Tangent ratioing is a method that is used to raise and lower the pavilion and or the crown of a design while keeping everything in proportion. It was a method that was used in older designs particularly (books). A lot of the Long & Steele material is set up this way.
There is an online Tangent Ratio converter at the AMFED website This is easy to use and free to anybody.
You might have a Quartz design that you want to cut in a piece of Sapphire. The critical angle for Quartz is 40.49 degrees, the critical angle for Sapphire is 34.62 degrees. By using Tangent Ratioing you can lower the angles of the Quartz design to 36 degrees for the Sapphire.
Note: You will need to do the conversion for all of the tiers in the design in either the pavilion and crown or just the pavilion if that is all you are adjusting.
Making charts for all of the Tangent Ratios for a design.
This was done a lot, like I said in the old days. It can be done now if somebody wants to take the time and effort that would be needed to do it. While it sounds like a good idea, there are a lot of limitations.
The main limitation is the lack of performance in the finished design, after Tangent Ratioing the design to more than a couple of degrees either way.
You can Tangent Ratio any design to about any refractive index, but getting a good quality performing design is another matter entirely. It can be done in some cases, depending on the design. But it is usually not worth the effort for most designs because as they get more complex the Tangent Ratio will dramatically impact and lower the performance of most designs (after Tangent Ratioing). This especially applies to changing the Pavilion of a design, and in particular high refractive index designs being lowered for lower refractive indexes.
Note: Think of lowering the refractive index of a design like trying to get 10 pounds of sugar in a 5 pound bag. You can go up in refractive index easily, but going down it is very difficult to ge a design that will perform.
In most cases, to get the design to work with any kind of good performance after it has been Tangent Ratio adjusted to more than a few degrees, the design basically needs to be redone/changed. If you do that, the design is not the same anymore and it defeats the whole purpose of the exercise.
That is one of the reasons that I started to design my cuts for specific uses. If you look you will notice that where it is practical I always try to list what other materials will cut well in that design. I have check them and often cut a stone in those other materials to be sure. Some designs will just not work in any refractive index but the one they were designed for.
There are exceptions, and in particular my Barion Addition #5 has a lot of designs that can be cut in about any refractive index with no changes. No need to bother with Tangent Ratioing. I really prefer things to be simple and in most cases, with these designs there is no angle changes required. Just sit down, cut, and get a pretty stone in almost any material, these are the types of designs I always try for.
I have never really cared for the Tangent Ratio method and very seldom use it . Aas some of you may have noticed. Everybody has their own ideas and opinions, so remember this is just my experience. I find that in the majority of cases that it is not necessary to make any adjustments to a design and that often if you do Tangent Ratio the design more than a couple of degrees, it will not work well in the new refractive index anyway. Light return performance almost always suffers, particularlly when putting a high refractive index into a low one.
In the example above. When using the Quartz design to cut a Sapphire (depending on the design and rough of course) in most cases you would be better off using the Quartz design without changes. Sapphires unusually make a better stone in a deeper design, and because most natural Sapphire rough is expensive you certainly do not want to lose any of it by cutting a shallower design than you have to.
Note: I generally will sacrifice a little bit of rough to get a deeper stone (unless the stone is dark) because I think it is a better looking stone with more dispersion.
If you ray trace the Quartz design with no changes in refractive index 1.76 (Sapphire) you will quite often find that the light return is as good and sometimes better than if you did Tangent Ratio it to the lower Sapphire angles. Quite frankly even if the ray trace is a little lower I feel that the dispersion and look gained by using a deeper design is well worth the trade off. Remember a better colored stone (by using a deeper design) can make a huge difference in the selling price of a finished stone. Especially a Sapphire. Also remember the look of a stone is not all in a high light return.
Note: For those of you familiar with the old style cutting charts, you will remember that 41 degrees was very commonly used for a lot of stones, including Sapphires. I suspect that while they did not have the tools to prove it, the old cutters knew what worked, probably from trial and error. As a designer it is really worth studying those old charts, they had a lot of things right.
Most low refractive index (R.I.) designs (Quartz, Beryl, Tourmaline…) will Tangent ration “OK” to a higher refractive index material. Not always well, but adequately. Although, like I said, I usually cut them as is, with no changes.
Most high refractive index designs will not Tangent Ratio down too well. There are a lot of reasons for this, but mainly, a high refractive index design has more latitude for angles and tiers because of the high R.I. of the design’s material. In other words because of the high refractive index used, the designer was able to do things that will not work in a lower refractive index. If you Tangent ration this type of design it will be cuttable, but the performance of the design will almost always suffer badly.
Every wonder why a lot of designers typically design for R.I. 1.73 or 1.76? The answer is. It is a lot easier to make designs for a higher refractive index work. Also using a high R.I. gives the designer a lot more room to work, there is a lot more angle or sweet spot to work in for additional facets and tiers. Working in low refractive material is always tough, because the low indexes are so limiting and you cannot do much to get around the problems.
If you take a design with a high refractive index like Sapphire and try to Tangent Ratio it to a lower refractive index, like Quartz. A lot of the facets and tiers will basically fall out of the sweet spots, or light return areas. Kind of like trying to put 10 pounds of sugar in a 5 pound bag. This is especially true if the high refractive index design is multi-tiered on the pavilion.
High refractive index designs, especially if they are multi-tiered, will very seldom Ratio Tangent to a lower index and work well (there are a few exceptions.), with good light return.
If you are taking a Sapphire (1.76) design and Tangent Ratioing it to a Peridot (1.67), you will be a lot more successful than if you try to make it into a Quartz (1.54) design which is a lot lower and there fore larger change. The closer your refractive indexes are to each other the better they Tangent Ratio performance wise.
Of course, the closer the refractive indexes are to each other the less need there is to even bother with Tangent Ratioing the design. Unless you are just adjusting the design to meet the critical angle of the lower refractive index material.
What I use Tangent Ratioing for. When I use it which is not often. If I am cutting the crown on a design and find that I do not have enough rooms to complete that design. To tell the truth I am a bit on the lazy side and the crown is usually not critical, so I usually just subtract a few degrees from all of the crown tiers on the fly rather than take the time to Tangent Ratio.
If I need to adjust the Pavilion angle slightly down on the design for a lower refractive index. A lot of times this is not necessary because most designs have some cushion above the critical angle. I usually will not Tangent Ratio more than a few degrees because of the performance problems mentioned above.
If I am trying to make minor adjustments for a particular piece of rough. I usually end up designing a new design truthfully.