Re: well, well, well...(tables)

To Ed Foote, Paul Erlich, Paul Bailey, and all hairy piano-tech bushmen:

I've gone ahead and put together some tables for the four temperaments I
talked about earlier.
---------------------------------------------------------------------------------------------------------------------------------------------------
Frequency measurements, and deviations in cents (rounded to nearest half-cent)
from 12-tet:

1. 2. 3. 4.

C 263.12 (+10) 262.69 (+7) 262.39 (+5) 262.17 (+3.5)
C# 277.49 (+2) 277.31 (+1) 277.18 (+/-0) 277.18 (+/-0)
D 294.33 (+4) 294.13 (+3) 293.97 (+2) 293.9 (+1.5)
Eb 312.18 (+6) 311.97 (+4.5) 311.83 (+4) 311.63 (+3)
E 329.26 (-2) 329.33 (-1.5) 329.4 (-1) 329.47 (-1)
F 351.2 (+10) 350.73 (+7.5) 350.33 (+5.5) 350.02 (+4)
F# 369.99 (+/-0) 369.74 (-1) 369.74 (-1) 369.82 (-1)
G 393.79 (+8) 393.24 (+5.5) 392.88 (+4) 392.63 (+3)
G# 416.24 (+4) 415.96 (+3) 415.77 (+2) 415.64 (+1.5)
A 440 440 440 440
Bb 468.27 (+8) 467.95 (+6.5) 467.53 (+5) 467.14 (+3.5)
B 493.32 (-2) 493.33 (-2) 493.44 (-2) 493.56 (-1)
-------------------------------------------------------------------------------------------------------------------------------------------------------------
Fifths in cents:

G-D, D-A 696.04 697.21 698.0 698.55
C-G, A-E 698.0 698.38 698.77 699.11
F-C, E-B 700 699.55 699.55 699.67
Bb-F, B-F# 701.9 700.73 700.34 700.22
Eb-Bb, F#-C# 701.9 701.9 701.12 700.78
Ab-Eb, C#-G# 701.9 701.9 701.9 701.34
--------------------------------------------------------------------------------------------------------------------------------------------------------------
Thirds in cents:

C-E 388.18 391.38 393.72 395.54
F-A, G-B 390.15 392.54 394.51 396.04
Bb-D, D-F# 396.06 396.06 396.84 397.75
Eb-G, A-C# 401.96 400.77 399.96 399.96
Ab-C, E-G# 405.86 404.26 403.09 402.18
Db-F, B-D# 407.79 406.6 405.43 403.9
Gb/F#-Bb/A# 407.79 407.79 406.23 404.41
----------------------------------------------------------------------------------------------------------------------------------------------------------------
Whew!
Well, there you have it.

Dale Scott

Dale writes:

>Frequency measurements, and deviations in cents (rounded to nearest
half-cent)
>from 12-tet:

> 1. 2. 3. 4.

>C 263.12 (+10) 262.69 (+7) 262.39 (+5) 262.17 (+3.5)
>C# 277.49 (+2) 277.31 (+1) 277.18 (+/-0) 277.18 (+/-0)
>D 294.33 (+4) 294.13 (+3) 293.97 (+2) 293.9 (+1.5)
>Eb 312.18 (+6) 311.97 (+4.5) 311.83 (+4) 311.63 (+3)
>E 329.26 (-2) 329.33 (-1.5) 329.4 (-1)
329.47 (-1)
>F 351.2 (+10) 350.73 (+7.5) 350.33 (+5.5) 350.02 (+4)
>F# 369.99 (+/-0) 369.74 (-1) 369.74 (-1) 369.82 (-1)
>G 393.79 (+8) 393.24 (+5.5) 392.88 (+4) 392.63 (+3)
>G# 416.24 (+4) 415.96 (+3) 415.77 (+2) 415.64 (+1.5)
>A 440 440 440
440
>Bb 468.27 (+8) 467.95 (+6.5) 467.53 (+5) 467.14 (+3.5)
>B 493.32 (-2) 493.33 (-2) 493.44 (-2) 493.56 (-1)

Greetings,
This is good info, I can actually tune a piano in any of these from the
cents deviation. #1 has good regularity in the thirds, and aside from
lacking a Just C-E sounds and behaves much like a Kirnberger III.
I don't find the need for a Just C-E on the modern piano, a 5 cent
width gives the sense of purity in the key, and by not having to dump the
whole comma in the remaining two thirds, the top of the circle of fifths
doesn't heat up too much.
I see in this #1 there are three keys (C#, F# and B) with full comma
thirds, which is not unusual, but that the next two adjacent keys are only
1.7 cents less than the most highly tempered , and on a piano, this degree of
separation is virtually indistinguishable.
The following is a listing of cent deviations from ET that I received
from one of my tuning mentors, it provides a different shape to the balances
among thirds as they progress by movements of a fourth or fifth.
C 4
C# 1
D 1
D# 1
E -2
F 5
F# -1
G 3
G# 1
A 0
A# 3
B -2

This is a more refined Victorian style of tuning than Jorgensen presents,
and I have been having a lot of commercial success replacing ET on pianos
with this tuning.
Regards,
Ed Foote
Precision Piano Works
Nashville, Tn.