the "Invisible Haircut" page itself,
  > you'll see that Joe says he could not apply this
  > kind of JI trascription to works of other composers
  > that he tried it on.
  
  I had been thinking about responding to this
  earlier and now you kind of pushed me into the ring.
  
  As far as JI working "magically" in my tune:
  I have to side with Carl on this.  I enjoy writing
  and especially improvising in different kinds of
  ETs and other tunings too, but with JI I can
  always find *exactly* the pitch I'm looking for.
  
  I should have been more specific when I talked
  about how this "justification" didn't work for
  other composers.  I've tried to use 7-limit ratios
  for Mozart and couldn't make it work, and there
  are 12-Eq tunes that I've tried to interpret
  rationally and I just can't do it.
  
  By it "working", what Paul and I mean is that
  in ETs you are able to write repeated phrases
  that cycle back to the same point, and in JI
  there are usually commatic shifts involved in
  repeated cycling.
  
  I suppose what the Mozart experiment proved is
  that Mozart was pretty much thinking in 5-limit/meantone,
  because higher-limit intervals just sounded "off".
  (I even tried 19s in Mozart)
  
  I think the main reason it "worked" in "Invisible
  Haircut" is because I unconsciously made use of
  the 3--19 bridge, so the root movement can be
  perceived basically as Pythagorean or 5-limit.
  The bridge is what made the magic.
  
  - Monzo
  

  ---

  Updated:

  2023.06.12
  2002.09.14
  2001.12.14
  2000.10.04
  2000.02.05
  1999.03.09

---

For many more diagrams and explanations of historical tunings, see my book. If you don't understand my theory or the terms I've used, start here or try some definitions.

I welcome feedback about this webpage: corrections, improvements, good links. Let me know if you don't understand something. return to my home page return to the Sonic Arts home page

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corrections, improvements, good links.
Let me know if you don't understand something.

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