<HTML>In a message dated 10/21/01 3:18:16 PM Central Daylight Time, dm.porritt@verizon.net (David M. Porritt) writes: <BLOCKQUOTE TYPE=CITE style="BORDER-LEFT: #0000ff 2px solid; MARGIN-LEFT: 5px; MARGIN-RIGHT: 0px; PADDING-LEFT: 5px">You've talked quite a bit of your "tempered octaves" but as one who has seen everything from 2:1 octaves to outrageously stretched octaves I don't know what kind of stretch you're talking about.  Could you give us some numbers so we can know what you mean by tempered octaves?   For example, when I tune a Steinway D my C6 is stretched to 5.06, C7 is 16.11, and C8 is 43.84.  This is what I consider a conservative tuning.  What kind of numbers do you get?  On any piano, just measure your C6, 7, & 8 and tell us what model piano it is, and how it stretches with your tempered octaves.  That would be very informative.</BLOCKQUOTE> Thanks for your question.  Although I developed the idea on my own, it is the same as an idea Steve Fairchild has on tuning octaves.  The essence of the idea is that the  octave is stretched to the point where the corresponding 5th is slowed down to where the two will beat equally.  On a piano with high inharmonicity, it seems very natural and very little.  On a piano with lower inharmonicity, it can seem exaggerated. On a piano tuned in ET, your figures are about what I would come up with if I equalized the beating between the double octave and the octave and 5th all the way to the top.  I often take it further than that, however.  Once I get to the 6th and 7th octaves, I compare the triple octave and double octave and 5th which will boost by C8 all the way up to about +75 cents.  The last half of the 7th octave will be in the +50's and +60's. The amount I choose to stretch depends upon the circumstances.  I like the idea I have, not only because I think it sounds really good (and many have confirmed this) but because, at the very least, I can *justify* the amount I stretch instead of simply pulling *numbers* out of thin air. Stretching octaves in this manner will naturally produce single octaves between the 5th, 6th & 7th octaves with audible beats.  To some, this is not tolerable.  It will also speed up considerably the 10th & 17ths although in ET, they can and will still progress smoothly.  Sometimes, when I have chosen the *higher road*, so to speak, (triple octaves), I find the dissonance in the single octaves to be overbearing, so I go back to a more moderated stretch. I feel a high amount of stretch is appropriate for a large piano in a large room, particularly if it is a concerto piano.  This gives the piano more projecting power and helps it *compete* with the strings who also have a tendency to play much sharper in their highest registers.  Conservative stretch is more appropriate for more intimate conditions.  There are probably many different opinions on this. When tuning a modern temperament (such as my EBVT) or late 19th Century HT, there will be some pure 5ths and some 5ths which are tempered more than ET (and some about the same).  When a double octave is compared with an octave and 5th, the amount of stretch of the upper note will be minimal.  When compared with a tempered 5th, equalizing the beats between the two will boost the upper note much more.  Therefore, a typical tuning of mine has octaves of considerably varying size up and down the piano. I came to call these octaves of irregular size, "Tempered Octaves".  You won't see those words in any book because I am the one who coined their use.  Immediately, many people raise eyebrows at the very concept.  Isn't smoothness and regularity a goal?  Wouldn't uneven octaves simply be bad tuning?  The fact is that they give the piano a beautiful melodic voice in the upper registers that cannot be achieved in any other way.  The irregular sizes of the octaves cannot be discerned when playing actual music.  They give the piano what I have called *texture* and *clarity*. Unfortunately, the style of stretch I use cannot be had using any of the typical ETD programs.  You'd have to program for a relatively high amount of stretch, then adjust by ear or use the Direct Interval method, as I do, to adjust them. I'll forward you and another person who asked about this my EBVT bearing plan and my article on Tempered Octaves. Bill Bremmer RPT Madison, Wisconsin</HTML>