Tukey correction in emmeans

I have a question about the Tukey correction in emmeans. See the example below. It says "P value adjustment: tukey method for comparing a family of 3 estimates." Does this mean that the error rate for all of these comparisons (which were run together) is 5%? What does "a family of 3 estimates" refer to?

The two factors here were "group," which had 3 levels, and "condition," which had 4 levels, so there are a total of 12 comparisons. If each comparison had been run separately, the error rate would have been 5% comparison. I'm hoping that because these were all run together with the Tukey correction, the error rate is down to 5% for all of them. What I'm confused about is the reference to "3 estimates," and I'm wondering if this means the error rate is 5% for each set of three group comparisons (one set per condition), meaning the error rate is actually 1 - 0.95^4 (about 18%)?

Thank you!

``````> emmeans(model_1, pairwise ~ group | condition)
\$emmeans
condition = EN-TVJT-R-BRF:
group emmean     SE  df lower.CL upper.CL
EN    0.0456 0.0174 219  0.01129   0.0799
HS    0.1515 0.0211 368  0.11004   0.1930
SB    0.0208 0.0246 500 -0.02745   0.0691

condition = EN-TVJT-R-BRT:
group emmean     SE  df lower.CL upper.CL
EN    0.9848 0.0174 219  0.95043   1.0191
HS    0.8712 0.0211 368  0.82974   0.9127
SB    0.9479 0.0246 500  0.89964   0.9962

condition = EN-TVJT-R-THF:
group emmean     SE  df lower.CL upper.CL
EN    0.0254 0.0174 219 -0.00891   0.0597
HS    0.1742 0.0211 368  0.13277   0.2157
SB    0.0417 0.0246 500 -0.00661   0.0899

condition = EN-TVJT-R-THT:
group emmean     SE  df lower.CL upper.CL
EN    0.9898 0.0174 219  0.95548   1.0241
HS    0.8939 0.0211 368  0.85247   0.9354
SB    0.9688 0.0246 500  0.92047   1.0170

Degrees-of-freedom method: kenward-roger
Confidence level used: 0.95

\$contrasts
condition = EN-TVJT-R-BRF:
contrast estimate     SE  df t.ratio p.value
EN - HS   -0.1059 0.0266 783 -3.983  0.0002
EN - SB    0.0248 0.0294 783  0.842  0.6768
HS - SB    0.1307 0.0317 783  4.117  0.0001

condition = EN-TVJT-R-BRT:
contrast estimate     SE  df t.ratio p.value
EN - HS    0.1136 0.0266 783  4.271  0.0001
EN - SB    0.0368 0.0294 783  1.252  0.4230
HS - SB   -0.0767 0.0317 783 -2.417  0.0420

condition = EN-TVJT-R-THF:
contrast estimate     SE  df t.ratio p.value
EN - HS   -0.1488 0.0266 783 -5.597  <.0001
EN - SB   -0.0162 0.0294 783 -0.552  0.8454
HS - SB    0.1326 0.0317 783  4.177  0.0001

condition = EN-TVJT-R-THT:
contrast estimate     SE  df t.ratio p.value
EN - HS    0.0959 0.0266 783  3.606  0.0010
EN - SB    0.0211 0.0294 783  0.716  0.7543
HS - SB   -0.0748 0.0317 783 -2.357  0.0489

Degrees-of-freedom method: kenward-roger
P value adjustment: tukey method for comparing a family of 3 estimates
``````

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