Numbers, numbers, numbers. A shaku is a Japanese unit of length, just a hair less than a foot. Traditionally, shakuhachi have been made in 1/10 shaku variations from, let's say, 1 to 3 shaku--the 1.8 shaku (THE shakuhachi) being the most popular and the unofficial standard.

Once Western music settled on 12 notes per octave, the question became how the twelve notes would be spaced. They could have equal spacing or various other schemes. The Even Tempered Scale, which won the day, employs a simple and unique spacing method--the twelfth root of two. What number multiplied times itself three times equals 27? It's three, the cube root of 27. What number multiplied five times itself equals 3125? Five or the fifth root of 3125. The spacing of the Even Tempered Scale reduces to finding what number when multiplied twelve times itself equals two. We want to fill the interval between one and two (an Octave) with twelve regularly, but not necessarily equally, spaced notes. That's what the twelfth root of two does. Numerically it equals 1.059463094. Multiply that number times itself twelve time and you'll end up with two. As the twelfth root of two steps through an Octave it slowly ‘picks up speed’. The first spaces are smaller than the last-- the first divisions are shorter than the last, but the ratio is constant.

Further, the frequency of any note on the scale (think piano) multiplied or divided by the twelfth root of two will be the frequency of the next or previous note. Thus any two keys which are 12 notes apart on a piano comprise an Octave. The Even Tempered Scale spacing is close to equal but slowly and steadily increases as the frequency rises. Its a percentage deal, each note is about 6% bigger or smaller than the previous or next.

With the tenth-shaku system each flute is 1/10 shaku longer or shorter than the next. We've got two systems going on here--one is mathematically linear (shaks) and the other isn't. But we can ask: At what point do the two systems coincide? This is the same as asking at what flute length does 1/10 shaku equal 5.9463% of the whole? It's the inverse of 6% or the 1.6817 flute. Either side of that length the two systems of measurement start getting out of step.

So the 1.7 flute is very close to the length where the two systems match. Let's double that length to 3.4. Should play one octave (12 notes) lower, right? But according to the shaku system it should be 17 notes lower. There's the rub--12 notes verus 17. The two systems match up best the closer you are to 1.7--the further away the worse.

Back to the Village. What to do? They wanted to retain the tenth-shaku system and have the flutes in tune with Western musical standards. Two irreconcilable systems--obviously they're going to have to fudge a little. After considerable contemplation the Gray Beards decided. The tenth-shaku system will remain in name only, not as a actual literal system of measurement. For literal measurement they'd switch to the metric system or do shaku fractions and be done with it. The tenth-shaku system will be retired and elevated to a ambassadorial position, the Even Tempered Scale holds its own and the metric system or shaku fractions will be used for measuring the length of flutes.

""Our flutes played just fine before the notes came to town.", exclaimed the villagers.

"How do these three systems relate? What's the deal?", they asked.

Well, flutes aren't usually measured in tenth shaku anymore, they're just named that and their length is such to create the note we want. Thus the name and length may be, and often are, independent. Monty Levenson reports that most of the D shakuhachi he has seen fall in the range of 54.5 cm plus-or-minus one cm as a host of factors can and do effect pitch--bore volume, air temperature and, most importantly, the method and habit of playing. Monty goes on to explain the phenomena of 'leap flutes':

It is analogous to the Earth's annual trip around the sun. Each calendar year is a bit out of phase with the actual rotation necessitating the addition of an extra day every four cycles. In this case, the shakuhachi "leaps" at 2.2'. Starting with a 1.3' (Key of G), each sun added to the length of the instrument lowers the pitch one semitone. This convenient pattern, however, breaks down when we get past the 2.1' (Key of B). Hence, a Key of B-flat shakuhachi can be 2.2' to 2.3' in length. Often, shakuhachi made in the Key of A (called 2.4') are actually closer to 2.5'. Add to this the different overall bore diameters used by shakuhachi makers as well as the relative thickness of the bamboo and resulting chimney heights (i.e. finger hole depths). These variations in bore parameters will determine the volume of air column inside the bamboo and account for small changes in pitch. Makers deal with this issue by slightly varying the overall length of the bamboo used. I would venture to guess that, upon close inspection, very few 1.8' shakuhachi exist that are exactly i shaku ha sun. The same is true for other lengths.

Tom Deaver in Japan, offers this:

Just now I'm finishing the cosmetics on some 9-sun flutes which were made at just 1 shaku 9 sun. They play a bit high. 2 shaku are made here at 2.03 shaku = C, 2 shaku 1 sun at 2.15 shaku = B, 2 shaku 3 sun at 2.3 shaku = B flat and 2 shaku 4 sun at between 2.43 and 2.46 shaku = A. The next time any 9-sun flutes are made here I will just make them a bit longer. It all seems to get worked out in the long haul. 8-sun flutes are now, and have always been, made at 1.8 shaku even though they seem to play lower or higher as the current fashion demands. People just kari up or meri down to the currently popular fashion.

Tom further mentions:

All the shakuhachi making folks I know here use the shaku system for the lengths and the metric system for bore measurements. There are two shaku systems in common use in Japan. One, called "kujira shaku" (same pronunciation means whale) which is about 20% longer than the other one, is used exclusively for kimono and cloth and every "wasai" (Japanese seamstress) knows it internally. The other, called "kane shaku" (kane means metal) is used for everything except kimono and cloth.

The first column, in the following table, is the flute model, then the note it should play and the frequency, then length in centimeters using a progression factor of 1.0594 (the twelfth root of two) and the length in shakus with the same factor. The last column uses a heuristically arrived at factor (1.0632) taking into account the comments of Deaver and applies most directly to his flutes. This is closer to reality (the builder's experience) than the twelfth root of two used in the progression of the notes. If this holds, notes and flute lengths progress at different rates--that is, using different factors. See Equivalent Timbre for more.

So what's the length of the Model 1.8, the D flute or any other for that matter? As Monty rightly points out, it depends on the bore profile and other factors. Aside from the note system not correlating with the shaku system there are a host of other factors complicating the correlation of frequency to length. So measure your D flute when you play 'D' on it--that's how long a D flute is. Or more correctly, that's how long your D flute is. Levenson makes it clear when he points out that shakuhachi these days are measured by frequency rather than length.

Measuring the instrument linearly made sense when the shakuhachi was essentially a solo flute used as a tool for mediation. Later on, when it evolved in a secular, i.e. "musical", direction and was used in ensemble with other instruments, an acoustical standard became paramount. The impact of western forms and paradigms no doubt reenforced this shift after Japan was forcibly opened after three-hundred years of isolation before the Meiji Restoration. The confluence of traditional and modern always makes for an interesting, if somewhat baffling, amalgam.