The toctave

Mike,

You asked to have a reason to admit that toctave might be or perhaps is a real option. This time I think I deserve to get your answer. Please see below. There are much other similar cases.

2^(1/12) = Equal tempered tone interval = 1.05946309436

Equal tempered C = 261.6255 Hz

Equal tempered G = 1.49830707688

Equal tempered 2C/G interval = 1.33483985417

Equal tempered 2C = 523.251 Hz

-----------------------------------------------------------------------------

&^(1/12) = Toctaved scale interval = 1.05976223699

C = 261.6255 Hz

Toctaved G = (3/2)*(261.6255) = 392.43825 Hz

(9/8)*2^(1/4) = (&/G) = 1.33785800437

C*& = 525.02665398476 Hz

2*MMJ = Toctave = & = 2.00678700656

Cmajor 261.6255, 392.43825, 525.0266.. that works with the toctave & sounds much better than tempered C major due to the round 3/2 and (9/8)*2^(1/4) intervals.

Regards

Mario

September, 14

--- In tuning@yahoogroups.com, "Mario Pizarro" <piagui@...> wrote:
> [Mike] asked to have a reason to admit that toctave might be or perhaps is a real option. ... Please see below. There are much other similar cases.
> 2^(1/12) = Equal tempered tone interval = 1.05946309436
> Equal tempered C = 261.6255 Hz
> Equal tempered G = 1.49830707688
> Equal tempered 2C/G interval = 1.33483985417
> ...
> &^(1/12) = Toctaved scale interval = 1.05976223699
> C = 261.6255 Hz
> Toctaved G = (3/2)*(261.6255) = 392.43825 Hz
> (9/8)*2^(1/4) = (&/G) = 1.33785800437
> ...
> [the second one] sounds much better than tempered C major due to the round 3/2 and (9/8)*2^(1/4) intervals.

Mario,
I have not listened to the two chords, but I am a bit confused about what they are intended to show.
(1) why add a third note ("G") into the comparison - can't we just simply compare the 1 - 2 dyad with the 1 - & dyad?
(2) if there is to be a third note, why make the above tuning choices? e.g. why not use the pure 3/2 interval in both cases? If you say the 3/2 is not available in the equal tempered system with 2/1, well it is not available in the toctave system either, is it? Just as the "equal tempered" G is 2^(7/12), so the "Toctaved G" is not 3/2*C but &^(7/12)*C, isn't it?

Steve.

Am 15.09.2011 22:45, schrieb martinsj013:
> --- In tuning@yahoogroups.com, "Mario Pizarro" <piagui@...> wrote:
>> [Mike] asked to have a reason to admit that toctave might be or perhaps is a real option. ... Please see below. There are much other similar cases.
>> 2^(1/12) = Equal tempered tone interval = 1.05946309436
>> Equal tempered C = 261.6255 Hz
>> Equal tempered G = 1.49830707688
>> Equal tempered 2C/G interval = 1.33483985417
>> ...
>> &^(1/12) = Toctaved scale interval = 1.05976223699
>> C = 261.6255 Hz
>> Toctaved G = (3/2)*(261.6255) = 392.43825 Hz
>> (9/8)*2^(1/4) = (&/G) = 1.33785800437
>> ...
>> [the second one] sounds much better than tempered C major due to the round 3/2 and (9/8)*2^(1/4) intervals.
>
> Mario,
> I have not listened to the two chords, but I am a bit confused about what they are intended to show.
> (1) why add a third note ("G") into the comparison - can't we just simply compare the 1 - 2 dyad with the 1 - & dyad?
> (2) if there is to be a third note, why make the above tuning choices? e.g. why not use the pure 3/2 interval in both cases? If you say the 3/2 is not available in the equal tempered system with 2/1, well it is not available in the toctave system either, is it? Just as the "equal tempered" G is 2^(7/12), so the "Toctaved G" is not 3/2*C but &^(7/12)*C, isn't it?
>
> Steve.
>
>
Hi there,
ever heared of the Octove (the German "Oktove") ;)
here http://xenharmonic.wikispaces.com/Oktove
I added some links to (octave) streching techniques

Best,
Wolf

Steve, you wrote:

>>(1) why add a third note ("G") into the comparison - can't we just simply >>compare the 1 - 2 dyad with the 1 - & dyad?

1) I had to do it by taking just C major triad to simplify the comparison, keeping in mind that this is a partial comparison since a complete one deals with tone relations within the octave or toctave range. I attached a toctaved scale where D/C# = P interval factor = (8/9)*2^(1/4) = 1.05705299111 and C#/C = K interval = (9/8)^(1/2) = 1.06066017178. Both K and P intervals are the generators of the three Piagui scales.

Consonant P and K factors are cells # 49 and # 52 respectively, their ratio gives MMJ product.

Starting with K factor, the tone intervals follow the sequence K P K K K K P P K K K K = (K^9)*(P^3) that surprisingly gives the toctave & = Cell # 615 = 2.00678700656.

>>>>>>>>>>>>>>>>>>>>>>>>>
2) > (2) if there is to be a third note, why make the above tuning choices? e.g. why not use the pure 3/2 interval in both cases? >If you say the 3/2 is not available in the equal tempered system with 2/1, well it is not available in the toctave system either, is >it? Just as the "equal tempered" G is 2^(7/12), so the "Toctaved G" is not 3/2*C but &^(7/12)*C, isn't it?

I didn�t say that 3/2 is not available in the equal tempered system with 2/1, I was misunderstood, I think.
----------------------------------------------------------
It sounds logic to me that the presence of the 2/1 octave in the 12 tet system opened the doors to hundreds of proposals which includes G = 3/2. Similarly, for a 2.006787....toctaved scale, many options with G = 3/2 can be proposals like the one you have on the attachment and also below. Therefore, both types of octaves can be designed to operate with G = 3/2.

Just as the "equal tempered" G is 2^(7/12), so the "Toctaved G" is not 3/2*C but &^(7/12)*C, isn't it?
In my opinion, 2^(7/12) can only be applied to equal tempered scales provided they work with 2/1 octaves since the tone frequencies of the toctave tones are not derived from the digit 2. Consequently, it would be too hard or impossible that one tone frequency of the many coupled toctaves takes the form 2^(2N), where N is an integer number.
I see that I should have remarked that G = 3/2 in a toctave scale and G = 1.49830707688 in a 2/1 tet intonation can take other relative frequencies regarding C = 1. On the other hand, if we decide on G tempered = 3/2, the remaining range from 3/2 to 2, forces to a some reduced value in tones Ab, or A, or ......2C; this effect is less critic when using the toctave I think.

Number Decimal cell Cell
of added frequencies frequencies
cells Cell # 12 Tone toctave scale (Cents)
0 0 C 1.00000000000 0
K
52 52 C# 1.06066017178 101.955
P
49 101 D 1.12119522033 198.045
K
52 153 Eb 1.189207115 300
K
52 205 E 1.26134462288 401.955
K
52 257 F 1.33785800438 503.91
K
52 309 F# 1.41901270074 605.865
P
49 358 G 1.5 701.955
P
49 407 Ab 1.58560948667 798.045
K
52 459 A 1.68179283051 900
K
52 511 Bb 1.78381067250 1001.955
K
52 563 B 1.89201693432 1103.91
K
52 615 & 2.00678700656 1205.865
Toctave

Best

Mario

September, 16
-------------------------------------------------------------------------
----- Original Message ----- From: "martinsj013" <martinsj@...>
To: <tuning@yahoogroups.com>
Sent: Thursday, September 15, 2011 3:45 PM
Subject: [tuning] Re: The toctave

> --- In tuning@yahoogroups.com, "Mario Pizarro" <piagui@...> wrote:
>> [Mike] asked to have a reason to admit that toctave might be or perhaps >> is a real option. ... Please see below. There are much other similar >> cases.
>> 2^(1/12) = Equal tempered tone interval = 1.05946309436
>> Equal tempered C = 261.6255 Hz
>> Equal tempered G = 1.49830707688
>> Equal tempered 2C/G interval = 1.33483985417
>> ...
>> &^(1/12) = Toctaved scale interval = 1.05976223699
>> C = 261.6255 Hz
>> Toctaved G = (3/2)*(261.6255) = 392.43825 Hz
>> (9/8)*2^(1/4) = (&/G) = 1.33785800437
>> ...
>> [the second one] sounds much better than tempered C major due to the >> round 3/2 and (9/8)*2^(1/4) intervals.
>
> Mario,
> I have not listened to the two chords, but I am a bit confused about what > they are intended to show.
> (1) why add a third note ("G") into the comparison - can't we just simply > compare the 1 - 2 dyad with the 1 - & dyad?
> (2) if there is to be a third note, why make the above tuning choices? > e.g. why not use the pure 3/2 interval in both cases? If you say the 3/2 > is not available in the equal tempered system with 2/1, well it is not > available in the toctave system either, is it? Just as the "equal > tempered" G is 2^(7/12), so the "Toctaved G" is not 3/2*C but &^(7/12)*C, > isn't it?
>
> Steve.
>
>
>
> ------------------------------------
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Steve,

When writing the message given below I intended to remark that not only 1.33483985417** and 1.33785800437**(*) which are the tempered and Piagui F tone frequencies respectively, but the tones of the TOCTAVE SCALE shown below, work with the derived K and P semitone factors of the Piagui scales, they fit the toctave range with exactness. There are some other toctave scales which present similar characteristics. I am not insisting on discussing again on the toctave, only want to state that it seems to me that the arithmetical tone frequencies of the equal tempered scale might be forced to fit the slightly short range of the 2/1 octave. On the other hand, the toctave 2.00678700656 makes the toctave calculations a rather complex matter.

I am considering the possibility of two complementary sets, each with 6 tones, C, C#, D, Eb, E, F and F#, G, Ab, A, Bb, B.
It seems a nonsense, perhaps it is.

The mathematical derivations of K and P Piagui semitone factors and tone frequencies demanded a rigurous analysis so it is a noticeable result that both elements satisfy the toctave with exactness. Information regarding the mentioned derivations can be found in folder /tuning/files/MarioPizarro/
C# = P, D = KP, Eb = (K^2)P, E = (K^3)P, F = (K^4)P, G = (K^4)(P^3), Ab = (K^5)(P^3), A = (K^6)(P^3), Bb = (K^7)(P^3), B = (K^8)(P^3), toctave = (K^9)(P^3) = 2.00678700656.

M 47 1.05468750000 M 99 1.11866502492
M 48 1.05587840080 M 100 1.11992816597
C# J 49 1.05707299111 96.09 cents D J 101 1.12119522034 198.045 cents

M 151 1.18652343750 M 203 1.25849815304
M 152 1.18786320090 + 203.91 = M 204 1.25991918672 + 203.91 =
Eb J 153 1.18920711500 300 cents E J 205 1.26134462288 401.955 cents

M 255 1.33483886719 M 307 1.41581042217
M 256 1.33634610102 + 203.91 = M 308 1.41740908506 + 203.91 =
F J 257 1.33785800438 503.91 cents F# J 309 1.41901270074 605.865 cents

M 353 1.49155336651 M 405 1.58203125000
M 354 1.49323755463 + 192.18 = M 406 1.58381760120 + 192.18 =
G J 355 1.49492696045 696.09 cents Ab J 407 1.58560948667 798.045 cents

M 457 1.67799753700 M 509 1.77978515600
M 458 1.67989224900 + 203.91 = M 510 1.78179480100 + 203.91 =
4A J 459 1.68179283051 900 cents 4Bb J 511 1.78381067300 1001.955 cents

M 561 1.88774722956 M 613 2.00225830078
M 562 1.88987878008 + 203.91 = M 614 2.00451915152 + 203.91 =
B J 563 1.89201693432 1103.91 cents Toct J 615 2.00678700656 1205.865 cents

Regards

Mario

October, 05
----------------------------------------
----- Original Message ----- From: "martinsj013" <martinsj@...>
To: <tuning@yahoogroups.com>
Sent: Thursday, September 15, 2011 3:45 PM
Subject: [tuning] Re: The toctave

> --- In tuning@yahoogroups.com, "Mario Pizarro" <piagui@...> wrote:
>> [Mike] asked to have a reason to admit that toctave might be or perhaps >> is a real option. ... Please see below. There are much other similar >> cases.
>> 2^(1/12) = Equal tempered tone interval = 1.05946309436
>> Equal tempered C = 261.6255 Hz
>> Equal tempered G = 1.49830707688***
>> Equal tempered 2C/G interval = 1.33483985417**
>> ...
>> &^(1/12) = Toctaved scale interval = 1.05976223699
>> C = 261.6255 Hz
>> Toctaved G = (3/2)*(261.6255) = 392.43825 Hz
>> (9/8)*2^(1/4) = (&/G) = 1.33785800437**(*)
>> ...
>> [the second one] sounds much better than tempered C major due to the >> round 3/2 and (9/8)*2^(1/4) intervals.
>
> Mario,
> I have not listened to the two chords, but I am a bit confused about what > they are intended to show.
> (1) why add a third note ("G") into the comparison - can't we just simply > compare the 1 - 2 dyad with the 1 - & dyad?
> (2) if there is to be a third note, why make the above tuning choices? > e.g. why not use the pure 3/2 interval in both cases? If you say the 3/2 > is not available in the equal tempered system with 2/1, well it is not > available in the toctave system either, is it? Just as the "equal > tempered" G is 2^(7/12), so the "Toctaved G" is not 3/2*C but &^(7/12)*C, > isn't it?
>
> Steve.
>
>
>
> ------------------------------------
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