*Kou*, *ko*, *gen*

*Slopes,* and the mysteries of the right triangle. Part 5 in a series.

This is part five in a series of posts:

- Sashigane 1: Introduction
- Sashigane 2: Traditional Units
- Sashigane 3: Physical design
- Sashigane 4:
*Hyoume*and*Urame* - Sashigane 5:
*Kou,**ko,**gen*(this post) - Sashigane 6: Simple usage
- Sashigane 7: Right triangle magic

## Slopes, not angles

It turns out that Japanese carpenters almost never use angles or talk about
“degrees”. All angles are considered as *slopes* (a rise over a run)
called *koubai* (勾配)
.

A run is called

*ko*(殳 ).A rise is called

*kou*(勾 ).It’s unfortunate that these two words sound so similar to western ears, but to Japanese natives,

*kou*sounds completely different.Enunciating slowly and distinctly,

*kou*sounds like “ko ooh,” with two distinct beats, but run together (like a half-note instead of two quarter notes in music!).*Kou*when spoken makes a long “o” sound (kind of like the word “boat” but expressly drawn out for two beats: “bo-oat”) while*ko*sounds for just one beat (a quarter-note) and is clipped short.A line connecting the start of the run to the top of the rise is called the

*gen*(玄 ).

By convention, slopes are named assuming a fixed *ko* (run) of 10 *sun*. The *kou*
(rise) is then used to express the slope. So a 4/10 slope would be referred to
as 4 *sun koubai.*

*Kanekoubai*

A 45° slope is a special case: the rise is the same as the run. A 45° angle could be
described as a slope of “10 *koubai*” or “10 *sun* *koubai*.” This special case
is more commonly referred to as *kanekoubai* (矩勾配
), however.

*Hirakoubai* and *kaeshikoubai*

By convention, slopes with a rise less than the run (angles less than 45°) are
considered the common or ordinary case. This is called *hirakoubai* (平購買)
.

Slopes with a greater rise than run (greater than 45° angles) are called
*kaeshikoubai* (返し購買)
or sometimes *koubai no korobi* (勾配の転び)
, which
means “upside down *koubai.*”

Note that *hirakouba* and *kaeshikoubai* slopes come in *pairs.*

For example, a slope of 5 *sun koubai* has a related *kaeshikoubai* slope of
10/5.

These are the two corners in a right triangle that aren’t 90°:

The slope labeled `a`

is *hirakoubai* (a rise of 5 over a run of 10) and `b`

is
the corresponding *kaeshikoubai* (with a “rise” of 10 over a “run” of 5).

Things get confusing soon, so it pays to ensure you fully understand the following five points:

The

*ko*(run) is always the*adjacent*side, and the*kou*(rise) is always the*opposite*side.By convention, slopes are normalized to a

*ko*(run) of length 10. If you had a right triangle with sides that were, say, 3*sun*and 6*sun*long, you’d refer to the normal/conventional slope (or*hirakoubai*) as “5*sun koubai*” (keeping the 3/6 ratio but scaling it to 5/10).Further, the shallower slope (smaller angle) is the common or ordinary slope (

*hirakoubai*) and is normally what gets called out.A steeper slope (greater than 45° angle) is called

*kaeshikoubai*and the*kou*now becomes a divisor rather than a dividend – you divide the*ko*(run) by the*kou*(rise) to get a*kaeshikoubai*slope (it’s the upside down slope).Normally, one doesn’t refer to slopes (

*koubai*) greater than 10.If, for some reason, you wanted to work with a slope of, say, 25/10, you should instead think of it as the

*kaeshikoubai*for a slope of 4*sun koubai*.That’s because the

*kaeshikoubai*for a common slope (*hirakoubai*) of 10/25, which is equivalent to 40/100, which is the same as a slope of 4/10 or 4*sun koubai.*)

Next, we’ll give all this theory and naming stuff a rest, and finally get to
some practical applications for the *sashigane*.